VARIATIONAL ANALYSIS: METHODS AND APPLICATIONS€¦ · are describable by set convergence,...

SOFIA UNIVERSITY ”ST. KLIMENT OHRIDSKI”

FACULTY OF MATHEMATICS AND INFORMATICS

VARIATIONAL ANALYSIS:METHODS AND APPLICATIONS

Nadia Peycheva Zlateva, PhD

DISSERTATION FOR DOCTOR OF SCIENCES DEGREE IN MATHEMATICS

S O F I A2 0 1 7

Dedicated to the bright memory of

MD Lidia Todorova Grigorova, my mother

and to MD Peycho Zlatev Zlatev, my father

Contents

Preface 3

1 Primal lower-nice functions and prox-regular sets 51.1 On primal lower-nice property . . . . . . . . . . . . . . . . . . . . . . 71.2 Subdifferential characterization of primal lower-nice functions on smooth

Banach spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.3 Characterizations of prox-regular sets in uniformly convex Banach

spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.3.1 Prox-regular sets . . . . . . . . . . . . . . . . . . . . . . . . . 271.3.2 Local Moreau envelopes . . . . . . . . . . . . . . . . . . . . . 331.3.3 Characterizations of prox-regular sets . . . . . . . . . . . . . . 401.3.4 Characterizations of uniformly prox-regular sets . . . . . . . . 48

1.4 Prox-regular sets and epigraphs in uniformly convex Banach spaces:various regularities and other properties . . . . . . . . . . . . . . . . . 571.4.1 Normal and tangential regularity properties of prox-regular sets 661.4.2 Epigraphs of J-primal lower regular functions . . . . . . . . . 721.4.3 Prox-regularity and N-hyporegularity . . . . . . . . . . . . . . 781.4.4 Comparison of normal cones . . . . . . . . . . . . . . . . . . 821.4.5 Preservation of hyporegularity and prox-regularity . . . . . . . 831.4.6 Conical derivative of the mapping PC . . . . . . . . . . . . . . 881.4.7 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

2 Integrability of subdifferentials of functions 992.1 Integrability of the subdifferential of a convex function through infi-

mal regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1002.2 Integrability of subdifferentials of directionally Lipschitz functions . . 108

2.2.1 Subdifferential properties of directionally Lipschitz functions . 1102.2.2 Local integrability . . . . . . . . . . . . . . . . . . . . . . . . 114

2.3 Integrability of subdifferentials of certain bivariate functions . . . . . 1182.3.1 Concepts of regularity . . . . . . . . . . . . . . . . . . . . . . 120

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2 Contents

2.3.2 Integrability of subdifferentials of certain locally Lipschitz bi-variate functions . . . . . . . . . . . . . . . . . . . . . . . . . 122

2.3.3 Concepts of directional Lipschitzness . . . . . . . . . . . . . . 1242.3.4 Local integrability of subdifferentials of certain directionally

Lipschitz bivariate functions . . . . . . . . . . . . . . . . . . . 1282.4 Partially ball weakly inf-compact saddle functions . . . . . . . . . . . 136

2.4.1 Saddle functions. Properties . . . . . . . . . . . . . . . . . . . 1402.4.2 Subdifferential of a saddle function. Partially ball weakly inf-

compact saddle functions. Definition and properties . . . . . . 1452.4.3 Integrability of the subdifferential of a proper closed partially

ball weakly inf-compact saddle function . . . . . . . . . . . . 153

3 Variational analysis of multivalued maps 1573.1 Parameterized minimax problem: on Lipschitz-like dependence of the

solution with respect to parameter . . . . . . . . . . . . . . . . . . . . 1583.1.1 Parameterized minimization problem . . . . . . . . . . . . . . 1603.1.2 Parameterized minimax problem . . . . . . . . . . . . . . . . 1723.1.3 Lipschitz continuity of the saddle points map in context of

two-player zero sum differential games . . . . . . . . . . . . . 1783.2 Aubin criterion for metric regularity . . . . . . . . . . . . . . . . . . . 180

3.2.1 An implicit mapping theorem . . . . . . . . . . . . . . . . . . 1853.2.2 Proof of Aubin criterion . . . . . . . . . . . . . . . . . . . . . 1893.2.3 Applications of the Aubin criterion . . . . . . . . . . . . . . . 192

3.3 Long orbit or empty value principle, fixed point and surjectivity the-orems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1983.3.1 Long Orbit or Empty Value (LOEV) principle . . . . . . . . . 1993.3.2 Caristi-Kirk fixed point theorem . . . . . . . . . . . . . . . . . 2003.3.3 Surjectivity theorems . . . . . . . . . . . . . . . . . . . . . . . 201

Bibliography 209

Preface

Variational Analysis covers a broad field of mathematical theory developed inconnection with the study of problems of optimization, equilibrium, control, and sta-bility of linear and nonlinear systems, as stated in the eponymous book of Rockafellarand Wets [152].

For a long time, “variational” problems have been identified mostly with the“calculus of variations” concerning minimization of integral functionals, where amajor point is to explore variations in order to characterize solutions and describethem in terms of “variational principles”.

In this connection, notions of perturbation, approximation, generalized differentia-bility were extensively investigated.

From decades are the attempts to free the term “variational” from the limitations ofits past and to use it for much larger area of modern mathematics. The contemporaryapproach is to consider “variations” not only in a classical sense: as a movement awayfrom a given point along rays or curves, and the geometry of tangent and normalcones associated with that, but also as forms of perturbation and approximation thatare describable by set convergence, set-valued mappings and so on. Subgradients andsubdifferentials of functions, convex and nonconvex, are crucial in analyzing such“variations”.

At the present time Variational Analysis is considered as a branch of Analysisproviding not only powerful tools for the problems that have motivated it so far butalso as a mathematical discipline with new applications.

In the dissertation we present several original results in the field of VariationalAnalysis obtained in the last 15 years and published in 11 journal articles – cited as[18, 19, 63, 94, 95, 96, 140, 162, 163, 164, 172] in the bibliography.

Content is organized into three chapters. Each chapter is divided in sections. Eachsection bears the title of the eponymous article and for the convenience of the readerit begins with the necessary notations and preliminaries even they were already usedbefore.

In Chapter 1 we study primal lower-nice functions and prox-regular sets. Primallower-nice functions possess good behaviour as convex functions related to their reg-ularization, integrability, etc. and they are intensively studied last years. We prove inany Banach space a characterization of primal lower-nice functions by hypomono-tonicity of certain truncations of its subdifferential (Section 1.1), and that for such

3

4 Preface

functions in smooth Banach spaces proximal subdifferential and Clarke subdifferen-tial coincide (Section 1.2). Primal lower-nice functions belong to the larger class ofprox-regular functions. The sets whose indicator functions belong to that class areprox-regular sets. We study their properties and find several characterizations of suchsets in uniformly convex Banach spaces in Section 1.3 and Section 1.4.

Chapter 2 contains results concerning integrability of subdifferentials of functions.The purpose of studying integration of subdifferentials is to answer the questionwhether or not the condition that the subdifferential of one function contains thesubdifferential of other function implies that these two functions differ by a constant.We give affirmative answer to that question for several classes of functions definedon Banach space – a new proof of integrability of a lower semicontinuous convexfunction in Banach space (well-known result of Rockafellar) using regularizations inthe spirit of the pioneering proof of Moreau in Hilbert space (Section 2.1); regularfunction continuous on its domain and strictly directionally Lipschitz at point of itsdomain is integrable near that point (Section 2.2); bivariate (resp. separately) lowersemicontinuous function, continuous on its domain which is (resp. separately) upper-upper regular and (resp. separately) strictly directionally Lipschitz at point of itsdomain is integrable near that point (Section 2.3); proper closed partially ball weaklyinf-compact saddle function (Section 2.4).

In Chapter 3 we study multivalued maps, as well as, their dependence on param-eter. Such maps are considered in optimization and are intensively studied recently.In Section 3.1 we present a sufficient condition, ensuring that the map which to anyvalue of the parameter assigns the set of solutions (possibly multi-valued, and un-bounded) of a parameterized minimax problem on a product Banach space possessesAubin property. In Section 3.2 we establish a derivative criterion for metric regular-ity of set-valued mappings that is based on works of J.-P. Aubin and co-authors. Arelated implicit mapping theorem is also obtained. A new proof of the radius theoremfor metric regularity based on Aubin criterion is given as well. In Section 3.3 a LongOrbit or Empty Value (LOEV) principle is proved and applied to provide unifiedapproach to several fixed point and surjectivity results.

I would like to express my sincere gratitude to all my co-authors with whom Iworked together over the years and who played a crucial role in my formation as amathematician – my husband Assoc. Prof. Milen Ivanov (Sofia University), my su-pervisor Prof. Pando Georgiev (Sofia University and currently University of Florida),Prof. Lionel Thibault and Dr. Frederic Bernard (Universite Montpellier II), Prof. MarcQuincampoix (Universite de Bretagne Occidentale), Prof. Asen Dontchev (Universityof Michigan), and Assoc. Prof. Boyan Zlatanov (Plovdiv University). I am also grate-ful to my colleagues from the Department of Probability, Operational Research andStatistics at the Faculty of Mathematics and Informatics of Sofia University, where Ihave been working for 17 years and to my colleagues from the Department of Op-erations Research at the Institute of Mathematics and Informatics for their collegialcollaboration, and especially to Venelin Chernogorov, who had been my support forall these years.

Chapter 1

Primal lower-nice functions andprox-regular sets

In this chapter we present some results concerning properties of an important classof lower semicontinuous extended real-valued functions called “primal lower-nice”and introduced in 1991 by Poliquin in [136] and prox-regular sets which indicatorfunctions are “prox-regular” functions – a class of functions firstly introduced in 1996by Poliquin and Rockafellar in [137].

The class of prox-regular functions firstly introduced in finite dimensional settingsenlarges significantly the class of primal lower-nice functions previously introducedin the same finite dimensional context. Convex functions and lower-C2 functionsbelong to these classes, as well as qualified convexly C2-composite functions alsocalled strongly amenable functions (see Poliquin [136] and Poliquin and Rockafellar[137]).

The class of primal lower nice functions considerably generalizes the scope offunctions that possess as good behaviour as convex functions concerning their reg-ularization, their integrability, their second-order properties, etc., see Poliquin [136],Thibault and Zagrodny [160], Levi, Poliquin and Thibault [111], Bernard, Thibaultand Zagrodny [16], Marcellin and Thibault [115] and the references therein.

In Poliquin and Rockafellar [137] (resp. Poliquin [136]) one of the key results isan important subdifferential characterization in the line of the well-known result stat-ing that a lower semicontinuous function is convex if and only if its subdifferential ismonotone. The latter result characterizing convex functions first obtained by Poliquinin Rn was later proved in any Banach space by Corea, Jofre and Thibault in [52]. Forprimal lower-nice functions characterization via hypomonotonicity of certain trunca-tions of their subdifferentials was given by Poliquin in [136] in Rn and extended tothe Hilbert space context by Levy, Poliquin and Thibault in [111]. In Section 1.1we prove the characterization of primal lower-nice functions by hypomonotonicity ofcertain truncations of their subdifferentials in any Banach space (see Theorem 1.1.6).

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6 Chapter 1. Primal lower-nice functions and prox-regular sets

The result is published by Ivanov and Zlateva in [94].

In [136] Poliquin showed that for a primal lower-nice function in Rn proximalsubdifferential and Clarke subdifferential coincide. The result was proved in Hilbertspace by Levy, Poliquin and Thibault in [111]. In Section 1.2 we establish this prop-erty for primal lower-nice functions in smooth Banach spaces (see Theorem 1.2.2).The result is published by Ivanov and Zlateva in [95].

The extension of second-order considerations was in fact the main motivationof Poliquin and Rockafellar to defining the class of prox-regular functions. Theyintroduced as a special case the concept of prox-regularity of sets. The study of thisconcept was developed in Hilbert space by Poliquin, Rockafellar and Thibault in[138], where they showed its rich geometric implications.

Prox-regular sets appear under different names in the literature, depending onthe point of view chosen by the authors who considered them often independently.The first one was Federer in [73], where he introduced these sets in Rn as the “setswith positive reach”, in order to extend the Steiner polynomial formula to a muchlarger class of subsets of Rn than those of convex sets or compact C2 manifolds.Later, motivated by different purposes other authors focused their analysis on distinctproperties of sets and considered the classes of p-convex sets (Canino in [37]), setswith 2-order tangential property (Shapiro in [153]), proximally smooth sets (Clarke,Stern and Wolenski in [46]), prox-regular sets (Poliquin, Rockafellar and Thibault in[138]), and so on. All these concepts are actually known to be the same and to beequivalent in Rn to the notion of positively reached sets. The class of prox-regularsets is much larger than that of convex sets, but it shares with the latter many goodproperties with regard to the applications in optimization, control theory, etc. and alsohas rich geometric implications; see, in addition to the works quoted above, Clarke,Ledyaev, Stern and Wolenski [45], Thibault [158], Marcellin and Thibault [115],Edmond and L. Thibault [68], Maury and Venel [116]. Such sets are also involved indifferential inclusions in mechanics (see, e.g., Colombo and Goncharov [48], Edmondand Thibault [68], Thibault [158]), in resource allocation mechanisms in economics(see, e.g., Thibault [158]), in crowd motion problems (Maury and Venel [116]), inthe theoretical study of viability for differential inclusions subject to constraints (see,e.g., Thibault [158]), etc. Concerning related concepts for functions, we refer toBernard and Thibault [15, 14, 17], Bernard, Thibault and Zagrodny [16], Degiovanni,Marino and Toques [56], Marcellin and Thibault [115], Poliquin [136], Poliquin andRockafellar [137], Rockafellar and Wets [152], Thibault and Zagrodny [160] and thereferences therein, and concerning the other similar concept of subsmoothness for setswe refer to Aussel, Daniilidis and Thibault [11].

In Section 1.3 and Section 1.4 we establish a lot of properties of prox-regular setsin uniformly convex Banach spaces. These results are published by Bernard, Thibaultand Zlateva in [18] and [19], respectively.

1.1. On primal lower-nice property 7

1.1 On primal lower-nice property

Poliquin in [136] shows that primal lower-nice functions in finite dimensionalspaces are completely characterized by their Clarke-Rockafellar subdifferential. Thisis the first large class of non-convex lower semicontinuous functions with this prop-erty. The properties of these functions in infinite dimensional Hilbert spaces areinvestigated in detail by Levi, Poliquin and Thibault in [111], Thibault and Zagrodnyin [160], Combary, Elhilali, Levi, Poliquin and Thibault in [47]. There are two naturaldefinitions of the primal lower-nice property: one relying on lower estimate of thefirst order Taylor approximation (see Definition 1.1.4) and other including only subd-ifferentials (see Definition 1.1.5). The second one states in fact hypomonotonicity ofcertain truncations of the subdifferential. Their equivalence could be considered alsoas subdifferential characterization of primal lower-nice property and is established byPoliquin in [136] in finite dimensional spaces and by Levy, Poliquin and Thibault in[111] in Hilbert spaces.

We fill the gap by proving, in more general setting of spaces and subdifferentials,that the two definitions are equivalent. In this way we show that they characterize thesame class of functions and answer the question posed by Combary, Elhilali, Levi,Poliquin and Thibault in [47].

The results from this section are published by Ivanov and Zlateva in [94].

We begin by fixing some notations. If it is not stated other, (X, ‖ · ‖) will be areal Banach space, i.e. completed normed space. X∗ will stand for its topological dualspace, i.e., the set of continuous linear functionals on X. If x∗ ∈ X∗, we will write〈x, x∗〉 for the value of x∗ at x ∈ X. Recall that the weak topology w(X, X∗) is thesmallest topology on X with respect to which all the functions 〈·, x∗〉 (x∗ ∈ X∗) arecontinuous and that the weak-star topology w(X∗, X) is the smallest topology on X∗

with respect to which all the functions 〈x, ·〉 (x ∈ X) are continuous. We denote byB[x, r] (resp. by B(x, r)) the closed (resp. open) ball with centre x ∈ X and radius r.The closed (resp. open) unit ball in X will be denoted by B (resp. by B◦).

A bornology β on X is a family of bounded subsets of X together with theproperties: {x} ∈ β for arbitrary x ∈ X; A ∈ β, D ⊂ A ⇒ D ∈ β. It is clear thatthe Gateaux bornology G consisting of all singletons is contained in any bornologyand the Frechet bornology F of all bounded sets contains any other bornology. TheBanach space X is said to be β-smooth with respect to certain bornology β if thereexists a Lipschitz continuous bump (i.e. with non empty bounded support) functionb ∈ C1

β(X), where

C1β(X) = { f : X → R : f is Gateaux differentiable and the derivative is a con-

tinuous mapping from X to the dual space X∗, equipped with thetopology of uniform convergence on the members of the bornology β}.


It is easy to see that any space possessing an equivalent norm, which is β differ-entiable on its unit sphere, is β-smooth. The inverse is not true as shown by Haydonin [83].

We consider lower semicontinuous functions from X to R ∪ {+∞}. Function f :X → R ∪ {+∞} is said to be lower semicontinuous at point x0 if f (x0) ≤ lim inf

x→x0f (x)

and is said to be lower semicontinuous if it is so at any x0 ∈ X. Function f : X →R ∪ {+∞} is proper if it is not equal to +∞ everywhere, that is dom f , ∅, wheredom f = {x ∈ X : f (x) ∈ R}.

Further we consider β-smooth subdifferential, a notion that goes back to Crandaland Lions [55].

If β is a bornology on X and f : X → R ∪ {+∞} is a proper and lower semicon-tinuous function then β-smooth subdifferential of f at x is the set

Dβ f (x) = {u′(x) : u ∈ C1β(X) and f − u has a local minimum at x}

if x ∈ dom f and Dβ f (x) = ∅ if f (x) = +∞.Following Thibault and Zagrodny [160], Correa, Jofre and Thibault [52], Ioffe [89],

Borwein and Ioffe [26] we will consider an abstract subdifferential operator:

Definition 1.1.1. Abstract subdifferential operator ∂ is an operator that associateswith each function f : X → R ∪ {+∞} and with each point x ∈ X a subset ∂ f (x) ofX∗, that is called an abstract subdifferential of f at x, and for which the followingproperties hold:

Property 1. ∂ f (x) = ∅ if x < dom f ;Property 2. ∂ f (x) = ∂g(x) whenever f and g coincide on a neighbourhood of x;Property 3. ∂ f (x) is equal to the subdifferential in the sense of convex analysis

whenever f is convex;Property 4. If g is convex and continuous, f is lower semicontinuous and f + g

has local minimum point at x0 then for arbitrary ε > 0 there exist x, y ∈ X andp ∈ ∂ f (x), q ∈ ∂g(y) such that:

‖x − x0‖ < ε, ‖y − x0‖ < ε, | f (x) − f (x0)| < ε and

‖p + q‖ < ε.

We will prove that β-smooth subdifferential on β-smooth Banach space is abstractsubdifferential in the above sense.

Let us recall the famousEkeland variational principle (e.g. Phelps [133, p.45]). Let f be a proper lowersemicontinuous function from a Banach space X into R ∪ {+∞}. Let f be boundedbelow and ε > 0, y ∈ dom f be such that f (y) ≤ inf f + ε. Then for each λ > 0 thereis x ∈ dom f such that:


(ı) λ‖x − y‖ ≤ f (y) − f (x),

(ıı) ‖x − y‖ ≤ ε/λ, and

(ııı) λ‖x − x‖ + f (x) > f (x) for all x , x.

We will use the following assertion that is straightforward implication of thedefinition of abstract subdifferential and Ekeland variational principle.

Proposition 1.1.2. Let X be a Banach space, ∂ be an abstract subdifferential, f : X →R∪{+∞} be a lower semicontinuous and g : X → R be a convex continuous function.Let µ, ε and δ be given positive numbers. If x0 is such that

( f + g)(x0) < inf( f + g)(x) + ε

then there exist x, y ∈ X and p ∈ ∂ f (x), q ∈ ∂g(y) such that

(i) ‖x0 − x‖ <ε

µ+ δ, ‖x0 − y‖ <

ε

µ+ δ,

(ii) ‖p + q‖ < µ + δ,

(iii) | f (x0) − f (x)| < ε + δ(2 + µ) + |g(x) − g(x0)|.

Proof. Applying Ekeland variational principle we find a point x1 ∈ X such that‖x0 − x1‖ < ε/µ and such that the function ( f + g)(x) + µ‖x − x1‖ attains its strongminimum at x1. Moreover, |( f + g)(x1) − ( f + g)(x0)| < ε.

According to Property 4 in Definition 1.1.1 for the sum of f (x) and convexcontinuous g(x) + µ‖x − x1‖ there exist x, y ∈ B(x1, δ) and p ∈ ∂ f (x), q ∈ ∂(g(·) + µ‖ ·−x1‖)(y) such that ‖p + q‖ < δ, | f (x) − f (x1)| < δ and |g(x) + µ||x − x1|| − g(x1)| < δ.We may represent q = q + µξ with some q ∈ ∂g(y) and some ξ ∈ ∂‖ · ‖(y − x1). Then‖p + q‖ = ‖p + q − µξ‖ < δ + µ, which is (ii).

Simple applications of the triangle inequality give (i) and (iii). �

From this result it follows that the domain of an abstract subdifferential is denselydefined in the domain of a lower semicontinuous function.

Now we are ready to prove that β-smooth subdifferentials are abstract subdiffer-entials on β-smooth Banach spaces. The statement and the proof will be given forGateaux smoothness only since its expose is the same for all types of smoothness ofwhich Gateaux smoothness is the weakest. Of course, Property 4 is the only one thathas to be verified.

In fact we prove a more general result, namely:


Theorem 1.1.3. Let X be a Gateaux smooth Banach space, f : X → R ∪ {+∞} be alower semicontinuous and g : X → R be a locally uniformly continuous function. Iff + g has a local minimum at x0 then for arbitrary ε > 0 there exist x, y ∈ X andp ∈ DG f (x), q ∈ DGg(y) such that:

(i) ‖x − x0‖ < ε, ‖y − x0‖ < ε,

(ii) | f (x) − f (x0)| < ε and

(iii) ‖p + q‖ < ε.

A similar result is called enhanced fuzzy sum rule by Borwein, Mordukhovichand Shao in [29].

Proof. We follow El Haddad and Deville [80]. By the trivial invariance of Gateauxsubdifferential DG with respect to translation and addition of a constant, there is noloss of generality in assuming that f (0) = g(0) = 0 and that 0 is a local minimum off + g.

Fix r > 0 such that on rB the function f is bounded below, g is uniformlycontinuous, and 0 is a minimum of f + g on rB.

Let l(x) be a Leduc function on X, that is l(x) is continuously Gateaux smoothaway from the origin, Lipschitz continuous, and b‖x‖ ≥ l(x) ≥ ‖x‖ for some constantb > 0, see Deville, Godefroy and Zizler [57].

The function f + l2 + g has a strong local minimum at 0.Consider the functions

wn(x, y) =

{f (x) + l2(x) + g(y) + nl2(x − y) , x, y ∈ rB∞ , otherwise.

For each n ∈ N the function wn is lower semicontinuous and bounded below onX × X, so according to the smooth variational principle, see Deville, Godefroy andZizler [57], there exists a function ϕn : X × X → R that is Lipschitz continuous,Gateaux smooth and such that ‖ϕn‖∞ < n−1, ‖ϕ′n‖∞ < n−1, and wn +ϕn attains its strongminimum at (xn, yn).

We claim that ‖xn − yn‖ −→n→∞

0 and xn is a minimizing sequence for f + l2 + g.

First, observe that

(wn + ϕn)(0, 0) > (wn + ϕn)(xn, yn)

and using wn(0, 0) = 0 we obtain

(1.1) ϕn(0, 0) > f (xn) + l2(xn) + g(yn) + nl2(xn − yn) + ϕn(xn, yn).


Let K be a lower bound of f and g on rB. From both ‖ϕn‖∞ < n−1 and l2(xn−yn) ≥‖xn−yn‖

2 we have that 2n−1 > 2K+n‖xn−yn‖2, hence ‖xn−yn‖ ≤

√2(1 − K)n−1 −→

n→∞0.

By the uniform continuity of g on rB◦ it follows that |g(xn) − g(yn)| −→n→∞

0. From

(1.1) we have 2n−1 > f (xn) + l2(xn) + g(yn) > ( f + l2 + g)(xn)− |g(xn)− g(yn)|, and then

(1.2) 2n−1 + |g(xn) − g(yn)| > ( f + l2 + g)(xn) ≥ 0,

hence ( f + l2 + g)(xn) −→n→∞

0 and xn is a minimizing sequence for f + l2 + g. As 0

is a strong local minimum point for f + l2 + g this implies that xn −→n→∞

0 and thus,

yn −→n→∞

0 too. Hence, for n large enough the points (xn, yn) are interior points for

rB × rB. Moreover, (1.2) implies that f (xn) −→n→∞

0.

It is clear that for large n the function f (·) + l2(·) + ϕn(·, yn) + nl2(· − yn) has a localminimum at xn and, from the definition of D−G it follows that pn = −2l(xn)l′(xn) −(ϕn)′x(xn, yn)− 2nl(xn − yn)l′(xn − yn) ∈ D−G f (xn). Similarly, the function g(·) +ϕn(xn, ·) +

nl2(xn − ·) has a local minimum at yn and qn = −(ϕn)′y(xn, yn) + 2nl(xn − yn)l′(xn − yn) ∈D−Gg(yn). Hence, pn + qn = −2l(xn)l′(xn) − (ϕn)′x(xn, yn) − (ϕn)′y(xn, yn) and ‖pn + qn‖ ≤

2n−1 + 2l(xn)‖l′(xn)‖, so we may take sufficiently large n to complete the proof. �

We shall state the two known definitions of primal lower-nice property beforeshowing that they are equivalent. They are given in arbitrary Banach space X and forarbitrary fixed abstract subdifferential ∂.

Definition 1.1.4. Let f : X → R∪{+∞} be a lower semicontinuous function. Functionf is said to be ∂ primal lower-nice (∂-pln in short) at x ∈ dom f if there exist λ > 0,c > 0, T > 0 such that

f (y) ≥ f (x) + 〈p, y − x〉 −t2‖y − x‖2

whenever t ≥ T , x ∈ x + λB◦, y ∈ x + λB◦, p ∈ ∂ f (x), and ‖p‖ ≤ ct.

Definition 1.1.5. Let f : X → R∪{+∞} be a lower semicontinuous function. Functionf is said to be ∂ primal lower-nice (∂-pln in short) at x ∈ dom f if there exist λ > 0,c > 0, T > 0 such that

〈p − q, x − y〉 ≥ −t‖x − y‖2

whenever t ≥ T , x, y ∈ x + λB◦, p ∈ ∂ f (x), q ∈ ∂ f (y) and max{‖p‖, ‖q‖} ≤ ct.

Theorem 1.1.6. Any function f that is ∂-pln at x ∈ dom f according to Defini-tion 1.1.4 is also ∂-pln at x according to Definition 1.1.5 and vice versa.


Proof. The fact that function which is ∂-pln according to Definition 1.1.4 is ∂-plnaccording to Definition 1.1.5 is well known (see for example Combary, Elhilali, Levi,Poliquin and Thibault [47]). To establish the opposite direction we will use someideas from Thibault and Zagrodny [160] and Levi, Poliquin and Thibault [111].

Let now f be ∂-pln at x ∈ dom f according to Definition 1.1.5 and the constantsλ, c, T be as they are therein. We will find positive constants λ′, c′, T ′ such that fsatisfies Definition 1.1.4 with these “prime” constants.

The proof is divided in two parts.Part 1. Localization. Take 0 < λ′ < max{λ/4, c/16} and such that f is bounded

below on B(x, 4λ′) and let f (x) ≥ K for x ∈ B(x, 4λ′).Fix c′ and T ′ in a way that

(1.3) 0 < c′ < λ′/8 and T ′ > max{2T, (λ′)−2(1 + f (x) − K)}.

We claim that for arbitrary x0 ∈ B(x, λ′) and p ∈ ∂ f (x0) such that ‖p‖ ≤ c′t, t ≥ T ′

and arbitrary y0 ∈ B(x, 4λ′) such that

f (y0) + 〈p, x0 − y0〉 +t2‖y0 − x0‖

2 < infy∈B(x,4λ′)

{f (y) + 〈p, x0 − y〉 +

t2‖y − x0‖

2}

+ 1

it follows that y0 ∈ B(x, 3λ′).Assume the contrary. That is, there are x′ ∈ B(x, λ′) and p′ ∈ ∂ f (x′) such that

‖p′‖ ≤ c′t, for some t ≥ T ′, and y′ ∈ B(x, 4λ′) \ B(x, 3λ′) such that

f (y′) + 〈p′, x′ − y′〉 +t2‖y′ − x′‖2 < inf

y∈B(x,4λ′)

{f (y) + 〈p′, x′ − y〉 +

t2‖y − x′‖2

}+ 1.

In particular

f (y′) + 〈p′, x′ − y′〉 +t2‖y′ − x′‖2 < f (x) + 〈p′, x′ − x〉 +

t2‖x − x′‖2 + 1.

Thus,

1 + f (x) − K > 〈p′, x − y′〉 +t2

(‖y′ − x′‖2 − ‖x − x′‖2).

Observe that ‖y′ − x′‖ − ‖x − x′‖ ≥ ‖y′ − x‖ − 2‖x − x′‖ ≥ 3λ′ − 2λ′ = λ′ to estimate‖y′−x′‖2−‖x−x′‖2 = (‖y′−x′‖−‖x−x′‖)(‖y′−x′‖+‖x−x′‖) ≥ λ′.‖y′−x‖ ≥ λ′.3λ′ = 3(λ′)2.

Also, 〈p′, x − y′〉 ≥ −‖p′‖.‖x − y′‖ ≥ −4λ′‖p′‖ ≥ −4λ′c′t, because ‖p′‖ ≤ c′t.Hence,

1 + f (x) − K > −4λ′c′t +t2

3(λ′)2 = tλ′(32λ′ − 4c′

)> t(λ′)2

using for the last inequality that c′ < λ′/8 (see (1.3)). Then we have that

1 + f (x) − K > t(λ′)2 > T ′(λ′)2


which is a contradiction to the choice of T ′ (see (1.3)).

Part 2. Variation. Let λ′, c′, T ′ be fixed as in Part 1. We claim that f is ∂-plnat x with λ′, c′, T ′ according to Definition 1.1.4.

Assume the contrary. Then, there are x0 ∈ B(x, λ′) and p ∈ ∂ f (x0) such that‖p‖ ≤ c′t, t ≥ T ′ and

infy∈B(x0,λ′)

{f (y) + 〈p, x0 − y〉 +

t2‖y − x0‖

2}< f (x0).

Now, it is clear that

(1.4) infy∈B(x,4λ′)

{f (y) + 〈p, x0 − y〉 +

t2‖y − x0‖

2}< f (x0).

Define the function

h(y) =

f (y) + 〈p, x0 − y〉 +t2‖y − x0‖

2 , y ∈ x + 4λ′B

+∞ , otherwise.

Let yn be such that h(yn) < inf h(y)+n−1. Because of (1.4) there is no subsequenceof {yn} norm convergent to x0. Hence, there exists r > 0 such that ‖yn − x0‖ ≥ r.

According to Part 1 the points yn are in B(x, 3λ′).Through applying Proposition 1.1.2 for ε = (2n)−1, µ = (4n)−1/2, δ = (2n)−1,

and sufficiently large n, we obtain points xn ∈ B(yn, n−1/2), zn ∈ B(yn, n−1/2) andpn ∈ ∂ f (xn), qn ∈ −p + 2−1t∂(‖ · −x0‖

2)(zn) such that ‖pn + qn‖ < n−1/2.

We can represent qn = −p + tξn with some ξn ∈ 2−1∂(‖ · −x0‖2)(zn) which implies

that 〈p + qn, x0 − zn〉 = −t‖x0 − zn‖2.

We need to estimate ‖pn‖. First, we have that∣∣∣∣‖pn‖ − ‖qn‖

∣∣∣∣ −→n→∞

0. Second, we

consider ‖qn‖ = ‖p − tξn‖ ≤ ‖p‖ + t‖x0 − zn‖ ≤ c′t + t(‖x0 − x‖ + ‖x − yn‖ + ‖yn − zn‖) ≤c′t + t(λ′ + 3λ′ + n−1/2) ≤ (ct)/3 for sufficiently large n.

Hence, for n large enough, we have that xn ∈ B(x, λ), pn ∈ ∂ f (xn) is such that‖pn‖ ≤ ct/2, also x0 ∈ B(x, λ), p ∈ ∂ f (x0) is such that ‖p‖ ≤ c′t < ct/2, andt/2 ≥ T ′/2 > T . According to Definition 1.1.5 we have that

(1.5) 〈p − pn, x0 − xn〉 ≥ −t2‖x0 − xn‖

2.


We estimate the left-hand side by the following sequence of inequalities:

〈p − pn, x0 − xn〉 = 〈p + qn, x0 − xn〉 − 〈pn + qn, x0 − xn〉

≤ 〈p + qn, x0 − zn〉 + 〈p + qn, zn − xn〉 + ‖pn + qn‖.‖x0 − xn‖

≤ −t‖x0 − zn‖2 + ‖p + qn‖.‖zn − xn‖ + ‖pn + qn‖.‖x0 − xn‖

≤ −t‖x0 − xn‖2 + 2t‖xn − x0‖.‖zn − xn‖+

+‖p + qn‖.‖zn − xn‖ + ‖pn + qn‖.‖x0 − xn‖

and using that ‖x0 − xn‖ and ‖p + qn‖ are boundedand ‖xn − zn‖ −→

n→∞0, ‖pn + qn‖ −→

n→∞0 we have that

≤ −t‖x0 − xn‖2 + αn, where αn −→

n→∞0.

Combining with (1.5) we obtain

t2‖x0 − xn‖

2 ≤ αn.

Passing, if necessary, to subsequence we have that ‖x0 − xn‖ −→n→∞

a ≥ r>0, while

the right-hand side tends to zero, sot2

a2 ≤ 0 and t ≤ 0, which is a contradiction. �

1.2 Subdifferential characterization of primal lower-nice functions on smooth Banach spaces

Poliquin was proved in [136] that Clarke-Rockafellar subdifferential and proximalsubdifferential of a primal lower-nice function on finite-dimensional space coincide.This means in particular that if the definition of primal lower-nice property (see Def-inition 1.1.4 and Definition 1.1.5) is taken with respect to Clarke-Rockafellar subdif-ferential, this will produce the same class of functions. Also, these functions are com-pletely characterized by their Clarke-Rockafellar subdifferential, see Poliquin [136].This was the first large class of non-convex lower semicontinuous functions with thisproperty.

The coincidence of proximal subdifferential and Clarke-Rockafellar subdifferen-tials of a primal lower-nice function defined on Hilbert space was proved by Levy,Poliquin and Thibault [111]. Their proof uses the representation of Clarke-Rockafellarsubdifferential in Hilbert space as a sequential limit of proximal subdifferentials dueto Loewen [114]. Since a similar representation is available in general smooth Banachspaces (see Borwein and Ioffe [26], and Ivanov [91]) it is possible to extend the resultof Levy, Poliquin and Thibault to such spaces.

Here we show that Clarke-Rockafellar subdifferential and proximal subdifferentialof a primal lower-nice function defined on a β-smooth Banach space coincide (see

1.2. Subdifferential characterization of primal lower-nice functions on . . . 15

Theorem 1.2.2). The result obtained demonstrates that the class of primal lower-nicefunctions does not depend on what reasonable subdifferential is used in defining theclass. This would mean that it is possible to characterize primal lower-nice propertyin terms not involving subdifferentials.

The results from this section are published by Ivanov and Zlateva in [95].

Let (X, ‖.‖) be a Banach space, X∗ be its dual space. Recall that the Clarke-Rockafellar subdifferential (or simply the Clarke subdifferential) of a proper lowersemicontinuous function f : X → R ∪ {+∞} at point x ∈ dom f is the set

∂C f (x) = {p ∈ X∗ : f ↑(x; v) ≥ 〈p, v〉, ∀v ∈ X},

where

f ↑(x; v) = limε↓0

lim supy→ f x

t↓0

infw∈v+εB

f (y + tw) − f (y)t

is the Clarke generalized derivative and y → f x means that (y, f (y)) tend to (x, f (x))in X × R. If f (x) = +∞ then ∂C f (x) = ∅.

Also, recall that p ∈ X∗ is said to be a proximal subgradient of the lower semi-continuous function f : X → R ∪ {+∞} at x ∈ dom f , written p ∈ ∂p f (x), if for somet > 0 the inequality

f (y) ≥ f (x) + 〈p, y − x〉 −t2‖y − x‖2

holds for all y in a neighbourhood of x. The set of all such p is called proximalsubdifferential of f at x.

Borwein and Ioffe in [26] mentioned that it is quite useful to split β-smoothsubdifferential in the following manner. For any k > 0 one defines

Dkβ f (x0) = { p ∈ X∗ : there is a β smooth function g : X → R with a Lipschitz

constant k such that the function f − g has a local minimum at x0

and g′(x0) = p}.

It is clear that for x0 ∈ dom f

Dβ f (x0) =⋃k>0

Dkβ f (x0).

If x0 < dom f then Dkβ f (x0) = Dβ f (x0) = ∅.

Representation formulae of Clarke-Rockafellar subdifferential of a lower semi-continuous function in terms of different smaller subdifferentals are obtained, forexample, in Rockafellar [151], Loewen [114], Ioffe [89], Borwein and Ioffe [26],Ivanov [91], etc. We use the following


Representation formulae (Ivanov [91]). Let X be a β-smooth Banach space andf : X → R ∪ {+∞} be a lower semicontinuous function. Then

∂C f (x) = co∗(∂G f (x) + ∂∞G f (x)

),

where

∂G f (x) =

∞⋃k=1

{w∗− lim

n→∞pn; pn ∈ Dk

β f (xn), xn → f x},

and

∂∞G f (x) =

∞⋃k=1

{w∗− lim

n→∞λ−1

n pn; pn ∈ Dλnkβ f (xn), xn → f x, λn → ∞

}.

Further we will work in a β-smooth Banach space and will consider β-smoothsubdifferential. In this context is natural to consider Dβ-pln functions.

A lower semicontinuous function f : X → R ∪ {+∞} defined on a β-smoothBanach space X is said to be Dβ primal lower-nice (Dβ-pln in short) at x0 ∈ dom f ifthere exist λ > 0, c > 0, T > 0 such that

f (y) ≥ f (x) + 〈p, y − x〉 −t2‖y − x‖2

whenever t ≥ T , x ∈ B(x0, λ), y ∈ B(x, λ), p ∈ Dβ f (x), and ‖p‖ ≤ ct.In fact this is Definition 1.1.4 taken for the subdifferential ∂ = Dβ.Obviously, in any Banach space X,

(1.6) ∂p f (x) ⊆ ∂C f (x),

and in β-smooth Banach space X,

∂p f (x) ⊆ Dβ f (x) ⊆ ∂C f (x).

By Definition 1.1.4 it is clear that if f is ∂-pln at x0 then

∂ f (x) ∩ ctB∗ ⊂ ∂p f (x) for all x ∈ B(x0, λ).

Lemma 1.2.1. Let X be a Banach space. Let f : X → R ∪ {+∞} be a ∂-pln atx0 ∈ dom f function with respect an abstract subdifferential ∂ including ∂p. Then∂p f (x0) is a convex and w∗ closed subset of X∗.

Proof. The convexity of ∂p f (x0) is obvious from its definition. The set ∂p f (x0)being a convex subset of X∗ has the same closures in w∗ and bw∗ topologies (seeHolmes [86, Corollary 2, p. 154]). Let us recall that a set in X∗ is bw∗ closed when itcontains the w∗ limits of all bounded and w∗ converging nets of its elements. Hence,it is enough to show that ∂p f (x0) is a bw∗ closed set.

1.2. Subdifferential characterization of primal lower-nice functions on . . . 17

To this end, let {pα}α∈A be a norm bounded net such that pα ∈ ∂p f (x0), α ∈ A andpα → p in w∗ topology.

Let λ, c and T be as in the definition of ∂ primal lower-nice property of f at x0.Since the net {pα}α∈A is norm bounded, we can take t > T such that ‖pα‖ < ct for

all α ∈ A. Since f is ∂-pln at x0 and ∂ f (x) ⊃ ∂p f (x) for all x, we have that

f (x) ≥ f (x0) + 〈pα, x − x0〉 −t2‖x − x0‖

2

for any x such that ‖x− x0‖ ≤ λ and for all α ∈ A. Passing to limit in w∗ topology weobtain that p ∈ ∂p f (x0). �

In particular, since in β-smooth Banach space Dβ is a subdifferential including∂p, then for a Dβ-pln at x0 ∈ dom f function f : X → R ∪ {+∞} the proximalsubdifferential ∂p f (x0) is a convex and w∗ closed subset of X∗.

When a function f is ∂-pln at all points in dom f , then f is said to be ∂-pln.Now we are ready to prove

Theorem 1.2.2. Let X be β-smooth Banach space and f : X → R∪ {+∞} be a Dβ-plnfunction. Then ∂p f ≡ Dβ f ≡ ∂C f .

Proof. Let x0 ∈ dom f and λ, c and T be as in the definition of Dβ primal lower-niceproperty of f at x0.

First, we show that ∂G f (x0) ⊆ ∂p f (x0). Let p ∈ ∂G f (x0).Let k ≥ 1 and p be a w∗ limit of a sequence {pn}, pn ∈ Dk

β f (xn), xn → f x0. Thesequence {pn} is norm bounded by k, so we can take t > T such that ‖pn‖ ≤ k < ct.Eventually, ‖xn − x0‖ ≤ λ. Since f is Dβ-pln at x0 for all large enough n and all xsuch that ‖x − x0‖ ≤ λ we have that

f (x) ≥ f (xn) + 〈pn, x − xn〉 −t2‖x − xn‖

2.

Passing to limit as n→ ∞ we obtain that p ∈ ∂p f (x0).Further, we show that ∂G f (x0) + ∂∞G f (x0) ⊆ ∂p f (x0).Let p ∈ ∂G f (x0) and p∞ ∈ ∂∞G f (x0).Let k ≥ 1 and p∞ be a w∗ limit of a sequence {λ−1

n pn}, pn ∈ Dλnkβ f (xn), xn → f x0,

λn → ∞. The sequence {λ−1n pn} is norm bounded by k, so ‖pn‖ ≤ kλn. Since λn → ∞

we have that kλn ≥ Tc for sufficiently large n. Eventually ‖xn − x0‖ ≤ λ. Since f isDβ-pln at x0 the inequality

f (x) ≥ f (xn) + 〈pn, x − xn〉 −kλn

2c‖x − xn‖

2


holds for any x such that ‖x − x0‖ ≤ λ. Dividing by λn and taking limit as n→ ∞ weobtain that

0 ≥ 〈p∞, x − x0〉 −k2c‖x − x0‖

2

for any x ∈ dom f ∩ B(x0, λ).We have from the first part of the proof the inequality

f (x) ≥ f (x0) + 〈p, x − x0〉 −t2‖x − x0‖

2

for any x ∈ dom f ∩ B(x0, λ).We only need to sum up the last two inequalities to obtain that

f (x) ≥ f (x0) + 〈p + p∞, x − x0〉 −

(t2

+k2c

)‖x − x0‖

2

for any x ∈ dom f ∩ B(x0, λ) (hence, for any x ∈ B(x0, λ)) which means thatp + p∞ ∈ ∂p f (x0).

Therefore,

(1.7) ∂G f (x0) + ∂∞G f (x0) ⊆ ∂p f (x0).

To complete the proof, we remind that ∂p f (x0) is a convex and w∗ closed set, seeLemma 1.2.1. From the latter, the representation formulae of Ivanov, (1.7) and (1.6)it follows that

∂C f (x0) = co∗(∂G f (x0) + ∂∞G f (x0)

)⊆

⊆ co∗∂p f (x0) = ∂p f (x0) ⊆

⊆ ∂C f (x0). �

Remark 1.2.3. Not every function f such that ∂p f ≡ ∂C f is necessarily primallower-nice. Consider for example f : R→ R defined as

f (x) :=∫ x

0t2 sin

1t2 dt.

It is easy to check that f is everywhere twice differentiable. So, ∂p = ∂C f = { f ′}, butf is not primal lower-nice at 0. Otherwise it must locally satisfy f ′′ ≥ −C(| f ′| + 1)for some constant C > 0. The latter will imply that f ′′ is bounded from below at aneighbourhood of 0, but lim inf

x→0f ′′(x) = −∞.

1.3. Characterizations of prox-regular sets in uniformly convex Banach spaces 19

1.3 Characterizations of prox-regular sets in uniformlyconvex Banach spaces

In this section we extend to the setting of uniformly convex Banach spaces severalresults obtained for prox-regular sets in Hilbert spaces. Prox-regularity of a set C ata point x ∈ C is a variational condition related to normal vectors and which iscommon to many types of sets. In the context of uniformly convex Banach spaces,the prox-regularity of a closed set C at x is shown to be still equivalent to theproperty of the distance function dC to be continuously differentiable outside of Con some neighbourhood of x. Additional characterizations are provided in termsof metric projection mapping. We also examine the global level of prox-regularitycorresponding to the continuous differentiability of the distance function dC over anopen tube of uniform thickness around the set C.

The differentiability of the distance function dC to a nonempty closed subset Cof a Banach space X and the single valuedness of the metric projection mapping PC

are longstanding subjects of study. For a convex set C, the differentiability of d2C and

the single valuedness and continuity of PC on the whole space X are well known insmooth Banach spaces. In the finite dimensional Euclidean case, Motzkin [130] seemsto be the first to prove that a nonempty closed set C is convex if and only if its metricprojection mapping PC is single-valued everywhere. In the Hilbert setting, Klee [103]proved that for weakly closed sets C, the convexity of C is also characterized in thisway.

The characterization in the Hilbert setting of the convexity of a norm closedset C by the metric projection mapping PC being single-valued and norm-to-weakcontinuous is due to Asplund [1]. When the dual space of X is rotund, Vlasov [167]extended in some sense Asplund’s result by showing that a norm closed set C of Xis convex if and only if the metric projection mapping is single-valued and norm-to-norm continuous. See also Vlasov [166] for the case of approximately compactsets.

Using the original result by Fitzpatrick [75] reducing the differentiability of dC

to its Gateaux directional derivability in a certain key direction, Borwein, Fitzpatrickand Giles [24] characterized closed convex sets C of a Banach space X with rotunddual as closed sets C for which the distance function dC is Gateaux differentiable onX \ C with ‖∇GdC(x)‖ = 1 for all x ∈ X \ C. This result of Fitzpatrick will be statedbelow as a theorem (see Theorem 1.3.20) because of its importance.

In order to extend the Steiner polynomial formula concerning the n-dimensionalmeasure of the r-neighbourhood (with respect to the Euclidean norm) of a closedconvex subset or a compact C2-submanifold of Rn to a much larger class of sets,Federer [73] introduced the concept of subsets of Rn with positive reach. For a


nonempty closed set C ⊂ Rn, denoting by Unp(C) the set of all points x ∈ Rn forwhich C contains a unique nearest point to x, Federer defined its reach (that hedenoted by reach(C)) as the largest r (possibly +∞) such that {x ∈ Rn : 0 < dC(x) <r} ⊂ Unp(C). Then, he declared C to be positively reached whenever reach(C) > 0and established, among other results, that dC is continuously differentiable on the set{x ∈ Rn : 0 < dC(x) < reach(C)}. Note that Federer also worked, for a fixed pointx ∈ C with reach(C, x), i.e., the supremum of all r > 0 such that the open ball centeredat x with radius r is included in Unp(C). Considering, in Hilbert space, the conceptof p-convex set C of Degiovanni, Marino and Tosques [56], Canino [37] establishedthat on a suitable open neighbourhood of C the metric projection mapping PC issingle-valued and locally Lipschitz continuous. Staying on the global level in Hilbertspace, Clarke, Stern and Wolenski [46] introduced and studied the proximally smoothsets. Such sets correspond to closed sets C for which the distance function dC iscontinuously differentiable on an open tube around C of the type

UC(r) := {u ∈ X : 0 < dC(u) < r}

for some r > 0. In view of Federer’s result recalled above, in finite dimensions thosesets are positively reached and vice versa. Clarke, Stern and Wolenski characterized,in Hilbert space, proximal smooth sets in several interesting ways, in particular interms of proximal normals and proximal mapping PC. They also provided (in finitedimensions) a detailed analysis of locally Lipschitz continuous functions for whichthe epigraph is proximally smooth. Another previous interesting result was obtainedby Shapiro [153] on the local level, in the Hilbert setting. He proved, for a closedset C and a point x ∈ C, that the metric projection mapping PC is single-valued on aneighbourhood of x whenever the distance to the general Boulingand contingent coneto C satisfies a property referred to as the Shapiro property by Poliquin, Rockafellarand Thibault [138].

On the local level, in the study of sets C for which dC is locally differentiableand its consequences for the metric projection mapping PC, Poliquin, Rockafellar andThibault [138] recently made advance in the Hilbert setting with a different point ofview, by making the link with the local property of C called prox-regularity. Thisproperty has been introduced as a new important regularity in variational analysisby Poliquin and Rockafellar [137]. They defined this concept for functions and forsets, and studied it in the finite dimensional setting. Rich geometric implicationsand characterizations of such a concept were obtained by Poliquin, Rockafellar andThibault. In their work [138], they relied, in Hilbert space, on the prox-regularity ofa closed set C at x ∈ C and characterized it in terms of dC and PC, e.g. dC beingcontinuously differentiable outside of C on a neighbourhood of x or PC being single-valued and norm-to-weak continuous on this same neighbourhood. They also gave asubdifferential characterization of such sets with the normal cone to C. Coming back


to the global level, they showed that proximally smooth sets are exactly uniformlyprox-regular sets and provided new insights on those sets.

We extend the scope of these results to the more general setting of uniformlyconvex Banach spaces (e.g. lp, Lp and W p

m with 1 < p < ∞) and find their analoguesin the context of such spaces. We will rely on a property that we introduce, analogousto the prox-regularity.

In Subsection 1.3.1 we consider the definition and the first properties of prox-regular sets on a uniformly convex Banach space X. We also introduce related defi-nitions for functions and for set-valued mappings. Subsection 1.3.2 is devoted to thestudy of properties of local Moreau envelopes of functions on X. In Subsection 1.3.3we establish several characterizations of prox-regular sets in X (see Theorem 1.3.25)extending in this way the results of Poliquin, Rockafellar and Thibault [138]. In thefinal Subsection 1.3.4 we use some techniques developed in the previous subsec-tions to obtain in Theorem 1.3.27 various results similar to the characteristic The-orem 1.3.25 but on the global level of proximally smooth sets. So, with Theorems1.3.25 and 1.3.27 we extend several results of Federer [73], Canino [37], Clarke,Stern and Wolenski [46], Poliquin, Rockafellar and Thibault [138], and Colombo andGoncharov [48].

The results from this section are published by Bernard, Thibault and Zlatevain [18].

We begin by recalling some of the properties of uniformly convex Banach spaceswhich can be found in the books of Diestel [58], Brezis [34], Beauzamy [12], Deville,Godefroy and Zizler [57].

For a Banach space X the following are equivalent:

(X1) X has an equivalent uniformly convex norm ‖.‖, i.e., such that its modulus ofconvexity

δ‖.‖(ε) := inf{1−

∥∥∥∥∥ x+y2

∥∥∥∥∥ : ‖x‖ = ‖y‖ = 1, ‖x−y‖≥ε}

satisfies δ‖.‖(ε)>0 for all ε∈]0, 2].

(X2) X has an equivalent uniformly convex norm ‖.‖ with modulus of convexity ofpower type q, i.e., for some k > 0 one has δ‖.‖(ε) ≥ kεq, for all ε ∈]0, 2].The function δ‖.‖ is increasing in ]0, 2]. From Dvoretzky’s theorem necessarilyq ≥ 2.

(X3) X has an equivalent uniformly smooth norm ‖.‖, i.e., such that its modulus ofsmoothness

ρ‖.‖(τ) :=12

sup {‖x + y‖ + ‖x − y‖ − 2 : ‖x‖ = 1, ‖y‖ ≤ τ} for τ ≥ 0

satisfies limτ↓0ρ‖.‖(τ)τ

= 0.


(X4) X has an equivalent uniformly smooth norm ‖.‖ with modulus of smoothness ofpower type s, i.e., such that for some c > 0 one has ρ‖.‖(τ) ≤ cτs for all τ ≥ 0.From Dvoretzky’s theorem necessarily 1 < s ≤ 2.

(X5) X has an equivalent norm which is both uniformly convex and uniformlysmooth and which has moduli of convexity and smoothness of power typeq and s, respectively.

It is well-known (see Diestel [58], Lindenstrauss and Tzafriri [112, 113]) that allHilbert spaces H and the Banach spaces lp, Lp, and W p

m (1 < p < ∞) all are (for theirusual norms) uniformly convex and uniformly smooth with moduli of convexity andsmoothness of power type. More precisely, for ε ∈]0, 2] and τ ≥ 0

δH(ε) = 1 −√

1 − (1/4)ε2 ≥ ε2/4,

δlp(ε) = δLp(ε) = δW pm(ε) =

p−1

8 ε2 + o(ε2) > p−18 ε2, 1 < p < 2,

1−[1−

(ε2

)p]1/p> 1

p

(ε2

)p, p ≥ 2,

ρH(τ) = (1 + τ2)1/2 − 1 < τ,

ρlp(τ) = ρLp(τ) = ρW pm(τ) =

(1 + τp)1/p − 1 < 1pτ

p, 1 < p < 2,p−1

2 τ2 + o(τ2) < p−12 τ2, p ≥ 2.

In current Section 1.3 we work in an uniformly convex Banach space X which isequipped with an equivalent norm ‖.‖ that satisfies (X5).

Such a norm is a Kadec norm, i.e., it satisfies the property that whenever xnw−→n→∞

x,

meaning that the sequence xn converges to x in the weak topology, with ‖xn‖ −→n→∞‖x‖,

then xn‖.‖−→n→∞

x, meaning that the sequence xn converges to x in the norm topology.

Let q and s be the power types of moduli of convexity and smoothness of ‖.‖,respectively. Then X∗ is also uniformly convex and its dual norm has modulus ofconvexity of power type q∗ = s(s − 1)−1 and modulus of smoothness of power types∗ = q(q − 1)−1.

The mapping J : X → X∗ defined by

J(x) := {x∗ ∈ X∗ : 〈x∗, x〉 = ‖x∗‖.‖x‖, ‖x∗‖ = ‖x‖}

is generally called the normalized duality mapping. Let us put together, in the contextof uniformly convex Banach space X satisfying (X5), some of its properties that wewill use hereafter (see Cioranescu [43]):


(J) The mapping J : X → X∗ is single-valued, bijective and norm-to-norm uni-formly continuous on bounded sets, J(λx) = λJ(x) for all λ ∈ R, ‖J(x)‖ = ‖x‖,and J(x) = ∇ 1

2‖.‖2(x) for all x ∈ X.

An analogous property (J∗) holds for the normalized duality mapping J∗ : X∗ → X.Moreover, J∗ = J−1.

It is known (see e.g. Xu and Roach [169]), that for r > 0 there exist positiveconstants Kr, K′r such that

(1.8) 〈J(x) − J(y), x − y〉 ≥ Kr‖x − y‖q, ∀x, y ∈ rB,

(1.9) ‖J(x) − J(y)‖ ≤ K′r‖x − y‖s−1, ∀x, y ∈ rB.

The space X × R will be endowed with the norm ||| · ||| given by |||(x, r)||| =√‖x‖2 + r2. So, for the normalized duality mapping JX×R : X ×R→ X∗ ×R associated

with the norm ||| · |||, one has the equality

(1.10) JX×R(x, r) = (J(x), r).

Recall that a normed vector space (Y, ‖·‖) is rotund or strictly convex provided thatfor any y, y′ ∈ Y with ‖y‖ = ‖y′‖ = 1 and y , y′ one has

∥∥∥12 (y + y′)

∥∥∥ < 1. According to(X1), the uniform convexity holds when this inequality is fulfilled in some uniformway. Recall (see for example Deville, Godefroy and Zizler [57] and Fabian, Habala,Hajek, Montesinos, Pelant and Zizler [71]) that the strict convexity of the norm ‖ · ‖is equivalent to require for any non zero y, y′ ∈ Y , y , y′ the equality

(1.11) ‖y + y′‖ = ‖y′‖ + ‖y‖

to entail y′ = µy for some µ > 0.We will need the following elementary result concerning nearest points of a closed

subset C to a point in Y . It can be found in Hilbert space for example in Clarke,Ledyaev, Stern and Wolenski [45, p.4]. It must also be known in the general strictlyconvex setting but we did not find it in the literature.

As usual, dC(u) denotes the distance from u to the set C, i.e. dC(u) := infx∈C‖u − x‖

and PC(u) := {x ∈ C : dC(u) := infx∈C ‖u − x‖} denotes the set of all nearest points ofC to u.

Lemma 1.3.1. Let (Y, ‖ · ‖) be a strictly convex normed vector space, C be a closedsubset of Y and u < C. Assume that PC(u) , ∅. Then for any p ∈ PC(u) and anyt ∈]0, 1], one has PC(u + t(p − u)) = {p}.


Proof. The case t = 1 being obvious, we may suppose t ∈ ]0, 1[ . Putting ut :=u + t(p − u) we have

‖ut − p‖ = ‖u + t(p − u) − p‖ = (1 − t)‖u − p‖ = (1 − t)dC(u).

Further, for any y ∈ C,

‖ut − y‖ = ‖u + t(p − u) − y‖ ≥ ‖u − y‖ − t‖u − p‖≥ dC(u) − t‖u − p‖= (1 − t)dC(u)= ‖ut − p‖,

and hence p ∈ PC(ut).Suppose that there exists pt , p with pt ∈ PC(ut). Then setting y = pt in the abovesequence of inequalities we obtain

dC(ut) = ‖ut − pt‖ = ‖u − pt + t(p − u)‖ ≥ ‖u − pt‖ − t‖u − p‖≥ dC(u) − t‖u − p‖= ‖ut − p‖ = dC(ut).

All the last inequalities are then equalities and hence

(1.12) ‖u − pt‖ = dC(u)

and‖u − pt‖ = ‖u − pt + t(p − u)‖ + ‖t(u − p)‖.

Further, obviously t(u − p) , 0, and one also has u − pt + t(p − u) , 0 since ‖u − pt +

t(p − u)‖ ≥ (1 − t)dC(u). So, because of the rotundity, the last equality above entails(see (1.11)) that there exists µ > 0 with

u − pt + t(p − u) = µt(u − p),

that is,

(1.13) u − pt = t(µ + 1)(u − p).

Using (1.12) and taking the norm of both members of (1.13) yield dC(u) = t(µ+1)dC(u)and since dC(u) > 0 we have t(µ + 1) = 1. Putting this value in (1.13) gives pt = p,which completes the proof. �

For a set C ⊂ X we will denote by cl C its norm closure in X. A vector p ∈ Xis said to be a primal proximal normal vector (shortly proximal normal vector) to


C at x ∈ cl C (see Borwein and Strojwas [31]) if there are u < cl C and r > 0 suchthat p = r−1(u − x) and ‖u − x‖ = dC(u). It is known according to Lau theorem (seeLau [108]) that in any reflexive Banach space endowed with a Kadec norm, the setof those points which have a nearest point to any fixed closed subset is a dense set.In the appropriately renormed space X we are working in, the above property holdsand hence there are proximal normal vectors at any point of some dense subset ofthe boundary of C. Observe that the proximal normality of a non-zero p ∈ X to C atx ∈ cl C corresponds to the existence of some r > 0 such that x ∈ Pcl C(x + rp). Thecone of all such vectors p, together with the origin, will be denoted by PNC(x) andcalled primal proximal normal cone of C at x.

The concept is local in the sense that for any u < cl C and any closed ballV := B[x, β] centered at x ∈ cl C such that ‖u − x‖ = dC∩V(u) one has u − x ∈ PNC(x).Indeed, put ρ := dC∩V(u) > 0 and ut := x+t(u−x) for any fixed positive t < min

(1, β

2ρ

).

Observe first that ut ∈ int V according to the inequality t < β

2ρ and hence ut < cl Cbecause otherwise one would get ut ∈ cl (C ∩ V) and

‖ut − u‖ = (1 − t)‖u − x‖ < dC∩V(u)

which would be a contradiction. Further ‖ut − x‖ = tρ and, on the one hand, for anyy ∈ C ∩ V one can write

‖ut − y‖ = ‖u + (1 − t)(x − u) − y‖ ≥ ‖u − y‖ − (1 − t)‖u − x‖ ≥ tρ = ‖ut − x‖.

On the other hand, for any y ∈ C \ V one has

‖ut − y‖ ≥ ‖y − x‖ − ‖ut − x‖ > β − t‖u − x‖ = β − tρ > tρ = ‖ut − x‖,

the last inequality being due to the choice of t. So, ‖ut − x‖ = dC(ut) and hence by thedefinition of PNC(x) we have

(1.14) u − x = t−1(ut − x) ∈ PNC(x).

A continuous linear functional p∗ ∈ X∗ is said to be a proximal normal functionalto C at x ∈ cl C (see Borwein and Strojwas [31]) if there are u < cl C, r > 0 suchthat p∗ = r−1J(u − x) and ‖u − x‖ = dC(u). Or, equivalently, a non-zero p∗ ∈ X∗

is a proximal normal functional to C at x ∈ cl C if there exists r > 0 such thatx ∈ Pcl C(x + rJ∗(p∗)). The cone of all such functionals p∗, together with the origin,will be denoted by NP

C(x). One easily verifies that if p ∈ PNC(x), then J(p) ∈ NPC(x),

and that if p∗ ∈ NPC(x), then J∗(p∗) ∈ PNC(x). Hence, PNC(x) and NP

C(x) completelydetermine each other.

A functional x∗ ∈ X∗ is said to be a Frechet normal functional (see Borwein andStrojwas [31]) to C at x if for any ε > 0 there exists a neighbourhood Uε of x suchthat the inequality 〈x∗, x′ − x〉 − ε‖x′ − x‖ ≤ 0 holds for all x′ ∈ C ∩ Uε.


As the norm of the space X we work in is Frechet differentiable away from theorigin, it is not difficult to verify that for any closed subset C ⊂ X and any x ∈ C,any proximal normal functional to C at x is also a Frechet normal functional to C atx (see Borwein and Strojwas [31, Corollary 3.1]).

Let f : X → R ∪ {+∞} be a lower semicontinuous function. By definition, theeffective domain of f is the set dom f := {x ∈ X : f (x) < +∞} and the epigraph off is the set epi f := {(x, r) ∈ X × R : f (x) ≤ r}. It is clear that epi f is non-emptywhenever f is proper, and epi f is closed exactly when f is lower semicontinuous.

Let x ∈ dom f . We say that p∗ ∈ X∗ is a proximal subgradient of f at x if (p∗,−1)is a proximal normal functional to the epigraph of f at (x, f (x)). The proximalsubdifferential of f at x, denoted by ∂p f (x), consists of all such functionals. Thus,we have p∗ ∈ ∂p f (x), if and only if, (p∗,−1) ∈ NP

epi f (x, f (x)). The functional x∗ ∈ X∗

is said to be a Frechet subgradient of f at x if (x∗,−1) is a Frechet normal functionalto the epigraph of f at (x, f (x)). The Frechet subdifferential of f at x, denoted by∂F f (x), consists of all such functionals. If x < dom f then all subdifferentials of f at xare empty, by convention. It is known that for a lower semicontinuous function f ona reflexive Banach space with a Kadec and Frechet differentiable norm (in particular,on X), the set dom ∂p f is dense in dom f (see Borwein and Strojwas [32, Theorem7.1]). Moreover, from what we saw above, ∂p f (x) ⊂ ∂F f (x) for all x ∈ X. The Frechetsubgradients are known (see Ioffe [89]) to have an analytical characterization in the

sense that x∗ ∈ ∂F f (x), if and only if, lim infy→x

f (y) − f (x) − 〈x∗, y − x〉‖y − x‖

≥ 0. When

∂F f (x) , ∅, one says that f is Frechet subdifferentiable at the point x.As usual, we will denote by ψC the indicator function of a closed set C ⊂ X, i.e.,

ψC(y) = 0 if y ∈ C and ψC(y) = +∞ otherwise. It is easily checked that ∂pψC(x) =

NPC(x) for any x ∈ C.

Like for the proximal normal cone in Hilbert space (see Clarke, Stern and Wolen-ski [46] and Bounkhel and Thibault [33]) one can express, in our uniformly convexspace X, the proximal normal functional cone to C in terms of the proximal subdiffer-ential of dC. We denote by B∗ the closed unit ball of X∗ and by C(ρ) the ρ-enlargementof the set C, i.e., C(ρ) := {u ∈ X : dC(u) ≤ ρ}.

Proposition 1.3.2. For any closed subset C of X and any x ∈ C,

∂pdC(x) = NPC(x) ∩ B∗.

Proof. The inclusion x∗ ∈ ∂pdC(x) means (x∗,−1) ∈ NPepi dC

(x, 0), or equivalently, forany t > 0 small enough,

(1.15) inf(y,λ)∈epi dC

{‖x + tv − y‖2 + (−t − λ)2} = t2‖v‖2 + t2,

where v = J∗(x∗). This entails that

infy∈C{‖x + tv − y‖2} = t2‖v‖2,


hence v ∈ PNC(x). Further (1.15) ensures for all y ∈ X that

‖x + tv − y‖2 + 2tdC(y) + d2C(y) ≥ t2‖v‖2

and since −2tJ(v) = ∇(−‖ · ‖2)(tv), for each ε > 0 there exists some positive numberr < ε such that for all y ∈ B[x, r]

2t〈−J(v), x − y〉 ≤ ε‖x − y‖ + 2tdC(y) + d2C(y)

and taking the inequality dC(y) ≤ ‖x − y‖ into account we see that

2t〈−J(v), x − y〉 ≤ (ε + 2t + ‖x − y‖)‖x − y‖ ≤ (2ε + 2t)‖x − y‖.

This easily yields ‖v‖ = ‖ − J(v)‖ ≤ 1 and hence x∗ ∈ NPC(x) ∩ B∗.

Conversely, take x∗ ∈ NPC(x) ∩ B∗. Put v = J∗(x∗) and choose t > 0 small enough

that d2C(x + tv) = t2‖v‖2. Then

infy∈X{‖x + tv − y‖2 + (t + dC(y))2} = inf

ρ≥0h(ρ),

where h(ρ) := infy∈C(ρ){‖x + tv − y‖2 + (t + ρ)2}. Obviously we have the equalityh(ρ) = d2

C(ρ)(x + tv) + (t + ρ)2. Consider two cases:— If dC(x + tv) ≤ ρ, then h(ρ) = (t + ρ)2 ≥ t2 + d2

C(x + tv).— If dC(x + tv) > ρ, using Bounkhel and Thibault [33, Lemma 3.1] or Lemma 1.3.28in forthcoming Subsection 1.3.4, we have dC(ρ)(x + tv) = dC(x + tv) − ρ and thus,

h(ρ) = d2C(x + tv) + t2 + 2ρ[ρ + t − dC(x + tv)] ≥ d2

C(x + tv) + t2.

The last estimation comes from the fact that ‖v‖ ≤ 1.Finally, making use of the above inequalities and of the equality d2

C(x+tv) = t2‖v‖2

we obtaininfy∈X{‖x + tv − y‖2 + (t + dC(y))2} = t2‖v‖2 + t2,

which entails that (v,−1) is in PNepi dC (x, 0) or, in other words, that x∗ ∈ ∂pdC(x). �

1.3.1 Prox-regular sets

The concept of prox-regularity was introduced for functions from Rn into R ∪{+∞} by Poliquin and Rockafellar in [137], extending the class of primal lowernice functions previously considered by Poliquin in [136]. The introduction of prox-regular functions in Poliquin and Rockafellar [137] has been motivated by the studyof second-order properties of some non-convex functions. A subset of Rn is defined to


be prox-regular in Poliquin and Rockafellar [137] when its indicator function is prox-regular. The concept of prox-regularity of sets then has been studied and developed inHilbert space in [138] by Poliquin, Rockafellar and Thibault who showed in particularits rich geometric implications.

The prox-regularity property for a closed set C is, like for functions, local anddirectional, concerning a point x ∈ C and a direction p ∈ PNC(x). If the property holdsfor all possible proximal normal vectors to C at x, the set is said to be prox-regular atx. Considering the prox-regularity at a point, Poliquin, Rockafellar and Thibault [138]showed it to be a localization of Federer’s positive reach concept (see Federer [73]) orproximal smoothness property of Clarke, Stern and Wolenski [46]. The localization ofthe mentioned behavior is made clear by the fact that the authors showed in Poliquin,Rockafellar and Thibault [138] the equivalence of the prox-regularity of a set C,of the local single valuedness and continuity of the metric projection mapping PC,and of the local C1 regularity of the square distance function d2

C to C among othercharacterizations. Their approach allowed them to retrieve also the important globallevel results of Clarke, Stern and Wolenski [46].

Before extending the above concepts to uniformly convex spaces, we must pointout that, in addition to Federer’s study of Steiner polynomial formula (see Fed-erer [73]) and Canino’s work related to geodesics, the first strong applications ofproximal smoothness of sets to Control theory has been provided by Clarke, Ledyaev,Stern and Wolenski [45]. For other recent applications to evolution problems withmoving sets putting in light the amenability of prox-regular and proximally smoothsets, we refer to Edmond and Thibault [68] and Thibault [158].

Our extension of the definition of a prox-regular set to our setting will use the du-ality mapping as follows, in order to find striking characterizations of prox-regularitylike in the Hilbert space setting.

Definition 1.3.3. A closed set C ⊂ X is called prox-regular at x ∈ C for p∗ ∈ NPC(x)

if there exist ε > 0 and r > 0 such that for all x ∈ C and for all p∗ ∈ NPC(x) with

‖x − x‖ < ε and ‖p∗ − p∗‖ < ε the point x is a nearest point of {x′ ∈ C : ‖x′ − x‖ < ε}to x + rJ∗(p∗). The set C is prox-regular at x if this property holds for all p∗ ∈ NP

C(x).

The following proposition shows that the prox-regularity concept for subsets of Xin fact does not depend on any direction. The first part of its proof reproduces ideasof the proof of Proposition 1.2 in Poliquin, Rockafellar and Thibault [138].

Proposition 1.3.4. A closed set C ⊂ X is prox-regular at x, if and only if, it isprox-regular at x for p∗ = 0. If the closed set C is prox-regular at x for p∗ = 0 withε and r, then for all x ∈ C with ‖x − x‖ < ε and for all p∗ ∈ NP

C(x) with ‖p∗‖ ≤ ε,

(1.16) 0 ≥ 〈J[J∗(p∗) − r−1(x′ − x)], x′ − x〉, ∀x′ ∈ C with ‖x′ − x‖ < ε.


Proof. Obviously, if C is prox-regular at x for all p∗ ∈ NPC(x) then it is so for p∗ = 0.

To establish the converse, let us assume that C is prox-regular at x for p∗ = 0 withε > 0 and r > 0. Take any p∗ ∈ NP

C(x) with p∗ , 0, and set ε′ := min{ε/2, ‖p∗‖/2}.For x ∈ C and p∗ ∈ NP

C(x) with ‖x − x‖ < ε′ and ‖p∗ − p∗‖ < ε′ we have that

ε

2‖p∗‖‖p∗‖ ≤

ε

2‖p∗‖[‖p∗ − p∗‖ + ‖p∗‖] ≤

ε

2‖p∗‖ε′ +

ε

2≤ε

4+ε

2< ε.

We may rewrite the latter as∥∥∥∥ εp∗

2‖p∗‖ − 0∥∥∥∥ < ε. By prox-regularity of C at x for 0, we

have that x is a nearest point of {x′ ∈ C : ‖x′−x‖ < ε} to x+rJ∗(εp∗

2‖p∗‖

)= x+ rε

2‖p∗‖ J∗(p∗).

This means that C is prox-regular at x for p∗ with constants ε′ and r′ = rε2‖p∗‖ .

To prove the second claim, we suppose that C is prox-regular at x for p∗ = 0 withε and r. Fix any x ∈ C with ‖x − x‖ < ε and any p∗ ∈ NP

C(x) with 0 < ‖p∗‖ < ε. Bydefinition, x is a nearest point of {x′ ∈ C : ‖x′ − x‖ < ε} to x + rJ∗(p∗), that is,

‖x + rJ∗(p∗) − x‖ ≤ ‖x + rJ∗(p∗) − x′‖, ∀x′ ∈ C with ‖x′ − x‖ < ε.

Setting p := J∗(p∗) we rewrite the latter as

(1.17) r‖p‖ ≤ ‖x − x′ + rp‖, ∀x′ ∈ C with ‖x′ − x‖ < ε.

Since J(u) is the derivative of 12‖.‖

2 at u, we have for all t > 0 and all x′ that

γ := 〈J[J∗(p∗) − r−1(x′ − x)], x′ − x〉 = 〈J(p − r−1(x′ − x)), x′ − x〉 ≤

[2t]−1{∥∥∥p − r−1(x′ − x) + t(x′ − x)

∥∥∥2−

∥∥∥p − r−1(x′ − x)∥∥∥2}.

In particular for t = r−1

γ ≤r2

{‖p‖2 − ‖p − r−1(x′ − x)‖2

}=

r2

{‖p‖2 − r−2‖x − x′ + rp‖2

}=

12r

{(r‖p‖)2

− ‖x − x′ + rp‖2}.

Taking (1.17) into account we deduce that γ ≤ 0 for all x′ ∈ C with ‖x′ − x‖ < ε,which is (1.16). The case ‖p∗‖ = ε is obtained via a limit process and hence the proofis complete. �

In what follows, saying that C is prox-regular at x with ε and r we will mean thatthe constants ε and r are taken from prox-regularity of C at x for p∗ = 0. It is clearthat if the closed set C is prox-regular at x for p∗ = 0 with some positive constants εand r then it is so for any constants 0 < ε′ ≤ ε and 0 < r′ ≤ r.

The notion corresponding to the inequality (1.16) in the case of functions isintroduced in the following Definition 1.3.5. Let us note that another definition is


considered in Bernard and Thibault [14], using the “proximal-type” estimation withthe square of the norm as in Poliquin and Rockafellar [137] instead of (1.18). In theHilbert setting, functions satisfying the definition given below, are primal lower-nicefunctions (see Poliquin and Rockafellar [137] or Bernard and Thibault [14]).

We will see in the next subsection that the J-plr concept introduced in Def-inition 1.3.5 below for functions also yields, concerning their Moreau envelopes,various important properties which have their own interest. Recall that the contexthere is broader than in Bernard and Thibault [14] since the power of the modulus ofconvexity is any q ≥ 2. These properties applied to the indicator functions of sets willbe among the keys of the development of our study.

Definition 1.3.5. A lower semicontinuous function f : X → R ∪ {+∞} is J-primallower regular (J-plr in short) at x ∈ dom f if there exist positive constants ε and rsuch that

(1.18) f (y) ≥ f (x) + 〈J[J∗(p∗) − t(y − x)], y − x〉

for all x, y ∈ B(x, ε), all p∗ ∈ ∂p f (x), and all t such that ‖p∗‖ ≤ εrt.

It is easily seen that if f is J-plr at x with some positive constants ε and r thenit is so for any constants 0 < ε′ ≤ ε and 0 < r′ ≤ r. If the lower semicontinuousfunction f is J-plr at x ∈ dom f with positive constants ε and r, one can derive that

(1.19) 〈J[J∗(p∗) − t(y − x)] − J[J∗(q∗) − t(x − y)], y − x〉 ≤ 0

for all x, y∈B(x, ε), for all p∗∈∂p f (x), q∗∈∂p f (y), and all t such that max{‖p∗‖, ‖q∗‖} ≤εrt. This is the analog of the hypomonotonicity of certain truncations of ∂p f , that isintrinsic to primal lower-nice functions in Hilbert spaces: see Poliquin [136], Levi,Poliquin and Thibault [111], Bernard, Thibault and Zagrodny [16] and the referencestherein. The hypomonotonicity is no more appropriate in our setting and therefore weintroduce the following closely related concept, that we call J-hypomonotonicity.

Definition 1.3.6. A set-valued mapping T : X ⇒ X∗ is said to be J-hypomonotone ofdegree t ≥ 0 if for any (xi, x∗i ) ∈ gph T := {(x, x∗) ∈ X × X∗ : x∗ ∈ T (x)}, i = 1, 2, onehas

〈J[J∗(x∗1) − t(x2 − x1)] − J[J∗(x∗2) − t(x1 − x2)], x2 − x1〉 ≤ 0.

For a set-valued mapping T : X ⇒ X∗, we denote by Dom T its domain, i.e.Dom T := {x ∈ X : T (x) , ∅}. We will also use the following concept of truncationof a set-valued mapping, as in Bernard, Thibault and Zagrodny [16].

Let T : X ⇒ X∗ be a set-valued mapping and let ε > 0 and t ≥ 0. Then itsε, t-truncation at a point x ∈ X is the set-valued mapping Tx,ε,t defined by

gph Tx,ε,t := {(x, x∗) ∈ gph T : ‖x − x‖ < ε, ‖x∗‖ ≤ t}.


Without ambiguity, Tx,ε,t will be denoted simply by Tt.

So, by (1.19) we see that if f is J-plr at x with ε and r, then (∂p f )x,ε,εrt is J-hypomonotone of degree t for any t ≥ 0. We are not far from sets since the nextproposition shows that the prox-regularity of a set C entails the J-plr property of itsindicator function ψC. The equivalence will be obtained later in Theorem 1.3.25, aswell as the equivalence with the J-hypomonotonicity of a certain truncation of thenormal cone NP

C .It will be convenient, for any σ ≥ 0, to denote below by NPσ

C the set-valuedmapping NP

C truncated with σB∗, i.e.,

(1.20) NPσC (x) := NP

C(x) ∩ σB∗ for all x ∈ X.

Proposition 1.3.7. If the closed set C ⊂ X is prox-regular at x ∈ C with ε and r, thenthe indicator function ψC of C is J-plr at x, and hence (1.19) yields, for any t ≥ 0,the J-hypomonotonicity of degree t on B(x, ε) of the set-valued mapping NPσ

C , whereσ := εrt.

Proof. As C is prox-regular at x for p∗ = 0 with ε > 0 and r > 0, from Proposi-tion 1.3.4 we have

ψC(x′) ≥ ψC(x) + 〈J[J∗(p∗) − r−1(x′ − x)], x′ − x〉,

whenever x′ ∈ B(x, ε), x ∈ C∩B(x, ε), and ‖p∗‖ ≤ ε with p∗ ∈ NPC(x). We have already

noticed that p∗ ∈ NPC(x) if and only if p∗ ∈ ∂pψC(x). If p∗ ∈ NP

C(x) and ‖p∗‖ ≤ εrt,then r−1t−1 p∗ ∈ NP

C(x) with ‖r−1t−1 p∗‖ ≤ ε, hence

ψC(x′) ≥ ψC(x) + 〈J[J∗(r−1t−1 p∗) − r−1(x′ − x)], x′ − x〉,

ψC(x′) ≥ ψC(x) + 〈J[J∗(p∗) − t(x′ − x)], x′ − x〉,

which means that the function ψC is J-plr at x. The proof is complete. �

We will need another result concerning J-hypomonotone set-valued mappings. Itwill be one of the key steps of our development of the proof of Theorem 1.3.13.First recall that a set-valued mapping T : X ⇒ X∗ is bounded when its range T (X) :=∪x∈XT (x) is a bounded set in X∗.

Lemma 1.3.8. Let T : X ⇒ X∗ be a bounded set-valued mapping which is J-hypomonotone of degree r. Then for any r > 2r we have that (I + r−1J∗ ◦ T )−1 is asingle-valued mapping on its domain which is 1

q -Holder continuous on its intersectionwith any bounded subset.


Proof. Let us denote by µ any upper bound of {‖z‖ : z ∈ T (x), x ∈ Dom T }. For anyr > 2r and ρ > 0, take xi ∈ Dom (I + r−1J∗ ◦ T )−1 with ‖xi‖ ≤ ρ, i = 1, 2. Choose anyyi ∈ (I + r−1J∗ ◦ T )−1(xi), i.e., J[r(xi − yi)] ∈ T (yi), i = 1, 2. Hence ‖xi − yi‖ ≤ µ/r. Byassumption, for any t ≥ r,

0 ≥ 〈J{J∗(J[r(x1 − y1)]) − t(y2 − y1)} − J{J∗(J[r(x2 − y2)]) − t(y1 − y2)}, y2 − y1〉

0 ≥ 〈J(rx1 − ty2 + (t − r)y1) − J(rx2 − ty1 + (t − r)y2), y2 − y1〉.

Now for any λ ∈ ]0, 1[ such that λr2 > r, replacing t in the above inequality by rtλ

where tλ := λ/2 we obtain

0 ≥ 〈J(x1 − tλy2 − (1 − tλ)y1) − J(x2 − tλy1 − (1 − tλ)y2), x1 − x2 + (1 − 2tλ)(y2 − y1)〉

+〈J(x1 − tλy2 − (1 − tλ)y1) − J(x2 − tλy1 − (1 − tλ)y2), x2 − x1〉 := (I) + (II).

Note also that

‖x1 − tλy2 − (1 − tλ)y1‖ ≤ (1 − tλ)‖x1 − y1‖ + tλ‖x1 − y2‖

≤ (1 − tλ)‖x1 − y1‖ + tλ(‖x1 − x2‖ + ‖x2 − y2‖)≤ (1 − tλ)

µ

r + tλ(2ρ +

µ

r

)≤ γ,

where γ := ρ +µ

r , and similarly ‖x2 − tλy1 − (1 − tλ)y2‖ ≤ γ. A first estimation of (I) isobtained by using (1.8), that is,

(I) ≥ Kγ‖x1 − x2 + (1 − 2tλ)(y2 − y1)‖q.

To proceed further in the estimation, we need to consider two cases.The first case is when (1 − 2tλ)‖y1 − y2‖ > ‖x1 − x2‖. In that case, we need

to estimate below ‖a − b‖q when ‖a‖ > ‖b‖. Since ‖a − b‖ ≥ ‖a‖ − ‖b‖ > 0, wederive ‖a − b‖q ≥ [‖a‖ − ‖b‖]q = ‖a‖q

[1 − ‖b‖

‖a‖

]q. This leads us to consider the real-

valued function g(s) = [1 − s]q + qs on the interval s ∈ [0, 1[ . As the derivativeg′(s) = −q[1−s]q−1+q is non-negative on this interval, the function g is non-decreasingon [0, 1[ and then g(s) ≥ g(0) = 1 for all s ∈ [0, 1[ . Finally, [1 − s]q ≥ 1 − qs fors ∈ [0, 1[ . We conclude that

‖a − b‖q ≥ ‖a‖q[1 −‖b‖‖a‖

]q

≥ ‖a‖q[1 − q

‖b‖‖a‖

]= ‖a‖q − q‖a‖q−1‖b‖.

Using the latter we continue to estimate (I) by

(I) ≥ Kγ

[(1 − 2tλ)q‖y1 − y2‖

q − q(1 − 2tλ)q−1‖y1 − y2‖q−1‖x1 − x2‖

]≥ γ1‖y1 − y2‖

q − γ2‖x1 − x2‖,


where γ1, γ2 are some nonnegative constants, depending on λ. On the other hand,

(II) ≥ −‖J(x1 − tλy2 − (1 − tλ)y1) − J(x2 − tλy1 − (1 − tλ)y2)‖.‖x2 − x1‖

≥ −K′γ‖x1 − x2 + (1 − 2tλ)(y2 − y1)‖s−1.‖x2 − x1‖

≥ −γ3‖x2 − x1‖,

where we used (1.9) for the second estimation, and γ3 is some nonnegative constantdepending on λ. Finally, 0 ≥ (I) + (II) ≥ γ1‖y1 − y2‖

q − (γ2 + γ3)‖x1 − x2‖ and hencefor some constant γ′ > 0 that depends on λ,

‖y1 − y2‖ ≤ γ′‖x1 − x2‖

1/q.

The second case is when (1 − 2tλ)‖y1 − y2‖ ≤ ‖x1 − x2‖. Observing that ‖x1 − x2‖ ≤

(2ρ)1− 1q ‖x1− x2‖

1q , we see that in both cases we have that ‖y1− y2‖ ≤ γ

′′‖x1− x2‖1/q for

some constant γ′′ > 0, so(I + r−1J∗ ◦ T

)−1is a single-valued mapping on its domain

and it is 1q -Holder continuous on the intersection of its domain with the set ρB. �

1.3.2 Local Moreau envelopes

Several properties of d2C and PC will be derived from corresponding ones (with

their own interest) concerning the so-called local Moreau envelope of a function.Here we will give the definition and properties of local Moreau envelopes. Let f :X → R∪ {+∞} be a lower semicontinuous function and W ⊂ X be a nonempty closedsubset where f is bounded from below and finite at some point. The local Moreauenvelope of index λ > 0 of f (relative to W), is defined as

(1.21) eλ,W f (x) := infy∈W

{f (y) +

12λ‖x − y‖2

}.

We fix W with the above property and we will write, when there is no risk ofconfusion, eλ f instead of eλ,W f . Note that the infimum in (1.21) may be seen as takenover all X for the function f given by f (x) = f (x) if x ∈ W and f (x) = +∞ otherwise.It is easy to see that the functions eλ f are everywhere defined and Lipschitz onbounded subsets. As usual we will consider the set

Pλ f (x) :={

y ∈ W : eλ f (x) = f (y) +1

2λ‖x − y‖2

}.

Whenever there exists some pλ(x) ∈ Pλ f (x) one has by Correa, Jofre and Thibault [51]

(1.22) ∂Feλ f (x) ⊂ {λ−1J(x − pλ(x))} ∩ ∂F f (pλ(x)),


hence Pλ f (x) is either empty or a singleton thanks to the one-to-one property of themapping J. We know by Theorem 11 of Borwein and Giles [25] that the infimumis attained whenever x is a point of Frechet subdifferentiability of eλ f . Denoting byGλ the subset of X where eλ f is Frechet subdifferentiable, we obtain for any x ∈ Gλ

that Pλ f (x) = {pλ(x)} and ∂Feλ f (x) = {λ−1J(x − pλ(x))}. Note that Gλ is dense in Xaccording to the result in Mordukhovich and Shao [126] and Preiss [139] concerningthe density of subdifferentiability points.

When Pλ f (x) = {pλ(x)} is a singleton, we make no difference between will writesometimes Pλ f (x) and pλ(x).

In the proof of the next lemma we follow an idea due to Borwein and Gilesfrom [25]. The lemma will be used in the proof of Proposition 1.3.19.

Lemma 1.3.9. Let f : X → R ∪ {+∞} be a lower semicontinuous function boundedfrom below on W and with W ∩ dom f , ∅. Then the following assertions areequivalent:(a) ∂Feλ f (x) , ∅;(b) eλ f is Frechet differentiable at x.

Further, in the case of (a) or (b), ∇Feλ f (x) = λ−1J(x − pλ(x)).

Proof. Obviously (b) implies (a). Now suppose that (a) holds. As we have seen above,Pλ f (x) is single-valued with a unique element pλ(x) and ∂Feλ f (x) = {λ−1J(x− pλ(x))}.Then, for any ε > 0 there exists some δ > 0 such that for any t ∈]0, δ[ and for anyy ∈ B

〈λ−1J(x − pλ(x)), ty〉 ≤ eλ f (x + ty) − eλ f (x) + εt,

hence,

(1.23) t−1[eλ f (x + ty) − eλ f (x)] − 〈λ−1J(x − pλ(x)), y〉 ≥ −ε.

At the same time, taking δ smaller if necessary and using the definition of eλ f andthe fact that the function 1

2‖ · ‖2 is Frechet differentiable with

J(x − pλ(x)) = ∇F

(12‖ · ‖2

)(x − pλ(x)),

we have for any y ∈ B

eλ f (x + ty) − eλ f (x) ≤ f (pλ(x)) +1

2λ‖x + ty − pλ(x)‖2 − f (pλ(x)) −

12λ‖x − pλ(x)‖2

≤ 〈λ−1J(x − pλ(x)), ty〉 + εt,

i.e.,

(1.24) t−1[eλ f (x + ty) − eλ f (x)] − 〈λ−1J(x − pλ(x)), y〉 ≤ ε.

Combining (1.23) and (1.24) we obtain the Frechet differentiability of eλ f at x aswell as the equality ∇Feλ f (x) = λ−1J(x − pλ(x)). �


The differentiability of the Moreau envelopes is also related to their regularity.This connection given by the equivalence between assertions (a) and (b) of the nextlemma will be needed in Proposition 1.3.19. The lemma also establishes, in view ofTheorem 1.3.13, the differentiability of eλ f under the single valuedness and continuityof Pλ f . Before giving its statement, let us recall that a function f is Frechet regularat x provided ∂F f (x) = ∂C f (x), where ∂C stands for the Clarke subdifferential.

Lemma 1.3.10. Under the assumptions of Lemma 1.3.9, for any open subset U of X,the equivalences (a)⇔(b) and (c)⇔(d) hold for the following properties:(a) eλ f is Frechet regular on U;(b) eλ f is Frechet differentiable on U and its Frechet derivative ∇Feλ f : U → X∗ isnorm-to-weak∗ continuous;(c) eλ f is continuously Frechet differentiable on U ( and hence (a) and (b) hold);(d) Pλ f is a single-valued norm-to-norm continuous mapping on U.

In any one of these cases, eλ f is Frechet differentiable on U with ∇Feλ f (x) =

λ−1J(x − Pλ f (x)).

Proof. (a)⇒(b): If ∂Feλ f = ∂Ceλ f on U, then we have that ∂Feλ f (x) , ∅ for anyx ∈ U and, hence, eλ f is Frechet differentiable on U from Lemma 1.3.9. Moreover,∂Ceλ f (x) = {∇Feλ f (x)} for any x ∈ U, and by the norm-to-weak∗ upper semicontinuityof ∂Ceλ f we have that ∇Feλ f is norm-to-weak∗ continuous.(b)⇒(a): Conversely, if eλ f is Frechet differentiable and norm-to-weak∗ continuouson U, we have that ∂Feλ f (x) = {∇Feλ f (x)} = ∂Leλ f (x) for any x ∈ U, where for alocally Lipschitz continuous function g : X → R the limiting subdifferential ∂Lg isdefined as the weak∗ sequential outer limit

(1.25) w∗ − lim supy→x

∂Fg(y) := {w∗ − lim x∗n : x∗n ∈ ∂Fg(xn), xn → x}.

By Mordukhovich and Shao [126], we know that ∂Cg(x) = co∗∂Lg(x), where co∗ de-notes the weak∗ closed convex hull in X∗. Thus, we obtain that ∂Feλ f (x) = ∂Ceλ f (x)for x ∈ U, which is the assertion (a).(c)⇒(d): The continuous differentiability of eλ f on U implies via Lemma 1.3.9 thesingle valuedness and norm-to-norm continuity of Pλ f , i.e., the implication holds.(d)⇒(c): Assume now that Pλ f is a single-valued norm-to-norm continuous map-ping on U. This continuity property along with (1.22) and (1.25) gives ∂Ceλ f (x) =

{λ−1J(x−Pλ f (x))} which entails that eλ f is Gateaux differentiable on U with ∇Geλ f (x) =

λ−1J(x−Pλ f (x)). The norm-to-norm continuity of Pλ f once again yields the existenceof ∇Feλ f as well as its norm-to-norm continuity on U. The proof of the lemma isthen complete. �

In the remainder of this subsection, we fix a point x ∈ dom f and ρ > 0 such thatf is bounded from below over B[x, 4ρ] and hence we also fix W = B[x, 4ρ]. Note


that according to the lower semicontinuity of f one always has some ρ > 0 with thedesired property. So it is natural to write eλ,ρ,x f (x) in place of eλ,W f (x) and when xand ρ with the above mentioned properties are fixed, it will be convenient to keep asabove for index only λ since this will not cause any confusion.

By the useful localization lemma (see Thibault and Zagrodny [160, Lemma 4.2]),there exists some λ0 > 0 such that for all λ ∈]0, λ0]

(1.26) Pλ f (x) ⊂ B(x, 3ρ) for all x ∈ U := B(x, ρ).

So, for any x ∈ U ∩ Gλ the unique element pλ(x) of Pλ f (x) belongs to B(x, 3ρ) andthen by (1.22)

(1.27) ∇Feλ f (x) = λ−1J(x − pλ(x)) ∈ ∂F f (pλ(x)) ∀x ∈ U ∩Gλ

and moreover

(1.28) ‖x − pλ(x)‖ ≤ ‖x − x‖ + ‖x − pλ(x)‖ < ρ + 3ρ = 4ρ.

In fact, we can make (1.27) more precise by proving in the following lemma thatthe stronger inclusion ∇Feλ f (x) ∈ ∂p f (pλ(x)) holds for x ∈ U ∩Gλ.

Lemma 1.3.11. For any λ ∈ ]0, λ0], x ∈ U ∩ Dom Pλ, and pλ(x) ∈ Pλ f (x), we havethat λ−1J(x − pλ(x)) ∈ ∂p f (pλ(x)). In other words, for any x ∈ U and any λ ∈ ]0, λ0],

Pλ f (x) ⊂ (I + λJ∗ ◦ ∂p f )−1(x).

Proof. Fix any 0 < λ ≤ λ0, x ∈ U ∩ Dom Pλ, and pλ(x) ∈ Pλ f (x). Then,

f (pλ(x)) + (2λ)−1‖x − pλ(x)‖2 ≤ f (y) + (2λ)−1‖x − y‖2, ∀y ∈ W,

that is,

(2λ)−1‖x − pλ(x)‖2 − (2λ)−1‖x − y‖2 ≤ f (y) − f (pλ(x)), ∀y ∈ W.

Since pλ(x) ∈ B[x, 3ρ] the last inequality holds true in particular for all y ∈B[pλ(x), ρ]. Let us set p := λ−1(x − pλ(x)). From the last inequality we have

2−1λ‖p‖2 − (2λ)−1‖x − y‖2 ≤ f (y) − f (pλ(x)), ∀y ∈ B[pλ(x), ρ],

which entails

λ2

2‖p‖2 −

12‖λp + pλ(x) − y‖2 ≤ λ[ f (y) − f (pλ(x))], ∀y ∈ B[pλ(x), ρ],

or,

λ2‖p‖2 − ‖λp + pλ(x) − y‖2 ≤ 2λ[β − f (pλ(x))], ∀(y, β) ∈ epi f with y ∈ B[pλ(x), ρ].


Adding λ2 to both sides yields

λ2‖p‖2−‖λp+pλ(x)−y‖2+λ2 ≤ 2λ[β− f (pλ(x))]+λ2, ∀(y, β) ∈ epi f with y ∈ B[pλ(x), ρ],

and using the inequality 2λ[β − f (pλ(x))] + λ2 ≤ [β − f (pλ(x)) + λ]2 we obtain that

λ2‖p‖2+λ2 ≤ ‖λp+pλ(x)−y‖2+[β− f (pλ(x))+λ]2, ∀(y, β) ∈ epi f with y ∈ B[pλ(x), ρ].

So, we obtain that for all (y, β) ∈ epi f with y ∈ B[pλ(x), ρ]

|||λ(p,−1)|||≤|||(pλ(x), f (pλ(x)))+λ(p,−1)−(y, β)|||.

By (1.14) this inequality entails that (p,−1) ∈ PNepi f (pλ(x), f (pλ(x))), which gives(see Subsection 1.3.1) that J(p) ∈ ∂p f (pλ(x)). This means that

λ−1J(x − pλ(x)) ∈ ∂p f (pλ(x)),

which entails the inclusion of the lemma. �

Remark 1.3.12. (a) In the case when the lower semicontinuous function f is theindicator function ψC of a non-empty closed set C, the above conclusions hold forW = X and any λ > 0 or for ρ = +∞, any x ∈ C, and any λ > 0. Further with ρ = +∞

one has eλ f (x) = 12λd2

C(x) and Pλ f (x) = PC(x) for all x ∈ X.(b) Still with f = ψC, for any x ∈ C, any ρ ∈ [0,+∞], and any λ > 0 one haseλ f (x) = 1

2λd2C∩W(x) and Pλ f (x) = PC∩W(x) for all x ∈ X.

But for any x ∈ B[x, 2ρ] and any y ∈ C \W,

‖x − y‖ ≥ ‖y − x‖ − ‖x − x‖ > 4ρ − 2ρ = 2ρ ≥ ‖x − x‖ ≥ dC∩W(x).

Hence, for x ∈ B[x, 2ρ], dC∩W(x) = dC(x) and PC∩W(x) = PC(x). Therefore, eλ f (x) =1

2λd2C(x) and Pλ f (x) = PC(x) for any x ∈ U := B(x, ρ).

Recall that a function g is of class C1,α on an open set U ⊂ X when it isdifferentiable on U and the derivative ∇g is locally α-Holder continuous on U.

Theorem 1.3.13. Let f : X → R ∪ {+∞} be a lower semicontinuous function whichis J-plr at x ∈ dom f with positive real numbers ε and r, such that ε < 1 < 1

r . Let

ρ ∈]0, rε

16

]be fixed in such a way that f is bounded from below on B[x, 4ρ]. Then

there exists λ0 > 0 such that for any λ ∈]0, λ0] the map x 7→ Pλ f (x) is single-valuedon U := B(x, ρ) with, for some constant γ ≥ 0,

(1.29) ‖pλ(x) − pλ(x′)‖ ≤ γ‖x − x′‖1q , ∀x, x′ ∈ U,

where pλ is given by Pλ f (y) = {pλ(y)} for any y ∈ U. Moreover, for each λ ∈]0, λ0] thefunction eλ f is of class C1,α on U with α := q−1(s−1) and ∇Feλ f (x) = λ−1J(x− pλ(x))for all x ∈ U.


Proof. Let λ0 be given by the analysis of (1.26) above for ρ fixed as in the state-ment of the theorem. We will work with arbitrary fixed λ ∈ ]0, λ0]. Put c := rε andTct := (∂p f )x,ε,ct for any t ≥ 0. The proof is divided in three steps.

Step 1. Let us prove that Pλ f is 1q -Holder continuous on U ∩ Dom Pλ f .

We have from Definition 1.3.6 and from (1.19) that the set-valued mapping Tct isJ-hypomonotone of degree t for any t ≥ 0. Hence, for tλ := c/(4λ), the set-valuedmapping Ttλ is J-hypomonotone of degree r := 1/(4λ). As λ−1 > 2r, Lemma 1.3.8entails that (I +λJ∗◦Ttλ)

−1 is a single-valued mapping on its domain and this mappingis 1

q -Holder continuous on the intersection of its domain with any bounded subset ofX. From Lemma 1.3.11, we have that Pλ f (x) ⊂ (I + λ∂p f )−1(x) for any x ∈ U. Weclaim that we even have

(1.30) Pλ f (x) ⊂ (I + λJ∗ ◦ Ttλ)−1(x) for any x ∈ U.

Indeed, fixing any x ∈ U ∩ Dom Pλ f and pλ(x) ∈ Pλ f (x), we know that pλ(x) ∈B[x; 3ρ]. So ‖pλ(x) − x‖ < ε, and

‖λ−1J(x− pλ(x))‖ ≤ λ−1‖x− pλ(x)‖ ≤ λ−1(‖x− x‖+‖x− pλ(x)‖) ≤ 4ρλ−1 ≤ (cλ−1)/4 = tλ.

Hence, as also λ−1J(x− pλ(x)) ∈ ∂p f (pλ(x)), we have that λ−1J(x− pλ(x)) ∈ Ttλ(pλ(x)),which proves the claim. Therefore we obtain that Pλ f is a single-valued 1

q -Holdercontinuous mapping on U ∩ Dom Pλ f , that is, there exists some constant γ ≥ 0 suchthat for all x, x′ ∈ U ∩ Dom Pλ f

(1.31) ‖pλ(x) − pλ(x′)‖ ≤ γ‖x − x′‖1q .

Step 2. Let us prove that U ⊂ Dom Pλ f .The proof now is similar to that of Bernard and Thibault [15, 14]. Take any x ∈ Uand fix some integer k ≥ 1 with B(x, 1/k) ⊂ U. According to the density of theFrechet subdifferentiability points of eλ f , for any integer n ≥ k there exists somepoint xn ∈ Gλ ∩ B(x, 1/n). By (1.31), for any integers n,m ≥ k,

‖pλ(xn) − pλ(xm)‖ ≤ γ‖xn − xm‖1q .

Hence, (pλ(xn))n is a Cauchy sequence. Denote by zλ its limit. By the definition ofpλ(xn) we have

f (pλ(xn)) +1

2λ‖xn − pλ(xn)‖2 ≤ f (y) +

12λ‖xn − y‖2, ∀y ∈ W.

Since f is lower semicontinuous, the latter implies

f (zλ) +1

2λ‖x − zλ‖2 ≤ f (y) +

12λ‖x − y‖2, ∀y ∈ W.


This means that zλ ∈ Pλ f (x), which yields U ∩ Dom Pλ f = U. Hence (1.31) holdsfor all x, x′ ∈ U and, through Step 1, Pλ f is a single-valued 1

q -Holder continuousmapping on U. Then by Lemma 1.3.10, the envelope eλ f is continuously Frechetdifferentiable on U with ∇Feλ f (x) = λ−1J(x′ − pλ(x)) for any x ∈ U.

Step 3. We will prove that eλ f is of class C1,α on U with α = q−1(s − 1). Let us takeany x, x′ ∈ U and, for r := 4ρ, use (1.9) to estimate

‖∇Feλ f (x) − ∇Feλ f (x′)‖ = λ−1‖J(x − pλ(x)) − J(x′ − pλ(x′))‖ ≤

λ−1K′r‖x − pλ(x) − x′ + pλ(x′)‖s−1 ≤ λ−1K′r[‖x − x′‖ + ‖pλ(x) − pλ(x′)‖]s−1 ≤

λ−1K′r[‖x − x′‖ + γ‖x − x′‖1q ]s−1 ≤ λ−1K′r(1 + γ)s−1‖x − x′‖

s−1q ,

where the third inequality is due to (1.29) and the last one to the fact that ‖x− x′‖ < 1.The proof of the theorem is then complete. �

We now state in the next proposition the relation obtained between the proximalmappings and some truncations of the subdifferential of a J-plr function.

Proposition 1.3.14. Under the assumptions of Theorem 1.3.13, one has for all λ ∈]0, λ0] and x ∈ U

Pλ f (x) = (I + λ−1J∗ ◦ Ttλ)−1(x),

where tλ := εr/(4λ) and Ttλ := (∂p f )x,ε,tλ .

Proof. The inclusion of the first member in the second one is (1.30) of Theo-rem 1.3.13, and the reverse one follows from the inclusion U ⊂ Dom Pλ f establishedin Step 2 of the same proof since (I + λ−1J∗ ◦ Ttλ)

−1 is at most single-valued as wesaw in Step 1 above. �

Corollary 1.3.15. Let C ⊂ X be a non-empty closed set such that its indicator functionψC is J-plr at x ∈ C with positive real numbers ε and r satisfying ε < 1 < 1

r . Then forρ = εr

16 the mapping x 7→ PC(x) is single-valued on U = B(x, ρ) and for some constantγ ≥ 0

(1.32) ‖PC(x) − PC(x′)‖ ≤ γ‖x − x′‖1q , ∀x, x′ ∈ U.

Further, the function d2C is of class C1,α on U with α = q−1(s − 1) and ∇Fd2

C(x) =

2J(x − PC(x)) for all x ∈ U.

Corollary 1.3.16. Under the assumptions of Corollary 1.3.15, we have that

PC(x) = (I + J∗ ◦ NPσC )−1(x) ∀x ∈ U,

where σ := εr/4 and the set-valued mapping NPσC is defined by (1.20).


Proof. Taking ρ = εr16 as in the statement of Corollary 1.3.15, since r < 1 it is easily

checked for T := ∂pψC that for all x ∈ U = B(x, ρ)(I + J∗ ◦ Tx,ε, εr

4

)−1(x) =

(I + J∗ ◦ NP εr/4

C

)−1(x).

Soq it suffices to apply Proposition 1.3.14 with λ = 1 to f = ψC, keeping in mindRemark 1.3.12(b). �

1.3.3 Characterizations of prox-regular sets

In this subsection we will give different characterizations of prox-regularity of aset.

Lemma 1.3.17. Let C ⊂ X be a closed subset. Then the single valuedness andnorm-to-weak continuity of the projection mapping PC over an open set U imply itsnorm-to-norm continuity on U.

Proof. Let un‖.‖−→n→∞

u and PC(un) w−→n→∞

PC(u). From the Lipschitz continuity of the

distance function, ‖un − PC(un)‖ = dC(un) −→n→∞

dC(u) = ‖u − PC(u)‖. By the Kadec

property of the norm, PC(un) ‖.‖−→n→∞

PC(u). �

The following proposition establishes that the continuity of the metric projectionmapping to a closed set C is equivalent to the continuous differentiability of thedistance function dC, as shown in the Hilbert setting in Poliquin, Rockafellar andThibault [138], where it is proved that those properties characterize the prox-regularityof a set in a Hilbert space. Its proof follows directly from Lemma 1.3.10 with f = ψC.

Proposition 1.3.18. Let C ⊂ X be a closed set and U ⊂ X be an open set. Then thefollowing are equivalent:(a) PC is single-valued and norm-to-norm continuous on U;(b) d2

C is of class C1 on U.

In fact these properties are equivalent to the only Frechet subdifferentiability ofthe distance function as we can see from the following proposition.

Proposition 1.3.19. For any closed set C ⊂ X and any open set U of X the followingare equivalent:(a) dC is continuously differentiable on U \C;(b) ∂FdC(x) is non-empty for all x ∈ U;(c) ∂Fd2

C(x) is non-empty at all points x in U;


(d) dC is Frechet differentiable on U \C;(e) dC is Frechet regular on U \C;(f) dC is Gateaux differentiable on U \C with ‖∇GdC(x)‖ = 1 for all x ∈ U \C.

Proof. (a) ⇒ (b) is obvious since one always has 0 ∈ ∂FdC(u) for any u ∈ C.(b) ⇒ (c) follows from the fact that for any x∗ ∈ ∂FdC(x), one has 2dC(x)x∗ ∈ ∂Fd2

C(x)according to Lemma 3.9 in Poliquin, Rockafellar and Thibault [138].(c) ⇒ (d): By Lemma 1.3.9 and Remark 1.3.12(a) we have that d2

C is Frechet differ-entiable on U, hence so is dC on U \C.(d) ⇒ (a): It is clear that (d) entails (b) and hence (c). From Lemma 1.3.9 and Re-mark 1.3.12(a) we get the Frechet differentiability of d2

C on U. The latter implies thatPC is single-valued on U and that

(1.33) ∇Fd2C(x) = 2J(x − PC(x)) for any x ∈ U.

So, it remains to prove the norm-to-norm continuity of PC over U and we will obtain

that of ∇Fd2C. Take any x0 ∈ U, and U 3 xn

‖.‖−→n→∞

x0. Then we also have that

‖xn − PC(xn)‖ = dC(xn) −→n→∞

dC(x0) = ‖x0 − PC(x0)‖,

which entails that the sequence (PC(xn))n is bounded. Taking if necessary a subse-

quence, we may suppose that PC(xn) w−→n→∞

z for some z ∈ X. As ‖x0−PC(xn)‖ −→n→∞‖x0−

PC(x0)‖, having in mind the Kadec property of the norm, it suffices to prove that

(1.34) ‖x0 − z‖ = ‖x0 − PC(x0)‖

to get that PC(xn) ‖.‖−→n→∞

z, and hence that z ∈ C. Then, using (1.34) and the fact

that PC(x0) is single-valued, we will have that PC(x0) = {z}, so PC is norm-to-normcontinuous at x0. To this end, following an idea of Borwein and Giles from [25],let us set t2

n := ‖x0 − PC(xn)‖2 − d2C(x0). If tn = 0 then PC(xn) = PC(x0) due to the

single-valuedness of PC, and there would be nothing to prove if this equality holdsfor infinitely many n, so we may suppose that tn > 0 for any integer n. Fix any ε > 0.From the Frechet differentiability of d2

C at x0 it follows that

〈∇Fd2C(x0), PC(xn) − x0〉 ≤

d2C(x0 + tn(PC(xn) − x0)) − d2

C(x0)tn

+ε

4

≤‖x0 + tn(PC(xn) − x0) − PC(xn)‖2 − d2

C(x0)tn

+ε

4

≤(1 − tn)2‖x0 − PC(xn)‖2 − d2

C(x0)tn

+ε

4


and hence for n large enough

〈∇Fd2C(x0), PC(xn) − x0〉 ≤

‖x0 − PC(xn)‖2 − d2C(x0)

tn− 2‖x0 − PC(xn)‖2 +

ε

2

= tn − 2‖x0 − PC(xn)‖2 +ε

2≤ −2d2

C(x0) + ε.

Passing to the limit, we obtain 〈∇Fd2C(x0), x0 − z〉 ≥ 2d2

C(x0), or,

〈2J(x0 − PC(x0)), x0 − z〉 ≥ 2‖x0 − PC(x0)‖2,

which entails that ‖x0−z‖ ≥ ‖x0−PC(x0)‖. Since ‖xn−PC(xn)‖ ≤ ‖xn−PC(x0)‖, one hasthat lim inf

n→∞‖xn − PC(xn)‖ ≤ lim

n→∞‖xn − PC(x0)‖, and hence, by the weak lower semicon-

tinuity of the norm, one gets ‖x0− z‖ ≤ ‖x0−PC(x0)‖. Finally, ‖x0− z‖ = ‖x0−PC(x0)‖,that is, (1.34), and hence the implication (d) ⇒ (a) holds.The implications (e) ⇒ (d) and (a) ⇒ (e) follow from Lemma 1.3.10 and the equiva-lence (d) ⇔ (f) is a direct consequence of Theorem 2.4 of Fitzpatrick [75]. The proofof the proposition is then complete. �

In addition to Proposition 1.3.19 we state the following theorem providing a weakderivability condition on dC under which PC is continuous, supposing it is nonempty-valued. It is a direct consequence of Theorem 2.4 in Fitzpatrick [75] as observedin Corollary 2 of Borwein, Fitzpatrick and Giles [24]. Note that in those works, theframework is beyond the uniform convexity. Further, Fitzpatrick’s important conditionof Frechet differentiability concerns general functions. The result of the theorem willbe used in the proof of Theorem 1.3.25.

Recall that, for v ∈ X, a function f : X → R has a Gateaux directional derivativeat a point x in the full direction v provided that the limit limt→0 t−1[ f (x + tv) − f (x)]exists and is finite.

Theorem 1.3.20. Let C ⊂ X be a closed set and x ∈ X \ C be such that PC(x) , ∅.If dC has a Gateaux directional derivative at x in the full direction x − p(x) for somep(x) ∈ PC(x), then dC is Frechet differentiable at x.

Another interest of this result will appear in the proof of Theorem 1.3.35.We now proceed to establish two lemmas. The first one is a key result proved in

Lemma 3.3 of Poliquin, Rockafellar and Thibault [138] in the Hilbert context. Theproof is valid in our setting, and we sketch the main parts below.

Lemma 1.3.21. Let C be a closed subset of X. Assume that dC is Frechet differ-entiable on a neighbourhood of a point u < C. Then there exists δ > 0 such thatwhenever u ∈ B(u, δ) and PC(u) = x, there exists some t > 0 such that the pointut := u + t(u − x) likewise has PC(ut) = x.


Proof. By Proposition 1.3.19 and Proposition 1.3.18, there exists ε > 0 such thatPC is single-valued and norm-to-norm continuous on B(u, 2ε), with dC continuouslyFrechet differentiable on this ball as well. For each u ∈ B(u, ε) and each t > 0 putut := u + t(u−PC(u)). Following the proof of Lemma 3.3 in Poliquin, Rockafellar andThibault [138], we find out some positive numbers δ < ε and s < 1 such that for allu ∈ B(u, δ) one has dC(u) ≥ δ, sdC(u) < δ and dC(us) > dC(u). Fix now u ∈ B(u, δ) andconsider the closed set D := {w ∈ X : dC(w) ≥ dC(us)}. As u < D, according to Lau’s

theorem (see Lau [108]) there is a sequence D = yn‖.‖−→n→∞

u with PD(yn) , ∅. Choosing

wn ∈ PD(yn) we have dC(wn) = dC(us) (because wn is a boundary point of D). For alln large enough, wn ∈ B(u, 2δ) since

(1.35) ‖yn − wn‖ = dD(yn) ≤ ‖yn − us‖ −→n→∞‖u − us‖ = s‖u − PC(u)‖ = sdC(u) < δ.

Consequently dC is Frechet differentiable at wn and by (1.33) we have

∇FdC(wn) = J(wn − PC(wn))/dC(wn) and ‖∇FdC(wn)‖ = 1.

Therefore, the half-space E := {v ∈ X : 〈−∇FdC(wn), v〉 ≤ 0} gives the Clarke tangentcone to D at wn (see Clarke [44]) and hence its negative polar cone −[0,∞[∇FdC(wn)is the Clarke normal cone to D at wn. The nonzero functional J(yn − wn) being aproximal normal functional to D at wn, it belongs to the Clarke normal cone to Dat wn. Consequently, there exists some λn > 0 such that J(yn − wn) = −λn∇

FdC(wn)which entails

yn − wn = −λn(wn − PC(wn))/dC(wn) and λn = ‖yn − wn‖.

For n large enough, we have by (1.35) that λn < δ and hence

λn < δ ≤ dC(u) < dC(us) = dC(wn).

It follows that for αn := λn/dC(wn) we have αn ∈]0, 1[ and yn = (1−αn)wn +αnPC(wn).Hence, PC(yn) = PC(wn) and

λn = ‖yn − wn‖ = dC(wn) − dC(yn) = dC(us) − dC(yn).

Putting tn :=αn

1 − αn=

dC(us) − dC(yn)dC(yn)

, we obtain wn = yn + tn(yn − PC(yn)). As (tn)n

converges to t := (dC(us) − dC(u))/dC(u) > 0, we have wn‖.‖−→n→∞

ut and ut ∈ B(u, 2δ) by

(1.35) and by the inclusion u ∈ B(u, δ). So, by continuity of PC over B(u, 2δ) we get

PC(wn) ‖.‖−→n→∞

PC(ut). But we also have PC(wn) = PC(yn) ‖.‖−→n→∞

PC(u). Finally, for this

number t we have PC(ut) = PC(u) and hence the proof is complete. �


The next lemma follows from the previous one.

Lemma 1.3.22. Let C ⊂ X be a closed subset of the space X and x ∈ C. If themapping PC is single-valued and norm-to-norm continuous in a neighbourhood U ofx, then there exists some ε > 0 such that for all x ∈ C ∩ B(x, ε) and all p ∈ PNC(x)with p , 0 the equality PC

(x + ε p

‖p‖

)= x holds.

Proof. Let PC be single-valued and norm-to-norm continuous in B(x, δ). Take ε < δ/2and consider any non-zero p ∈ PNC(x) with ‖x− x‖ < ε. By definition of the proximalnormal cone and by Lemma 1.3.1, there exists λ > 0 such that PC(x + λp) = x. Setλs := sup

{λ ≤ ε : PC

(x + λ p

‖p‖

)= x

}. By the continuity of PC on B(x, δ) we have that

PC

(x + λs

p‖p‖

)= x. Suppose that λs < ε. As x + λs

p‖p‖ belongs to the open set B(x, δ)

where dC is Frechet differentiable according to Proposition 1.3.18, by Lemma 1.3.21there exists η > 0 with λs + η ≤ ε such that PC

(x + (λs + η) p

‖p‖

)= x. This gives a

contradiction with the definition of λs. Hence λs = ε. �

Lemma 1.3.22 allows us to establish the following proposition which prepares thetheorem on characterizations of prox-regularity.

Proposition 1.3.23. Let C ⊂ X be a closed subset of the space X. The followingassertions are equivalent:(a) C is prox-regular at x ∈ C;

(b) there exists ε>0 such that the conditionx=PC(u), x , u0 < ‖u−x‖ < ε

}implies that x=PC(u′)

for u′:=x+ε u−x‖u−x‖ ;

(c) there exists ε > 0 such that p ∈ PNC(x) with x ∈ B(x, ε) and p , 0 imply that

PC

(x + ε

p‖p‖

)= x.

Proof. (a) ⇒ (b): If C is prox-regular at x, then PC is single valued and Holdercontinuous in a neighbourhood U of x according to Corollary 1.3.15. From thecontinuity of PC, for the positive number ε of Lemma 1.3.22, there exists ε′ ∈ ]0, ε[

such thatx = PC(u), x , u0 < ‖u − x‖ < ε′

}implies that ‖x−x‖ = ‖PC(u)−PC(x)‖ < ε. Lemma 1.3.22

and Lemma 1.3.1 ensure for p = u − x that PC

(x + ε′ u−x

‖u−x‖

)= x.

(b) ⇒ (c): Let us suppose that (b) holds with some ε > 0. If ‖x − x‖ < ε/2 andp ∈ PNC(x) with p , 0, then by definition of PNC(x) and by Lemma 1.3.1 thereexists some η ∈]0, ε/2[ such that x = PC(u), where u := x + η p

‖p‖ . We have

‖u − x‖ ≤ ‖u − x‖ + ‖x − x‖ < ε/2 + ε/2,

and from (b), PC

(x + ε u−x

‖u−x‖

)= x. Hence, one obtains (c) with ε/2.

(c) ⇒ (a): We suppose that (c) holds with some ε > 0. Let 0 , p∗ ∈ NPC(x) with


‖x − x‖ < ε and ‖p∗‖ < ε. There exists u < C such that⟨

p∗

‖p∗‖ , u − x⟩

= ‖u − x‖ = dC(u).

Thus, u − x ∈ PNC(x) and p∗

‖p∗‖ = J(

u−x‖u−x‖

). We have by (c) that PC

(x + ε u−x

‖u−x‖

)=

x, so PC

(x + ε

‖p∗‖ J∗(p∗)

)= x. Now, for all s ≤ 1 (since 1 ≤ ε

‖p∗‖ ), we have thatPC(x + sJ∗(p∗)) = x. By Definition 1.3.3 and by Proposition 1.3.4, the set C isprox-regular at x. �

Now, following Poliquin and Rockafellar [137] and Poliquin, Rockafellar andThibault [138] we give a subdifferential characterization of the prox-regularity of aset in terms of the truncated cone (see (1.20)) of proximal normal functionals.

Proposition 1.3.24. A set C ⊂ X is prox-regular at x ∈ C, if and only if, for someε, ρ > 0, the set-valued mapping NP ε

C : X ⇒ X∗ that assigns to each x ∈ X thetruncated cone of proximal normal functionals NP ε

C (x) is J-hypomonotone of degreeρ on B(x, ε).

Proof. By Proposition 1.3.7, if C is prox-regular at x then, for some ε > 0 andρ > 0, the truncated normal functional cone mapping NP ε

C is J-hypomonotone ofdegree ρ on B(x, ε). Conversely, suppose that NP ε

C is J-hypomonotone of degree ρ.Then the argument of Theorem 1.3.13 or Corollary 1.3.15 works as well (since itonly makes use of the J-hypomonotonicity of the truncation of ∂p f ), to get that PC

is single-valued and continuous on a neighbourhood of x. It just remains to invokeLemma 1.3.22 and Proposition 1.3.23 to conclude. �

Now we can state the theorem giving several characterizations of the prox-regularity of a set. Recall first that the (lower) Dini subdifferential of a locallyLipschitz continuous function f : X → R at a point x is defined by

∂− f (x) :={

x∗ ∈ X∗ : 〈x∗, h〉 ≤ lim inft↓0

t−1[ f (x + th) − f (x)], ∀h ∈ X}.

Theorem 1.3.25. Let C ⊂ X be a closed set. The following are equivalent:(a) C is prox-regular at x;(b) PC is single-valued and norm-to-norm 1

q -Holder continuous on some neighbour-hood U of the point x;(c) PC is single-valued and norm-to-weak continuous on some neighbourhood U of x;(d) there exists ε > 0 such that p ∈ PNC(x) with x ∈ B(x, ε) and p , 0 implies thatPC

(x + ε p

‖p‖

)= x;

(e) there exists ε>0 such that the conditionx=PC(u), x , u0 < ‖u−x‖ < ε

}implies that x=PC(u′)

for u′=x+ε u−x‖u−x‖ ;

(f) d2C is of class C1,α on some neighbourhood U of x with α = q−1(s − 1);

(g) dC is Frechet differentiable on U \C for some neighbourhood U of x;


(h) for some neighbourhood U of x, the function dC is Gateaux differentiable on U \Cwith ‖∇GdC(x)‖ = 1 for all x ∈ U \C;(i) dC is Frechet subdifferentiable on U for some neighbourhood U of x;(j) dC is Frechet regular on U \C for some neighbourhood U of x;(k) PC is nonempty-valued on U and dC is Dini subdifferentiable on U for someneighbourhood U of x;(l) the indicator function ψC is J-plr at x;(m) there exist ε, ρ > 0 such that the truncated normal functional cone mapping NP ε

Cis J-hypomonotone of degree ρ on B(x, ε).

If C is weakly closed, one has one more equivalent condition(n) PC is single-valued on some neighbourhood U of x.

Proof. First we will establish all the equivalences without specifying the Holdercharacter of the continuity in (b) and (f). The proof follows the scheme:

(m) ⇔ (a) ⇒ (l) ⇒ (f) ⇔ (j) ⇔ (h) ⇔ (i) ⇔ (g) ⇔ (k)

m

(c) ⇔ (b) ⇒ (d) ⇔ (e) ⇔ (a).

(m) ⇔ (a) is Proposition 1.3.24.(a) ⇒ (l) is established in Proposition1.3.7.(l) ⇒ (f) follows from Corollary 1.3.15.(f) ⇔ (j) ⇔ (h) ⇔ (i) ⇔ (g) is Proposition 1.3.19.(g) ⇒ (k): Assume that (g) holds. This obviously ensures the Dini subdifferentiabilityof dC on U \ C and since one always has 0 ∈ ∂−dC(x) for all ∈ C, we obtain that dC

is Dini subdifferentiable on U. The nonvacuity of PC on U follows from the aboveimplication (g) ⇒ (f) and from Proposition 1.3.18.(k) ⇒ (g): For any x ∈ U \C, there exists some x∗ in the Dini subdifferential ∂−dC(x)of dC at x. Choose p(x) ∈ PC(x). By the definition of Dini subdifferential, for anyε > 0, there is some δ > 0 such that, for any t ∈]0, δ[, one has

(1.36) 〈x∗, p(x) − x〉 ≤ t−1[dC(x + t(p(x) − x)) − dC(x)] + ε

and hence

〈x∗, x − p(x)〉 ≥ t−1[dC(x) − dC(x + t(p(x) − x))] − ε≥ t−1[‖x − p(x)‖ − ‖x + t(p(x) − x) − p(x)‖] − ε.

Then, for any ε > 0, using the equality ∇F(‖ · ‖2)(x − p(x)) = 2J(x − p(x)) and takingsome t > 0 small enough, we obtain

〈x∗, x − p(x)〉 ≥⟨

J(x − p(x))‖x − p(x)‖

, x − p(x)⟩− 2ε

= ‖x − p(x)‖ − 2ε,


the last inequality being due the fact that 〈J(y), y〉 = ‖y‖2. Therefore,

‖x − p(x)‖ ≤ 〈x∗, x − p(x)〉 ≤ lim inft↓0

t−1[dC(x + t(x − p(x)) − dC(x)] ≤

lim supt↓0

t−1[dC(x + t(x − p(x)) − dC(x)] ≤ ‖x − p(x)‖

(the second inequality coming from x∗ ∈ ∂−dC(x)), so

limt↓0

t−1[dC(x + t(x − p(x)) − dC(x)] = ‖x − p(x)‖.

Further observe (as in the proof of Corollary 2 in Borwein, Fitzpatrick and Giles [24])that for each t ∈ [−1, 0[ one has p(x) ∈ PC(x + t(x − p(x)) and hence

t−1[dC(x + t(x − p(x))) − dC(x)] = ‖x − p(x)‖.

So, limt→0

t−1[dC(x + t(x − p(x)) − dC(x)] = ‖x − p(x)‖, that is, dC has a Gateaux di-

rectional derivative in the full direction x − p(x). Consequently, (g) follows fromTheorem 1.3.20.(f) ⇔ (b) is Proposition 1.3.18.(b) ⇔ (c) is Lemma 1.3.17.(b) ⇒ (d) is Lemma 1.3.22.(d) ⇔ (e) ⇔ (a) is Proposition 1.3.23.

Under the additional assumption, to see that we have (n) ⇔ (c) we need to provethe implication (n) ⇒ (c).

Let us take any u ∈ U and un‖.‖−→n→∞

u. By weak compactness, taking if necessary a

subsequence, we may suppose that PC(un) w−→n→∞

v. From the weak lower semicontinuity

of the norm, we have ‖v − u‖ ≤ lim infn→∞ ‖un − PC(un)‖ and hence ‖v − u‖ ≤ dC(u).

Therefore, as C is weakly closed, v ∈ C and PC(u) = v. Thus, PC(un) w−→n→∞

PC(u),

which gives the norm-to-weak continuity of PC on U.

To conclude, we apply Corollary 1.3.15 to obtain (a) ⇔ (b) ⇔ (f) but now withthe Holder character of the continuity that holds on (possibly smaller) neighbourhoodof x. The proof is then complete. �


1.3.4 Characterizations of uniformly prox-regular sets

In this final subsection we proceed to the study of the global setting of positivelyreached or proximally smooth set C, corresponding (see Clarke, Stern and Wolen-ski [46]) to the continuous Frechet differentiability of the distance function dC overall an open tube of uniform thickness around the set C. We still have most of thecharacterizations for these sets given in Federer [73] for finite dimensional space andin Clarke, Stern and Wolenski [46] and Poliquin, Rockafellar and Thibault [138] inHilbert space. We also add the characterizations (c), (e), and (g).

Definition 1.3.26. Following our definition of prox-regular set (see Definition 1.3.3)and modifying slightly Definition 2.4 of Poliquin, Rockafellar and Thibault [138],we will say that the closed set C is uniformly r-prox-regular if whenever x ∈ C andp∗ ∈ NP

C(x) with ‖p∗‖ < 1, then x is the unique nearest point of C to x + rJ∗(p∗).

For the subset C of X, let us first recall the definitions ofthe r-enlargement of C

C(r) := {x ∈ X : dC(x) ≤ r},

the open r-tube around C

UC(r) := {x ∈ X : 0 < dC(x) < r},

and let us define the set of r-distance points to C

DC(r) := {x ∈ X : dC(x) = r}.

Theorem 1.3.27. Let C ⊂ X be a closed set and r > 0. The following are equivalent:(a) C is uniformly r-prox-regular;(b) dC is continuously differentiable on UC(r) \C;(c) dC is Frechet regular on UC(r) \C;(d) dC is Frechet differentiable on UC(r) \C;(e) dC is Gateaux differentiable on UC(r)\C with ‖∇GdC(x)‖ = 1 for all x ∈ UC(r)\C;(f) ∂FdC is nonempty-valued at all points in UC(r);(g) PC and ∂−dC are nonempty-valued at all points in UC(r);(h) d2

C is C1 on UC(r) with locally Holder continuous derivative mapping;(i) PC is single-valued and locally Holder continuous on UC(r);(j) PC is single-valued and norm-to-weak continuous on UC(r);(k) For any non-zero p ∈ PNC(x) with x ∈ C one has x ∈ PC

(x + r p

‖p‖

);

(l) If u ∈ UC(r) and x = PC(u), then x ∈ PC(u′) for u′ = x + r u−x‖u−x‖ .

If C is weakly closed, one has the additional equivalent condition(m) PC is single-valued on UC(r).


Proof. We will follow the scheme:

(a) ⇔ (k) ⇔ (l) ⇒ (i) ⇒ (h)

⇑ ⇓

(j) ⇔ (b) ⇔ (c) ⇔ (d) ⇔ (e) ⇔ (f).

m

(g)

(k) ⇔ (l) is obvious and here below both will be considered.(k) ⇒ (a) follows from Lemma 1.3.1.(a) ⇒ (k): Fix any nonzero p ∈ PNC(x) and take pn := p/

(‖p‖ + 1

n

). For any

x′ ∈ C, (a) entails ‖x′ − (x + rpn)‖ ≥ r‖pn‖ which yields after taking the limit∥∥∥∥x′ − (x + r p‖p‖ )

∥∥∥∥≥r. The latter means that x ∈ PC

(x + r p

‖p‖

), which is (k).

(b) ⇔ (c) ⇔ (d) ⇔ (e) ⇔ (f) follows from Proposition 1.3.19.(d) ⇔ (g) results from Theorem 1.3.20 like in the proof of the similar equivalence inTheorem 1.3.25.(b) ⇔ (j) is a consequence of Proposition 1.3.18 and Lemma 1.3.17.(b) ⇒ (l): Let u ∈ UC(r) and x = PC(u). Since (b) holds, according to Lemma 1.3.21there exists some t0 > 0 such that PC(ut) = x for all ut := u + t(u − x)/‖u − x‖ with0 < t < t0. As in Poliquin, Rockafellar and Thibault [138] one may consider thenumber λ0 given by the supremum over all t ∈ [0, r − dC(u)] such that x ∈ PC(ut).Using the equivalence (note that x ∈ C and ‖x − ut‖ = dC(u) + t)

x ∈ PC(ut)⇔ ∀x′ ∈ C, ‖x′ − ut‖ ≥ dC(u) + t,

it is easily seen that the supremum λ0 is attained. We now claim that λ0 = r − dC(u).Assume the contrary, i.e., λ0 < r − dC(u). Then one would have on the one handx ∈ UC(r) and on the other hand x = PC(uλ0) because of the assumptions (b) andProposition 1.3.18. Applying Lemma 1.3.21 again one would obtain a contradictionwith the supremum property of λ0. So the equality λ0 = r − dC(u) holds. As ut can bewritten in the form ut = x + (dC(u) + t)(u − x)/‖u − x‖, taking t = λ0 gives (l).(l) ⇒ (i): We will proceed in five steps.

Step 1. For any x∗ ∈ NP rC (x) = NP

C(x) ∩ rB∗ with x ∈ C and any α ∈ ]0, 1], from (k)⇔ (l) one has PC

(x + rα J∗x∗

‖x∗‖

)3 x. This means that, for any x′ ∈ C,∥∥∥∥∥x + rαJ∗x∗

‖x∗‖− x

∥∥∥∥∥ ≤ ∥∥∥∥∥x + rαJ∗x∗

‖x∗‖− x′

∥∥∥∥∥ .Besides, as J = ∇F

(12‖ · ‖

2), one also has

12

∥∥∥∥∥x + rαJ∗x∗

‖x∗‖− x′

∥∥∥∥∥2

+

⟨J(x − x′ + rα

J∗x∗

‖x∗‖

), x′ − x

⟩≤

12

∥∥∥∥∥rαJ∗x∗

‖x∗‖

∥∥∥∥∥2

.


So,⟨J(J∗(x∗) − ‖x

∗‖

rα (x′ − x)), x′ − x

⟩≤ 0. It is possible to take any α in

]0, ‖x

∗‖

r

].

Hence, whenever xi ∈ C, x∗i ∈ NP rC (xi), i = 1, 2, and t ≥ 1, one has

〈J(J∗(x∗1) − t(x2 − x1)), x2 − x1〉 ≤ 0and 〈J(J∗(x∗2) − t(x1 − x2)), x1 − x2〉 ≤ 0.

By adding, one obtains

〈J(J∗(x∗1) − t(x2 − x1)) − J(J∗(x∗2) − t(x1 − x2)), x1 − x2〉 ≤ 0

which is the J-hypomonotonicity of NP rC of degree t for any t ≥ 1 .

Step 2. For any α ∈ ]0, 1/2[ , we have by Step 1 and by Lemma 1.3.8 that the set-valued mapping (I + J∗ ◦ NPαr

C )−1 is 1q -Holder continuous on the intersection of its

domain with any bounded subset. We also have, for any r′ > 0,

(1.37) PC(x) ⊂ (I + J∗ ◦ NP r′C )−1(x) for any x ∈ UC(r′).

Indeed, for any x ∈ UC(r′), the inclusion y ∈ PC(x) entails that J(x − y) ∈ NPC(y), and

‖y− x‖ < r′ so J(x− y) ∈ NP r′C (y). So we have that for any α ∈]0, 1/2[, PC is 1

q -Holdercontinuous on the intersection of any bounded set with Dom PC ∩ UC(αr). Then bythe arguments of Step 2 in the proof of Theorem 1.3.13, PC is also nonempty, single-valued on UC(αr). As α can be made as close as one wants to 1

2 , PC is nonempty,single-valued, locally 1

q -Holder continuous on UC(r/2).

Step 3 corresponds to the two following lemmas. The first one completes the resultof Lemma 3.1 of Bounkhel and Thibault [33]. As usual the line segment between twopoints u, v ∈ X will be denoted by [u, v], that is, [u, v] := {tu + (1 − t)v : t ∈ [0, 1]}.

Lemma 1.3.28. Let C be a nonempty closed subset of a normed vector space (Y, ‖ · ‖).Let ρ > 0 and u < C(ρ). Then the following hold:(a) dC(u) = ρ + dC(ρ)(u) = ρ + dDC(ρ)(u);(b) If u0 ∈ PC(u) and y0 ∈ [u0, u] ∩ DC(ρ), then y0 ∈ PC(ρ)(u);(c) If y ∈ PC(ρ)(u) and z ∈ PC(y), then z ∈ PC(u). Further, if PC(ρ)(u) = {y} andz ∈ PC(y), then y ∈ [z, u] and PC(u) = {z}.

Proof. (a) For all y ∈ C(ρ)

dC(u) ≤ dC(y) + ‖u − y‖ ≤ ρ + ‖u − y‖,

hence

(1.38) dC(u) ≤ ρ + dC(ρ)(u) ≤ ρ + dDC(ρ)(u).


Fix any ε > 0 and choose uε ∈ C with ‖u − uε‖ ≤ dC(u) + ε. Since dC(uε) = 0 anddC(u) > ρ we may choose yε ∈ [uε, u] ∩ DC(ρ) and hence

dC(u) + ε ≥ ‖u − uε‖ = ‖u − yε‖ + ‖yε − uε‖ ≥ dDC(ρ)(u) + dC(yε) = dDC(ρ)(u) + ρ.

Combining this with (1.38) we obtain

(1.39) dC(u) = ρ + dC(ρ)(u) = ρ + dDC(ρ)(u).

(b) Assume now that u0 ∈ PC(u) and y0 ∈ [u0, u] ∩ DC(ρ). Then u0 ∈ PC(y0) andby (1.39) we have

‖u − y0‖ + ρ = ‖u − y0‖ + dC(y0) = ‖u − y0‖ + ‖y0 − u0‖ = ‖u − u0‖ = ρ + dC(ρ)(u)

and hence ‖u − y0‖ = dC(ρ)(u), i.e., y0 ∈ PC(ρ)(u).(c) Assume that y ∈ PC(ρ)(u) and z ∈ PC(y). Then

‖u − z‖ ≤ ‖u − y‖ + ‖y − z‖ = dC(ρ)(u) + dC(y) ≤ dC(ρ)(u) + ρ = dC(u)

(the last equality being due to (1.39)) and hence z ∈ PC(u).Assume now that PC(ρ)(u) = {y}. Taking y′ ∈ [z, u] ∩ DC(ρ) , ∅, we have by (b)

that y′ ∈ PC(ρ)(u) and hence y = y′ ∈ [z, u]. If there exists z′ , z with z′ ∈ PC(u),then one sees that z′ < u + [0,+∞[(z − u) and by (b) for y′′ ∈ [z′, u] ∩ DC(ρ) , ∅(hence y′′ , u) one would have y′′ ∈ PC(ρ)(u) and hence y′′ = y ∈ [z, u] which wouldcontradict z′ < u + [0,+∞[(z − u). This completes the proof of the lemma. �

Lemma 1.3.29. If C satisfies the assertion (l) of Theorem 1.3.27 with parameter r andPC is nonempty, single-valued on UC(αr) for some α ∈ ]0, 1], then for any α′ ∈ ]0, α[ ,the set C(α′r) := {x ∈ X : dC(x) ≤ α′r} satisfies (l) with parameter r(1 − α′).

Proof. Take u ∈ UC(α′r)(r(1 − α′)) and put r′ := dC(u). Note that 0 < dC(α′r)(u) <r(1 − α′) and hence by (a) of Lemma 1.3.28 one has α′r < dC(u) < r, which im-plies in particular u ∈ UC(r). Suppose that PC(α′r)(u) = {y}. We have to prove thaty ∈ PC(α′r)

(y + r(1 − α′) u−y

‖u−y‖

). Observing that y ∈ UC(αr), we may put z := PC(y)

according to the assumption on PC over UC(αr). By (c) of Lemma 1.3.28, we havez ∈ PC(u). Since dC(z) = 0 and dC(u) > α′r, we may take y1 ∈ [z, u] ∩ DC(α′r) , ∅.The assertion (b) of Lemma 1.3.28 says that y1 ∈ PC(α′r)(u) and hence y1 = y. Now,since y1 ∈ UC(r) and z = PC(y1) we have by (l) that PC(u′) 3 z for

u′ := z + ry1 − z‖y1 − z‖

= z + ru − z‖u − z‖

.

This entails by (b) of Lemma 1.3.28 again that PC(α′r)(u′) 3 y since (recall that y = y1)

y ∈ [z, u] ∩ DC(α′r) ⊂ [z, u′] ∩ DC(α′r).


Further, since ‖u′ − y‖ = ‖u′ − z‖ − ‖y − z‖ = r − α′r (the second equality being due tothe definition of u′ and to the inclusion y ∈ DC(α′r)) we see that

u′ = y + (r − α′r)u − y‖u − y‖

.

So, we obtain that C(α′r) satisfies (l) with parameter r(1 − α′), and the proof of thelemma is complete. �

Step 4. For α ∈ ]0, 1], let us consider the property

P(α){

C satisfies (l) with parameter r andPC is single-valued, locally Holder continuous on UC(αr).

We claim that P(α)⇒ P(α+1

2

).

Suppose that P(α) holds. From Steps 1, 2 and 3, we have that for any α′ ∈ ]0, α[ ,

(1.40) PC(α′r) is single-valued, locally Holder continuous on UC(α′r)

(r(1 − α′)

2

).

Take any u ∈ UC

(α′r +

r(1−α′)2

)such that r′ := dC(u) > α′r. By (a) of Lemma 1.3.28

we have dC(α′r)(u) = r′ − α′r, and so

(1.41) u ∈ UC(α′r)

(r(1 − α′)

2

)since r′ − α′r < r(1−α′)

2 because of the inclusion u ∈ UC

(α′r +

r(1−α′)2

). We may then

put y := PC(α′r)(u) according to (1.40) and put z := PC(y) according to the sec-ond assumption in P(α). Then by (c) of Lemma 1.3.28 we have z = PC(u) and soPC(u) = z = PC ◦PC(α′r)(u). So for any α′ ∈ ]0, α[ , (1.40), (1.41), and P(α) ensure thatPC is single-valued and locally Holder continuous on UC

(α′r +

r(1−α′)2

)\ C(α′r). By

assumption it is also locally Holder continuous on UC(αr). So PC is single valued,locally Holder continuous on UC

(α′r +

r(1−α′)2

)for any α′ ∈ ]0, α[ and hence also

on UC

(αr +

r(1−α)2

)= UC

(α+1

2 r). This establishes the claim and finishes the proof of

Step 4.

Step 5. Define (αn)n by α0 = 1/2, αn+1 = (αn + 1)/2. We have αn → 1 and by Step 4,P(αn)⇒ P(αn+1). As P(α0) is true by Step 1, we have P(1), that entails (i).(i) ⇒ (h): This implication follows from Lemma 1.3.10 and Remark 1.3.12.(h) ⇒ (d) is obvious.

Since the additional equivalence (m) can be established like in Theorem 1.3.25,the proof is now complete. �


The following proposition gives the expression of the metric projection mappingin terms of the normal cone.

Proposition 1.3.30. Under the assumptions of Theorem 1.3.27, one has for all x ∈UC(r)

∇FdC(x) = J(x − PC(x))/dC(x) and PC(x) = (I + J∗ ◦ NP rC )−1(x).

Proof. The equality concerning the derivative follows easily from the theorem aboveand from the expression of ∇Feλ f (x) in Lemma 1.3.9. Let us prove the equalityconcerning the metric projection mapping. We know by (1.37) that PC(x) ⊂ (I + J∗ ◦NP r

C )−1(x) for all x ∈ UC(r), so it is enough to check that (I + J∗ ◦ NP rC )−1 is at most

single-valued on UC(r). Suppose that y1, y2 are in (I + J∗ ◦ NP rC )−1(x) with x ∈ UC(r),

that is, J(x − yi) ∈ NP rC (yi), i = 1, 2. Then PC(x) = y1 = y2 by (k). �

The uniform prox-regularity also entails the J-hypomonotonicity of the truncatednormal functional cone.

Proposition 1.3.31. Under the assumptions of Theorem 1.3.27, we also have(n) The truncated normal functional cone mapping NP r

C is J-hypomonotone of degreet for any t ≥ 1.Conversely, (n) entails the assertions of Theorem 1.3.27 with parameter r/2 insteadof r.

Proof. See Steps 1 and 2 in the proof of Theorem 1.3.27. �

The following corollary gives several characterizations of convex sets. The con-dition (k) in the corollary is Vlasov’s extension (see Vlasov [167]) to Banach spacewith rotund dual of the known result proved by Asplund in Hilbert space (see As-plund [1]). The characterization (f) is exactly Theorem 18 of Borwein, Fitzpatrick andGiles [24]. Theorem 18 in Borwein, Fitzpatrick and Giles [24] was established in themore general case where the dual space is merely rotund and it was, in some sense, ageneralization of Theorem 3.6 of Fitzpatrick [75] where both the norm and the dualnorm were assumed to be Frechet differentiable away from the origins. Note also thatthe characterization (h) was explicitly given by Borwein, Fitzpatrick and Giles [24,Theorem 17] (with directional Gateaux derivability as in Theorem 1.3.20 instead ofthe Dini subdifferentiability), in the larger context of Banach space Y with rotunddual Y∗. Both Theorem 17 and Theorem 18 in Borwein, Fitzpatrick and Giles [24]are derived by the authors by proving that any closed subset of a Banach space Ysuch that

lim sup‖y‖→0

dC(x + y) − dC(x)‖y‖

= 1 for all x ∈ Y \C


is almost convex and hence, according to a result due to Vlasov [167], convexwhenever Y∗ is rotund for its dual norm.

In our corollary, the requirement (X5) is imposed for a large part because ofcharacterizations (b), (i), (j), (m), (n).

Corollary 1.3.32. Let C ⊂ X be a closed set. The following are equivalent:(a) C is convex;(b) C is uniformly ∞-prox-regular, i.e., uniformly r-prox-regular for any real numberr > 0;(c) dC is continuously differentiable on X \C;(d) dC is Frechet regular on X \C;(e) dC is Frechet differentiable on X \C;(f) dC is Gateaux differentiable on X \C with ‖∇GdC(x)‖ = 1 for all x ∈ X \C;(g) ∂FdC(x) , ∅ for all x ∈ X;(h) PC(x) , ∅ and ∂−dC(x) , ∅ for all x ∈ X;(i) d2

C is C1 on X with locally Holder continuous derivative mapping on X;(j) PC is single-valued and locally Holder continuous on X;(k) PC is single-valued and norm-to-norm continuous on X;(l) PC is single-valued and norm-to-weak continuous on X;(m) For any p ∈ PNC(x) with x ∈ C one has x ∈ PC(x + p);(n) If u ∈ X \C and x = PC(u), then x ∈ PC(u′) for u′ = x + r(u − x) and any r > 0.

Proof. The equivalence between all the assertions from (b) to (n) is easily seen tofollow from Theorem 1.3.27. The implication (a) ⇒ (g) is obvious according tothe convexity of the continuous function dC under (a). By Proposition 1.3.31, thecondition (g) entails that NP r

C is J-hypomonotone of degree 1 on UC(r) for any r > 0.Fix any x, y ∈ C and any x∗ ∈ NP

C(x), y∗ ∈ NPC(y). By definition of J-hypomonotonicity

of degree 1 for all s > 0 large enough

〈J[J∗(sx∗) − (y − x)] − J[J∗(sy∗) − (x − y)], x − y〉 ≥ 0,

or ⟨J[J∗(x∗) −

1s

(y − x)]− J

[J∗(y∗) −

1s

(x − y)], x − y

⟩≥ 0.

Using the continuity of J and J∗ and passing to the limit when s goes to +∞ weobtain

〈x∗ − y∗, x − y〉 ≥ 0.

This means that NPC is monotone and hence by Correa, Gajardo and Thibault [49] (see

also Correa, Jofre and Thibault [51, 52]) the set C is convex, that is, (a). The proofof the corollary is then complete. �


In the remainder of the subsection, we will give, for an r-prox-regular set C, someproperties of the ρ-enlargement C(ρ) with ρ ∈ ]0, r[ and of the set of ρ-external pointsto C

EC(ρ) := {u ∈ X : dC(u) ≥ ρ}.

The first result concerns the proximal normal cone to the ρ-enlargement of C. Inits statement we will denote by NCl

S the Clarke normal cone to a set S and by R+ theset of all non negative real numbers.

Proposition 1.3.33. Assume that C is uniformly r-prox-regular. Then, for any ρ ∈]0, r[ and any y ∈ DC(ρ),

NPC(ρ)(y) = NCl

C(ρ)(y) = R+∇FdC(y) ⊂ NP

C(PC(y)).

Proof. Fix y ∈ DC(ρ). From Lemma 1.3.29 and Theorem 1.3.27, the set C(ρ) isuniformly (r−ρ)-prox-regular and PC(ρ) is single-valued on UC(r). To see that ∇FdC(y)actually belongs to NP

C(ρ)(y), put

(1.42) u := y + ε(y − PC(y)),

where ε is small enough that dC(u) < r. As PC(u) = PC(y) by Theorem 1.3.27 (l), wehave

dC(u) = ‖u − PC(y)‖ = ‖y + ε(y − PC(y)) − PC(y)‖ = (1 + ε)‖y − PC(y)‖ = (1 + ε)ρ > ρ.

Then by Lemma 1.3.28 (b), since PC(ρ) is single-valued on UC(r), one has y = PC(ρ)(u).So u− y ∈ PNC(ρ)(y) and hence y−PC(y) ∈ PNC(ρ)(y) which entails ∇FdC(y) ∈ NP

C(ρ)(y).Further, as C(ρ) = {x ∈ X : dC(x) ≤ dC(y)} because y ∈ DC(ρ), we know byClarke [44, Theorem 2.4.7 and Corollary 1] that NCl

C(ρ) = R+∇FdC(y) and hence

R+∇FdC(y) ⊂ NP

C(ρ)(y) ⊂ NClC(ρ)(y) = R+∇

FdC(y).

So it remains to see that the inclusion R+∇FdC(y) ⊂ NP

C(PC(y)) follows from ∇FdC(y) =y − PC(y)‖y − PC(y)‖

∈ NPC(PC(y)). �

Before establishing the second result, let us prove the following lemma which hasits own interest.

Lemma 1.3.34. Assume that C is uniformly r-prox-regular. Then for any ρ ∈ ]0, r]and y ∈ UC(ρ) one has

(1.43) dC(y) + dEC(ρ)(y) = ρ.


Proof. Fix y ∈ UC(ρ) and put x := PC(y) according to (i) of Theorem 1.3.27. Then foru := x+ρ y−x

‖y−x‖ one has x ∈ PC(u) by Theorem 1.3.27 (k) and hence u ∈ DC(ρ) ⊂ EC(ρ)and

(1.44) dC(y) + dEC(ρ)(y) ≤ ‖y − x‖ + ‖y − u‖ = ‖u − x‖ = ρ.

For any z ∈ EC(ρ) we have

‖z − y‖ ≥ ‖z − PC(y)‖ − ‖y − PC(y)‖ ≥ dC(z) − dC(y) ≥ ρ − dC(y)

hence

(1.45) dEC(ρ)(y) ≥ ρ − dC(y).

It follows from (1.44) and (1.45) that dC(y) + dEC(ρ)(y) = ρ. �

From this lemma we see, on the one hand, through Theorem 1.3.27 (d) that ifC is uniformly r-prox-regular, then for any ρ ∈ ]0, r[ , the set EC(ρ) is uniformly ρ-prox-regular, because dEC(ρ)(.) = ρ − dC(.) and UC(ρ) = UEC(ρ)(ρ). On the other hand,the lemma allows us to retrieve the following characterization of uniform r-prox-regularity given by Clarke, Stern and Wolenski [46] in the context of Hilbert space.Their proof is different and it relies on the analysis of an appropriate infimum valuefunction. In our characterization below, the additional nonvacuity of PC(y) is notrequired (compare with Clarke, Stern and Wolenski [46, Theorem 4.1 (c)]).

Theorem 1.3.35. A closed set C is uniformly r-prox-regular for some r > 0, if andonly if,

(1.46) dC(y) + dEC(r)(y) = r for all y ∈ UC(r).

Proof. The fact that (1.46) is implied by the uniform r-prox-regularity of C followsfrom Lemma 1.3.34 with ρ = r. Assume now that (1.46) holds and consider anyy ∈ UC(r) for which PC(y) is a singleton, say PC(y) = x. Then, for any y′ ∈ ]x, y[ , onehas PC(y′) = x. This yields for any non zero t ∈

]−1, ‖y−y′‖

‖y′−x‖

[t−1[dC(y′ + t(y′ − x))) − dC(y′)] = t−1[‖y′ + t(y′ − x) − x‖ − ‖y′ − x‖] = ‖y′ − x‖,

which entails that dC has a Gateaux directional derivative in the full direction y′ − xand hence by Theorem 1.3.20 the function dC is Frechet differentiable at y′. Thus by(1.46) the function dEC(r) is also differentiable at y′. We then deduce (see the formulafor ∇Feλ f in Lemma 1.3.9) that PEC(r)(y′) is a singleton that will be denoted by u′.Using successively the inclusion u′ ∈ DC(r) and (1.46) we obtain

r ≤ ‖x − u′‖ ≤ ‖y′ − x‖ + ‖y′ − u′‖ = r,

so y′ ∈ ]x, u′[ (see (1.11)) and PC(u′) 3 x. By Theorem 1.3.27 (l), this means that Cis uniformly r-prox-regular. �

1.4. Prox-regular sets and epigraphs in uniformly convex Banach spaces: . . . 57

1.4 Prox-regular sets and epigraphs in uniformly con-vex Banach spaces: various regularities and otherproperties

In this section we continue the study of prox-regular sets that we began in theprevious section in the setting of uniformly convex Banach space endowed with anorm both uniformly smooth and uniformly convex (like Lp, Wm,p spaces).

Here in uniformly convex Banach space we prove normal and tangential regularityproperties for these sets, and in particular the equality between Mordukhovich andproximal normal cones for such sets. We also compare in this setting the proximalnormal cone with different Holderian normal cones depending on the power typess, q of moduli of smoothness and convexity of the norm. In the case of sets that areepigraphs of functions, we show that J-primal lower regular functions have prox-regular epigraphs and we compare these functions with Poliquin’s primal lower nicefunctions depending on the power types s, q of the moduli. The preservation of prox-regularity for the intersection of finitely many sets and of the inverse image areobtained under a calmness assumption. A conical derivative formula for the metricprojection mapping of prox-regular set is also established. Among other results of thesection it is proved that the Attouch-Wets convergence preserves the uniform r-prox-regularity property and that the metric projection mapping is continuous with respectto this convergence for such sets.

The results from this section are published by Bernard, Thibault and Zlatevain [19].

In this section we base our considerations on those in the previous one where weextended in some sense the study made in Poliquin, Rockafellar and Thibault [138] ofprox-regular sets C in Hilbert spaces. Here we continue our study mainly consideringnormal cones and subdifferentials. In the context of Hilbert space the coincidence fora prox-regular set C at x between the Mordukhovich normal cone and the proximalnormal one follows directly from the fact (due to the Hilbert structure) that in such acase there exists some non negative number γ and some neighbourhood U of x suchthat for any proximal normal x∗ of C at a point x ∈ U ∩C with ‖x∗‖ ≤ 1 one has

〈x∗, y − x〉 ≤ γ‖y − x‖2 for all y ∈ U ∩C.

The case of uniformly convex Banach space is not so obvious. Our aim is on theone hand to show that this important property of equality between the Mordukhovichnormal cone and the proximal normal one still holds for prox-regular sets of uni-formly convex Banach spaces and on the other hand to take advantage of this prop-erty to provide in the same context several new results including in particular the


conical derivative of the metric projection mapping to C and the preservation ofprox-regularity under the Attouch-Wets convergence.

In Subsection 1.4.1 we show the normal and tangential regularity properties ofprox-regular sets of a uniformly convex Banach space X and deduce a proximalnormal formula for these sets.

In Subsection 1.4.2 we prove that the epigraphs of the J-primal lower regularfunctions are prox-regular, and compare these functions with Poliquin’s primal lowernice functions.

In Subsection 1.4.3, we draw a comparison between the prox-regularity con-cept considered in the previous section and taken from Poliquin, Rockafellar andThibault [138], and another one from Poliquin and Rockafellar [137] and adapted byBernard and Thibault in [14] to the general Banach context. We also compare theprox-regularity notions when the norm varies in certain families.

Subsection 1.4.4 concerns some relationships between different normal cones (ofclosed sets) related to the moduli of power type of the norm of X.

Subsection 1.4.5 deals with the preservation of prox-regularity under the inter-section and the inverse image. Under a calmness qualification condition, it is shownthat the intersection of finitely many prox-regular sets and the inverse image of aprox-regular set by a C1,1 mapping inherit the prox-regularity property.

In Subsection 1.4.6 we take advantage of the result of tangential regularity ofSubsection 1.4.1 to establish a conical derivative formula for the metric projectionmapping of prox-regular set.

Finally Subsection 1.4.7 studies the behaviour of the metric projection mappingfor a family (Ct)t of uniformly r-prox-regular closed sets of X which convergesin the sense of Attouch-Wets to a closed set C. It is proved that C inherits theuniform r-prox-regularity property and that, for any x0 with d(x0,C) < r, one hasPCt(x0)−→

tPC(x0), for the metric projection mapping PC onto the closed set C.

We will use most of the notations and preliminaries from the previous section.However, we will recall or refine some of them.

Throughout this section we work in an uniformly convex Banach space (X, ‖ · ‖)whose norm ‖ · ‖ is both uniformly convex and uniformly smooth. In some statements,the moduli of uniform convexity and uniform smoothness of the norm ‖ · ‖ will berequired to be of power type q, and of power type s, respectively and this will beexplicitly stated when needed. One knows that such a renorm of the uniformly convexspace always exists.

Some of the properties of uniformly convex Banach spaces were discussed inthe beginning of the previous section, see also Diestel [58], Beauzami [12], Deville,Godefroy and Zizler [57].


For any real number σ > 1, consider the mapping Jσ : X → X∗ defined by

Jσ(x) = {x∗ ∈ X∗ : 〈x∗, x〉 = ‖x∗‖.‖x‖, ‖x∗‖ = ‖x‖σ−1}.

For σ = 2, J2 will be simply denoted by J and it is generally called the normalizedduality mapping associated with the norm ‖ · ‖. As X is reflexive we have that Jis surjective. The mapping Jσ is the subdifferential of the convex function 1

σ‖ · ‖σ,

i.e., Jσ = ∂

(1σ‖ · ‖σ

). With the additional uniformly convex property of X and by the

choice of the norm that we made, for any σ > 1, Jσ is single-valued, bijective andnorm-to-norm continuous. The inverse mapping J−1 (of J) will be denoted by J∗, itis the normalized duality mapping for the dual norm on X∗. Observe also (by thedefinitions of Jσ and J := J2) that for R+ := [0,+∞[

(1.47) Jσ(x) ∈ R+J(x) and Jσ(tx) = tσ−1Jσ(x) for all x ∈ X and t ≥ 0.

We recall (see Deville, Godefroy and Zizler [57]) that the mapping J is uniformlycontinuous over each bounded subset of X (in fact, this property characterizes theuniform smoothness of the norm).

Moreover, from Xu and Roach [169, (2.16) p. 201 with p=2], there is a constantK′ > 0 such that for all nonzero pairs (x, y) ∈ X × X

(1.48) 〈J(x) − J(y), x − y〉 ≥ K′(max{‖x‖, ‖y‖})2δ‖·‖

(‖x − y‖

2 max{‖x‖, ‖y‖}

)and by (3.1)′ of Xu and Roach [169, p. 208] one also has some constant L′ > 0 suchthat for all pairs (x, y) ∈ X × X with x , y

(1.49) ‖J(x) − J(y)‖ ≤ L′(max{‖x‖, ‖y‖})2

‖x − y‖ρ‖·‖

(‖x − y‖

max{‖x‖, ‖y‖}

).

When the modulus of uniform convexity of the norm ‖ · ‖ is of power type q, fromXu and Roach [169, (2.17)′ p. 202] again, there exists some constant K > 0 such thatfor every (x, y) ∈ X × X,

(1.50) ‖x + y‖q ≥ ‖x‖q + q〈Jq(x), y〉 + K‖y‖q.

Similarly, whenever the modulus of smoothness of the norm ‖ · ‖ is of power types, there exists, according to Xu and Roach [169, Remark 5, p. 208], some constantL > 0 such that, for all x, y ∈ X,

(1.51) ‖x + y‖s ≤ ‖x‖s + s〈Js(x), y〉 + L‖y‖s.


Observe that (1.50) and (1.51) respectively entail that for all x, y ∈ X,

(1.52) 〈Jq(x) − Jq(y), x − y〉 ≥2Kq‖x − y‖q

and

(1.53) 〈Js(x) − Js(y), x − y〉 ≤2Ls‖x − y‖s.

Inequalities in the lines of (1.52) and (1.53) also hold with the normalized dualitymapping J in place of Jq or Js when they are restricted to bounded subsets of X.Indeed, if the norm ‖ · ‖ has modulus of convexity (resp. smoothness) of power typeq (resp. s), then for any r > 0 according to (1.48) (resp. (1.49)) there exist somepositive constant Kr (resp. Lr) such that

(1.54) 〈J(x) − J(y), x − y〉 ≥ Kr‖x − y‖q, ∀x, y ∈ rB, (see (1.8))

(resp.

(1.55) ‖J(x) − J(y)‖ ≤ Lr‖x − y‖s−1, ∀x, y ∈ rB, see (1.9)).

As in the previous section, the space X × R will be considered with the norm||| · ||| given by |||(x, r)||| =

√‖x‖2 + r2. So, for the normalized duality mapping JX×R :

X × R→ X∗ × R associated with the norm ||| · |||, one has the equality

(1.56) JX×R(x, r) = (J(x), r).

When there is no risk of confusion, JX×R will be simply denoted by J.Recall that for a closed set C ⊂ X, a nonzero vector p ∈ X is said to be a

primal proximal normal vector to C at x ∈ C if there are u < C and r > 0 such thatp = r−1(u− x) and ‖u− x‖ = dC(u). Equivalently, a nonzero p ∈ X is a primal proximalnormal vector to C at x ∈ C if there exists r > 0 such that x ∈ PC(x + rp). We alsotake by convention the origin of X as a primal proximal normal vector to C at x. Thecone of all primal proximal normal vectors to C at x is denoted by PNC(x) and calledthe primal proximal normal cone of C at x. The concept is local in the sense that

(1.57)

{for any u < C and any closed ball V := B[x, β] centred at x ∈ Cand such that ‖u − x‖ = dC∩V(u), one has u − x ∈ PNC(x).

A continuous linear functional p∗ ∈ X∗ is said to be a proximal normal functionalto C at x ∈ C if J∗(p∗) ∈ PNC(x). This means for p∗ , 0 that there are u < C,r > 0 such that p∗ = r−1J(u − x) and ‖u − x‖ = dC(u). Or, equivalently, a nonzerop∗ ∈ X∗ is a proximal normal functional to C at x ∈ C if there exists r > 0 such that


x ∈ PC(x + rJ∗(p∗)). The cone of all proximal normal functionals to C at x will bedenoted by NP

C(x).One easily verifies that if p ∈ PNC(x), then J(p) ∈ NP

C(x), and that if p∗ ∈ NPC(x),

then J∗(p∗) ∈ PNC(x) (keep in mind that J∗ = J−1 is the normalized duality mappingfor X∗ endowed with the dual norm of ‖ · ‖). Hence, PNC(x) and NP

C(x) completelydetermine each other.

We also recall the concept of Frechet normal cone NFC (x). A continuous linear

functional x∗ ∈ X∗ is said to be a Frechet normal functional to C at x ∈ C if forany ε > 0 there exists a neighbourhood U of x such that the inequality 〈x∗, x′ − x〉 ≤ε‖x′ − x‖ holds for all x′ ∈ C ∩ U.

Since the norm of the space X we work in is Frechet differentiable away from theorigin, for any closed subset C ⊂ X and any x ∈ C, any proximal normal functionalto C at x is also a Frechet normal functional to C at x.

We will also use the β-Holder normal cone NβC(·) defined at x ∈ C by x∗ ∈ Nβ

C(x)when there exist ε, γ > 0 such that for all x′ ∈ B(x, ε) ∩C, 〈x∗, x′ − x〉 ≤ γ‖x′ − x‖β.

In the context where the norm ‖ · ‖ is associated with an inner product (· | ·), thatis, (X, ‖ · ‖) is a Hilbert space, then it is straightforward to verify that x ∈ PC(x + rp)if and only if

(p|y − x) ≤ (2r)−1‖y − x‖2 for all y ∈ C.

The same description says that for a nonzero vector p there exists some positive rsatisfying the above property if and only if there are positive ε and γ such that

〈Jp, y − x〉 = (p|y − x) ≤ γ‖y − x‖2 for all y ∈ B(x, ε) ∩C,

that is, for β = 2 one has Jp ∈ NβC(x), and hence NP

C(x) = NβC(x). Since Nβ

C(x) is ofcourse independent of any equivalent norm to ‖ · ‖, so is the cone NP

C(x) in the Hilbertsetting.

The above notions can be translated in the context of functions.Let f : X → R ∪ {+∞} be a lower semicontinuous function. By definition, the

effective domain of f is the set dom f := {x ∈ X : f (x) < +∞} and the epigraph of fis the set epi f := {(x, r) ∈ X × R : f (x) ≤ r}. Let x ∈ dom f .

We say that p∗ ∈ X∗ is a proximal subgradient of f at x if (p∗,−1) is a proximalnormal functional to the epigraph of f at (x, f (x)). The proximal subdifferential of fat x, denoted by ∂p f (x), consists of all such functionals. Thus, we have p∗ ∈ ∂p f (x),if and only if, (p∗,−1) ∈ NP

epi f (x, f (x)).The functional x∗ ∈ X∗ is said to be a Frechet subgradient of f at x if (x∗,−1) is a

Frechet normal functional to the epigraph of f at (x, f (x)). The Frechet subdifferentialof f at x, denoted by ∂F f (x), consists of all such functionals.

If x < dom f then all subdifferentials of f at x are empty, by convention. It isknown that for a lower semicontinuous function f on a reflexive Banach space with


a Frechet differentiable Kadec norm (in particular, on the space X we work in), theset dom ∂p f is dense in dom f . Moreover, from what we saw above, ∂p f (x) ⊂ ∂F f (x)for all x ∈ X. When ∂F f (x) , ∅, one says that f is Frechet subdifferentiable at thepoint x.

Similarly, a β-Holder subgradient of f at x is any functional x∗ ∈ X∗ suchthat (x∗,−1) ∈ Nβ

C(x, f (x)). Denoting by ∂β f (x) the set of such subgradients, wehave x∗ ∈ ∂β f (x) if there exists γ, ε > 0 such that for all y ∈ B(x, ε), f (y) ≥f (x) + 〈x∗, y − x〉 − γ‖y − x‖β.

It is easily checked that ∂pψC(x) = NPC(x) for any x ∈ C. The Frechet and β-Holder

normal cones of C are related to the indicator function ψC in a similar way.We continue to study prox-regularity property of a set in the context of uniformly

convex Banach spaces we began in Section 1.3.First of all we will refine the notion of prox-regularity of sets. Taking into account

Definition 1.3.3 and Proposition 1.3.4, it is easy to get

Proposition 1.4.1. For a closed subset C ⊂ X the following are equivalent:(a) C is prox-regular at x ∈ C;(b) there exist ε > 0 and r > 0, such that for all x ∈ C with ‖x − x‖ < ε and for allp∗ ∈ NP

C(x) such that ‖p∗‖ ≤ ε,

0 ≥ 〈J[J∗(p∗) − r−1(x′ − x)], x′ − x〉, ∀x′ ∈ C such that ‖x′ − x‖ < ε.

(c) for any Θ > 0 there exist ε > 0, r > 0 such that for all x ∈ C with ‖x − x‖ < ε andfor all p∗ ∈ NP

C(x) such that ‖p∗‖ ≤ Θ,

0 ≥ 〈J[J∗(p∗) − r−1(x′ − x)], x′ − x〉, ∀x′ ∈ C such that ‖x′ − x‖ < ε.

Proof. The equivalence (a) ⇔ (b) is proved in Proposition 1.3.4.We will prove that (b) ⇔ (c).Let (b) holds with ε′ > 0 and r′ > 0. Fix Θ > 0. Let p∗ ∈ NP

C(x) be such that‖x − x‖ < ε′ and ‖p∗‖ ≤ Θ. Then ε′

Θp∗ ∈ NP

C(x) is such that∥∥∥ ε′

Θp∗

∥∥∥ ≤ ε′ and form (b)we have

0 ≥⟨J[J∗

(ε′

Θp∗

)− (r′)−1(x′ − x)

], x′ − x

⟩, ∀x′ ∈ C such that ‖x′ − x‖ < ε′,

or equivalently

0 ≥ 〈J[J∗(p∗) − (r′ε′Θ−1)−1(x′ − x)], x′ − x〉, ∀x′ ∈ C such that ‖x′ − x‖ < ε′,

which means that (c) holds for Θ with ε = ε′ and r = r′ε′Θ−1.Let now (c) holds for some Θ > 0 with ε′ > 0 and r > 0. Let ε > 0 be such that

ε ≤ min{ε′,Θ}. Let p∗ ∈ NPC(x) be such that ‖x − x‖ < ε and ‖p∗‖ ≤ ε ≤ Θ. From the

choice of ε ≤ ε′ from (c) it holds that

0 ≥ 〈J[J∗(p∗) − r−1(x′ − x)], x′ − x〉, ∀x′ ∈ C such that ‖x′ − x‖ < ε,


which means that (b) holds with ε and r. �

To the rest of this section as prox-regularity of a set C at a point x will beconsidered Proposition 1.4.1 (c) for Θ = 1, namely the property

(1.58)

there exist ε > 0, r > 0 such that for all x ∈ C with ‖x − x‖ < εand for all p∗ ∈ NP

C(x) with ‖p∗‖ ≤ 1,0 ≥ 〈J[J∗(p∗) − r−1(x′ − x)], x′ − x〉, ∀x′ ∈ C such that ‖x′ − x‖ < ε.

In this regard, we refine the definition of prox-regularity as follows

Definition 1.4.2. A closed set C ⊂ X is called (metrically) prox-regular (with respectto the uniformly convex norm ‖ · ‖) or ‖ · ‖-prox-regular at x ∈ C provided there existε > 0 and r > 0 such that for all x ∈ C and for all p∗ ∈ NP

C(x) with ‖x − x‖ < ε and‖p∗‖ < 1 the point x is a nearest point of {x′ ∈ C : ‖x′ − x‖ < ε} to x + rJ∗(p∗).

The metrical aspect is due to the fact that the proximal normal cone NPC(·) is related

to the norm ‖ · ‖ and depends on it in general when the latter is not a Hilbert norm.Whenever there is no ambiguity concerning either the norm ‖ · ‖ or the involvementof the proximal normal cone NP

C(·), we will merely say that C is prox-regular at x.The crucial fact which needs to be emphasized here is that the real number r of

the definition (for which the closed ball B[x+rJ∗(p∗), r‖p∗‖] touches the set C∩B(x, ε)at the point x, when x is a boundary point of C with ‖x − x‖ < ε) does not dependon either the neighbouring point x or the proximal normal functional p∗ ∈ NP

C(x) with‖p∗‖ < 1.

Let us recall that for prox-regular sets in uniformly convex Banach space withnorm with moduli of uniform convexity and smoothness of power type a list ofequivalent characterizations is given in Theorem 1.3.25.

A set-valued mapping T : X ⇒ X∗ is said to be J-hypomonotone of degree t ≥ 0on a subset U ⊂ X (see Definition 1.3.6) if for any (xi, x∗i ) ∈ gph T := {(x, x∗) ∈U × X∗ : x∗ ∈ T (x)}, i = 1, 2, one has

〈J[J∗(x∗1) − t(x2 − x1)] − J[J∗(x∗2) − t(x1 − x2)], x2 − x1〉 ≤ 0.

We know that prox-regular property (1.58) means that the indicator function ψC

of C is J-primal lower regular at x. Another characterization can be given in termsof the mapping Jσ in place of the normalized duality mapping J.

Proposition 1.4.3. Let σ > 1 be a real number. The prox-regular property (1.58) isequivalent to any one of the following:(iσ) there exist ε > 0 and r > 0 such that for all x ∈ C and all p∗ ∈ NP

C(x) with‖x − x‖ < ε and ‖p∗‖ ≤ 1

0 ≥ 〈Jσ[J−1σ (p∗) − r−1(x′ − x)], x′ − x〉 ∀x′ ∈ C with ‖x′ − x‖ < ε;


(i′σ) there exist ε > 0 and r > 0 such that for all x ∈ C, t ≥ 0, and p∗ ∈ NPC(x) with

‖x − x‖ < ε and ‖p∗‖ ≤ rσ−1tσ−1

0 ≥ 〈Jσ[J−1σ (p∗) − t(x′ − x)], x′ − x〉 ∀x′ ∈ C with ‖x′ − x‖ < ε.

Proof. Note first that property (1.58) is equivalent to the following one:for any x, x′ ∈ C ∩ B(x, ε), any t ≥ 0, p∗ ∈ NP

C(x) with ‖p∗‖ ≤ rt,

(1.59) 0 ≥ 〈J[J∗(p∗) − t(x′ − x)], x′ − x〉.

Indeed, taking any t > 0, x, x′ ∈ C ∩ B(x, ε) and p∗ ∈ NPC(x) with ‖p∗‖ ≤ rt we have

by (1.58)0 ≥ 〈J[J∗(r−1t−1 p∗) − r−1(x′ − x)], x′ − x〉,

which is easily seen to be equivalent to (1.59) according to the positive homogeneityof J and J∗. Further it is obvious that (1.59) still holds for t = 0. The implication from(1.58) to (1.59) is then established. The converse follows from similar arguments.

With this new formulation, we are able to prove that (1.58) is equivalent to (i′σ).Suppose that (i′σ) holds. Observe first that for any non zero p∗ ∈ X∗, one has

J−1σ (p∗) = J∗

(‖p∗‖

2−σσ−1 p∗

)and for any non zero u ∈ X, one has Jσ(u) = J(‖u‖σ−2u).

Hence, putting σ′ := 2−σσ−1 , for any non zero p∗ ∈ X∗ and any t ≥ 0 one has the

equivalences

〈J[J∗(p∗) − t(x′ − x)], x′ − x〉 ≤ 0 ⇔

〈J[‖p∗‖−σ′

J−1σ (p∗) − t(x′ − x)], x′ − x ≤ 0 ⇔(1.60)

〈J[J−1σ (p∗) − t‖p∗‖σ

′

(x′ − x)], x′ − x〉 ≤ 0 ⇔

〈Jσ[J−1σ (p∗) − t‖p∗‖σ

′

(x′ − x)], x′ − x〉 ≤ 0.

Fix now any t > 0, any x, x′ ∈ C ∩ B(x, ε), and any non zero p∗ ∈ NPC(x) such that

rt ≥ ‖p∗‖. The latter inequality ensures that rt‖p∗‖σ′

≥ ‖p∗‖1

σ−1 , i.e., ‖p∗‖ ≤ rσ−1t′σ−1

for t′ := t‖p∗‖σ′

, thus (i′σ) with t′ = t‖p∗‖σ′

in place of t yields (1.60). By the aboveequivalences we obtain the inequality (1.59).

Conversely, suppose that (1.59) is fulfilled. Fix any t ≥ 0, x, x′ ∈ C ∩ B(x, ε), andany non zero p∗ ∈ NP

C(x) such that ‖p∗‖ ≤ rσ−1tσ−1. One has ‖p∗‖1

σ−1 ≤ rt and hence‖p∗‖ ≤ rt‖p∗‖1−

1σ−1 , which by (1.59), with t′ := t‖p∗‖1−

1σ−1 = t‖p∗‖−σ

′

in place of t(where as above σ′ := 2−σ

σ−1 ), yields

0 ≥ 〈J[J∗(p∗) − t′(x′ − x)], x′ − x〉.

By (1.60) this is equivalent to

0 ≥ 〈Jσ[J−1σ (p∗) − t′‖p∗‖σ

′

(x′ − x)], x′ − x〉,


which is exactly0 ≥ 〈Jσ[J−1

σ (p∗) − t(x′ − x)], x′ − x〉.

The latter being still true for p∗ = 0, we obtain (i′σ).The equivalence between (i′σ) and (iσ) can be argued like in the first part of the

proof. �

Anticipating Definition 1.4.4, we call property (i′σ) Jσ-primal lower regularity ofthe function ψC.

In the line of Definition 1.4.2 and Definition 1.3.26, the concept of prox-regularityon a global point of view is defined as follows: a closed subset C of X is (metrically)uniformly r-prox-regular or r-uniformly prox-regular if whenever x ∈ C and p∗ ∈NP

C(x) with ‖p∗‖ < 1, then x is the unique nearest point of C to x + rJ∗(p∗).Let us recall that for uniformly prox-regular sets in uniformly convex Banach

space with norm with moduli of uniform convexity and smoothness of power type alist of equivalent characterizations is given in Theorem 1.3.27.

In Definition 1.3.5 we introduced the J-plr concept for functions. Here we slightlyextend this concept keeping the name. Now it reduces in the Hilbert setting to thePoliquin’s primal lower nice concept, see Poliquin [136], and Levi, Poliquin andThibault [111].

Definition 1.4.4. A lower semicontinuous function f : X → R ∪ {+∞} is J-primallower regular (J-plr) at x ∈ dom f if there exist positive constants ε, r and Θ suchthat

(1.61) f (y) ≥ f (x) + 〈J[J∗(p∗) − t(y − x)], y − x〉

for all x, y ∈ B(x, ε), all p∗ ∈ ∂p f (x), and all t ≥ Θ such that ‖p∗‖ ≤ rt.

It is easily seen that a J-plr function according to Definition 1.3.5 is a J-plrfunction according to Definition 1.4.4 (with arbitrary Θ > 0). Further, Definition 1.3.5and Definition 1.4.4 are equivalent for indicator functions of closed sets (see the proofof Proposition 1.4.1).

Below we will consider J-plr functions according to Definition 1.4.4. If f is J-plrat x with some positive constants ε and r then it is so for any constants 0 < ε′ ≤ εand 0 < r′ ≤ r. If the lower semicontinuous function f is J-plr at x ∈ dom f withpositive constants ε, r and Θ, then obviously

(1.62) 〈J[J∗(p∗) − t(y − x)] − J[J∗(q∗) − t(x − y)], y − x〉 ≤ 0

for all x, y ∈ B(x, ε), for all p∗ ∈ ∂p f (x), q∗ ∈ ∂p f (y), and all t ≥ Θ such thatmax{‖p∗‖, ‖q∗‖} ≤ rt. This is the analog of the hypomonotonicity of certain trun-cations of ∂p f that characterizes primal lower-nice functions in Hilbert spaces: seePoliquin [136], Poliquin, Rockafellar and Thibault [111], Bernard, Thibault and Za-grodny [16] and the references therein.


1.4.1 Normal and tangential regularity properties of prox-regularsets

For a closed set C ⊂ X, the Mordukhovich limiting (or basic) normal cone NLC(x)

is defined (see Mordukhovich [124]) as the weak∗ sequential outer limit

(1.63) NLC(x) = w∗−seq Lim sup

y→xNP

C(y) := {w∗ − lim x∗n : x∗n ∈ NPC(xn), xn ∈ C → x}.

The following result appears in Ioffe [89, p. 188] with an approximation in X∗

with respect to the weak star topology, but merely under the local uniform convexityof the norm. For completeness we sketch below how Ioffe’s arguments also yield tothe following approximation with respect to the strong topology.

Proposition 1.4.5. Let C be a closed subset of X with x ∈ C and let x∗ ∈ NFC (x).

Then for any ε > 0 there exist uε ∈ C and u∗ε ∈ NPC(uε) such that

(1.64) ‖uε − x‖ < ε and ‖u∗ε − x∗‖ < ε.

In fact the result uses only the Frechet differentiability outside of zero of the norm‖ · ‖ and of its dual norm.

Proof. We may suppose that ‖x∗‖ = 1. By definition there exists some function ρfrom [0,+∞[ into [0,+∞[ with lim

t↓0ρ(t) = 0 and such that

(1.65) 〈x∗, y − x〉 ≤ ρ(‖y − x‖) ‖y − x‖ for all y ∈ C.

As this will not to be confusing for the reader, we denote the dual norm also by ‖ · ‖.Putting h := J∗x∗ we have

(1.66) 〈x∗, h〉 = ‖x∗‖‖h‖ = 1.

Further, by (1.65) we see that x + th < C for positive t small enough. By Lau theorem(see Lau [108]) for any such t we may choose some ht ∈ X such that ‖ht − h‖ < t andsuch that the nearest point of x + tht in C exists, say ut ∈ C. Writing ut in the formut = x + tvt we have

(1.67) t‖vt − ht‖ = dC(x + tht) ≤ t‖ht‖ and hence ‖vt‖ ≤ 2‖ht‖ ≤ 2(1 + t).

Then taking (1.65) into account we have

〈x∗, tvt〉 ≤ ρ(t‖vt‖) ‖tvt‖ ≤ 2tρ(t‖vt‖)(1 + t)

which yields

(1.68) 〈x∗, vt〉 ≤ 2ρ(t‖vt‖)(1 + t).


Now observe by the first inequality of (1.67) that

‖h − vt‖ ≤ ‖h − ht‖ + ‖ht − vt‖ ≤ ‖h − ht‖ + ‖ht‖ ≤ ‖h‖ + 2‖h − ht‖ < 1 + 2t

and hence for wt := (1 + 2t)−1(h − vt) we have ‖wt‖ ≤ 1. Therefore

(1 + 2t)−1[1 − 2ρ(t‖vt‖)(1 + t)] ≤ 〈x∗,wt〉 ≤ 1,

the first inequality being due to (1.66) and (1.68) and the second to the inequality‖wt‖ ≤ 1. Consequently we have

〈x∗,wt〉 → 1 = 〈x∗, h〉

and then since the dual norm ‖ · ‖ of X∗ is Frechet differentiable at the point x∗ of theunit sphere of X∗ and since ‖wt‖ ≤ 1 and ‖h‖ = 1, Smulian lemma (see, e.g., Fabian,Habala, Hajek, Montesinos, Pelant and Zizler [71, Lemma 8.4]) says that ‖wt−h‖ → 0as t ↓ 0, which is equivalent to ‖vt‖ → 0.

For positive t sufficiently small, the functional u∗t := ‖x + tht − ut‖−1J(x + tht − ut)

is by definition a unit functional in NPC(ut) and 〈u∗t , x + tht − ut〉 = ‖x + tht − ut‖, which

is equivalent, by the equality ut = x + tvt, to 〈u∗t , ht − vt〉 = ‖ht − vt‖. Since ‖vt‖ → 0and ht → h, we derive that 〈u∗t , ht〉 → ‖h‖ = 1 and hence

〈u∗t , h〉 → 1 = 〈x∗, h〉.

Remembering that ‖u∗t ‖ = 1 and ‖x∗‖ = 1 and that the norm is Frechet differentiableat the point h of the unit sphere of X, Smulian lemma again entails that ‖u∗t − x∗‖ → 0as t ↓ 0. The proof is then complete because obviously one also has ‖ut − x‖ → 0 ast ↓ 0. �

Taking the latter proposition and (1.63) into account, we see (as in Ioffe [89])that the Mordukhovich limiting normal cone above coincides with the (sequential)limiting normal cone obtained as above by replacing NP

C(·) with NFC (·), i.e.,

(1.69) NLC(x) = w∗−seq Lim sup

y→xNF

C (y).

(We must emphasize that one of the important advantages of the expression of NLC(x)

in the form of (1.69) is that it makes available the Mordukhovich normal cone in themore general context of Asplund space, see Mordukhovich [124], Mordukhovich andShao [126]).

We will also need in Subsection 1.4.3 below the strong outer limit

(1.70) NL,sC (x) := Lim sup

y→xNP

C(y) := {lim x∗n : x∗n ∈ NPC(xn), xn ∈ C → x}


for x ∈ C. From Proposition 1.4.5 we also have NL,sC (x) = Lim sup

y→xNF

C (y).

The concept of tangential regularity (see Clarke [44]) involved below is relatedto the Bouligand and Clarke tangent cones. A vector v of X is in the Bouligandtangent cone (or contingent cone) KC(x) to C at x ∈ C if there exists a sequence ofpositive numbers (tn)n converging to 0 and a sequence (vn)n of X converging to vsuch that x + tnvn ∈ C for all n. The Clarke tangent cone TC(x) can be also definedin a sequential way. A vector v ∈ TC(x) provided that for any sequence (xn)n in Cconverging to x and for any sequence of positive numbers (tn)n converging to 0 thereexists a sequence (vn)n of X converging to v with xn + tnvn ∈ C for all n. One alwayshas the inclusion TC(x) ⊂ KC(x). When TC(x) ≡ KC(x), one says that the set C isClarke or tangentially regular at x.

Through the Clarke tangent cone, the Clarke normal cone NClC (x) of C at x ∈ C is

defined as the negative polar of the latter, that is,

NClC (x) = {x∗ ∈ X∗ : 〈x∗, v〉 ≤ 0, ∀v ∈ TC(x)}.

In any Asplund space (hence in particular in our context), we have (see Mor-dukhovich [124], Mordukhovich and Shao [126])

(1.71) NClC (x) = co∗( w∗−seq Lim sup

y→xNF

C (y) ),

where co∗ denotes the weak star closed convex hull.When the Mordukhovich limiting normal cone of C at x coincides with the Frechet

(resp. proximal) normal cone at x, one says that C is normally regular at x with re-spect to the Frechet (resp. the proximal) normal cone. Obviously the normal regularitywith respect to the proximal normal cone implies the normal regularity with respectto the Frechet one (because of the inclusion NP

C(x) ⊂ NFC (x)).

We recall (see Bounkhel and Thibault [33]) that any one of the two above nor-mal regularities entails the (Clarke) tangential regularity. We refer to Bounkhel andThibault [33] for the development of a detailed comparison between the above con-cepts of normal and tangential regularities and various others.

The next theorem is one among the results at the heart of this subsection.

Theorem 1.4.6. Assume that the closed set C is prox-regular at x ∈ C. Then thereexists a neighbourhood U of x such that for any x ∈ U ∩ C one has the followingnormal regularity

NPC(x) = NF

C (x) = NLC(x) = NCl

C (x)

and hence∂pdC(x) = ∂FdC(x) = ∂LdC(x) = ∂CdC(x),

that is, the distance function itself is subdifferentially regular at all points of U ∩C.


So, the set C is in particular tangentially regular at x ∈ U ∩ C. Further, one hasNβ

C(x) ⊂ NPC(x) for β = 2.

Proof. By assumption, there exist positive real numbers ε, r with ε < 1/2 such that forevery x ∈ C∩B(x, ε) and every x∗ ∈ NP

C(x) with ‖x∗‖ ≤ 1, we have ‖x + tJ∗(x∗)− x′‖ ≥‖tJ∗(x∗)‖ for any x′ ∈ C ∩ B(x, ε) and t ∈]0, r]. Fix any x, x′ ∈ B(x, ε) ∩ C andx∗ ∈ NP

C(x) with ‖x∗‖ ≤ 1. Then

(1.72) ‖x + tJ∗(x∗) − x′‖2 ≥ ‖tJ∗(x∗)‖2 for any t ∈]0, r].

On the other hand for any t ∈]0, r], according to the Frechet differentiability of ‖ · ‖2,we have

‖x + tJ∗(x∗) − x′‖2

= ‖tJ∗(x∗)‖2 + 2∫ 1

0〈J[tJ∗(x∗) + θ(x − x′)], x − x′〉 dθ

= ‖tJ∗(x∗)‖2 + 2t〈x∗, x − x′〉 + 2∫ 1

0〈J[tJ∗(x∗) + θ(x − x′)] − J[tJ∗(x∗)], x − x′〉 dθ.

Combining this with (1.72) we obtain

(1.73) 〈x∗, x′ − x〉 ≤1t‖x′ − x‖

∫ 1

0‖J[tJ∗(x∗) + θ(x − x′)] − J[tJ∗(x∗)]‖ dθ.

In the properties of the duality mapping J recalled in the beginning of the section wesaw that J is uniformly continuous over bounded subsets of X. Therefore, denotingby ωr+1 the modulus of uniform continuity of J over the bounded set (r + 1)B, that is,

ωr+1(τ) := sup{‖J(u) − J(u′)‖ : u, u′ ∈ (r + 1)B, ‖u − u′‖ ≤ τ} for τ > 0,

we have ωr+1(τ)−→τ↓0

0 and (1.73) entails

(1.74) 〈x∗, x′ − x〉 ≤1t‖x′ − x‖ωr+1(‖x − x′‖).

Fix now x ∈ C ∩ B(x, ε) and x∗ ∈ NLC(x) and fix also any η > 0. Let x∗n ∈ NP

C(xn)

(see (1.63)) such that xn ∈ C −→n→∞

x and x∗n w∗−→n→∞

x∗. Choose a real number γ > 0

such that ‖x∗n‖ ≤ γ for all integers n and choose a positive α < ε − ‖x − x‖ such thatωr+1(τ) ≤ rη

γfor all positive τ < α. Take any x′ ∈ B(x, α) ∩ C. We have x′ ∈ B(x, ε)

and, for n large enough, xn ∈ B(x, ε) ∩ C and ‖x′ − xn‖ < α. By (1.74) for n largeenough we then have

〈x∗n, x′ − xn〉 ≤

1r‖x∗n‖ ‖x

′ − xn‖ωr+1(‖x′ − xn‖) ≤1rγ‖x′ − xn‖

rηγ


and hence passing to the limit for n→ ∞ we obtain

〈x∗, x′ − x〉 ≤ η‖x′ − x‖.

The latter inequality being true for all x′ ∈ B(x, α) ∩ C, it means that x∗ ∈ NFC (x).

So far we have proved that for any x ∈ B(x, ε) ∩ C, we have NLC(x) ⊂ NF

C (x). As thereverse inclusion is also true according to (1.69), we have in fact the equality

(1.75) NLC(x) = NF

C (x) for all x ∈ B(x, ε) ∩C.

Moreover, by (1.71), we know that NClC (·) = co∗(NL

C)(·), where co∗ denotes the weak∗

closed convex hull in X∗. Since NFC (x) is convex and (strongly) closed (see Bounkhel

and Thibault [33]), we deduce from the equality (1.75) that we even have

(1.76) NFC (x) = NL

C(x) = NClC (x) for all x ∈ B(x, ε) ∩C.

Let us now prove that the three cones in (1.76) are also equal to the cone of proxi-mal normal functionals to C. In our uniformly convex setting where the norm ‖·‖ of Xis uniformly convex and uniformly smooth, we know according to Proposition 1.4.5that for any x ∈ B(x, ε) ∩ C, x∗ ∈ NF

C (x) with ‖x∗‖ < 1, there exists xn −→n→∞

x with

xn ∈ C, and x∗n ∈ NPC(xn) such that x∗n ‖.‖

−→n→∞

x∗. For n large enough we have xn ∈ B(x, ε)

and ‖x∗n‖ < 1. For any such integer n, for any t ∈]0, r] and x′ ∈ B(x, ε) ∩ C, we haveby (1.72)

‖xn + tJ∗(x∗n) − x′‖2 ≥ ‖tJ∗(x∗n)‖2

which gives, by passing to the limit and by the continuity of J∗,

‖x + tJ∗(x∗) − x′‖2 ≥ ‖tJ∗(x∗)‖2,

i.e, x∗ ∈ NPC(x) thanks to the local character (1.57) of primal proximal normal vector.

So for any fixed x ∈ B(x, ε) ∩ C we obtain that NFC (x) ⊂ NP

C(x) ⊂ NLC(x), which

combined with (1.76) gives the equalities

NPC(x) = NF

C (x) = NLC(x) = NCl

C (x).

These equalities also ensure

∂CdC(x) ⊂ NClC (x) ∩ B∗ = NP

C(x) ∩ B∗ = ∂pdC(x),

the last equality being due to Proposition 1.3.2 (see also Clarke [44] for the firstinclusion). Consequently, we have

∂pdC(x) = ∂FdC(x) = ∂LdC(x) = ∂CdC(x).

Finally, on the one hand the equality between NFC (x) and NCl

C (x) ensures that C istangentially regular at x (see Bounkhel and Thibault [33]), and on the other hand, forβ = 2, since one always has Nβ

C(x) ⊂ NFC (x), the inclusion Nβ

C(x) ⊂ NPC(x) follows. �


We deduce from Theorem 1.4.6 a proximal normal formula; Lim sup below is likein (1.70) the strong outer limit, i.e., for a multivalued mapping T : X ⇒ X∗

Lim supy

y∈D→ x

T (y) := {lim y∗n : y∗n ∈ T (yn), yn ∈ D→ x}.

Proposition 1.4.7. Assume that the moduli of uniform convexity and of uniformsmoothness of the norm ‖ · ‖ of X are of power type. If C is prox-regular at x ∈ X,then there exists a neighbourhood U of x such that for any x ∈ U ∩C,

NPC(x) = NCl

C (x) = Lim supy

y<C→ x

R+∇FdC(y)

or equivalently,∂pdC(x) = ∂CdC(x) = [0, 1] Lim sup

yy<C→ x

∇FdC(y),

where ∇F denotes the Frechet derivative and ∂C is the Clarke subdifferential.

Proof. Remember that the prox-regularity of C entails the Frechet differentiability ofd2

C and the single-valuedness and continuity of PC on an open neighbourhood U ofx, see Theorem 1.3.25. Fix any x ∈ U ∩C.

Let us prove that NPC(x) ⊂ Lim sup

yy<C→ x

R+∇FdC(y).

Recall first that a nonzero continuous linear functional x∗ is in NPC(x) if and only

if there exists u ∈ U \C such that x ∈ PC(u) and x∗ = λJ(u− x) for some λ > 0. Thenfor any y ∈ U ∩ { tu + (1 − t)x : t ∈]0, 1] }, one has

J(u − x) ∈ R+J(y − x) = R+∇FdC(y)

and thus for n large enough x∗ ∈ R+∇FdC(yn) with yn := x + (u − x)/n, which proves

the desired inclusion.Now for the reverse inclusion, note that for any y ∈ U \C,

R+∇FdC(y) = R+J(y − PC(y)) ⊂ NP

C(PC(y)).

Making y → x, we have by continuity of PC that PC(y) → x and hence,x∗ ∈ Lim sup

yy<C→ x

R+∇FdC(y) entails that x∗ ∈ Lim sup

x′→xNP

C(x′) = NPC(x), the last equal-

ity coming from the normal regularity established in Theorem 1.4.6. So the reverseinclusion is proved and of course, by normal regularity with respect to the proximalnormal cone (see Theorem 1.4.6) we obtain NCl

C (x) = NPC(x).

The equivalent formulation with the subdifferential of dC comes from the equality∂pdC(x) = NP

C(x)∩B∗ (see Proposition 1.3.2) and from the equality ∂pdC(x) = ∂CdC(x)in Theorem 1.4.6. �


Remark 1.4.8. The proximal normal formula of the above proposition is more precisethan the more general one found in Borwein and Giles [25, Theorem 4] for any closedset C in a reflexive Banach space with weaker assumptions on the norm.

1.4.2 Epigraphs of J-primal lower regular functions

The following proposition and its proof essentially reproduces ideas of Bernardand Thibault from [17, Proposition 4.8].

Proposition 1.4.9. Assume that the moduli of uniform convexity and uniform smooth-ness of the norm ‖ · ‖ of X are of power type. If a lower semicontinuous function fis J-plr at x ∈ dom f , then its epigraph epi f is prox-regular at (x, f (x)).

We will need the following lemma, given in Clarke, Ledyaev, Stern and Wolen-ski [45, Exercise 2.1 (d)] in the Hilbert setting. The proof is essentially the same inour general context (without the power types of the norm), but the proximal normalcone is no more identical to Nβ

C(·) with β = 2.

Lemma 1.4.10. For any x ∈ dom f and α > f (x), the implication

(x∗, 0) ∈ NPepi f (x, α)⇒ (x∗, 0) ∈ NP

epi f (x, f (x))

holds true.

Proof. Recall that by convention the square of the norm in X × R is defined as|||(x, r)|||2 = ‖x‖2 + r2. Let (x∗, 0) ∈ NP

epi f (x, α). This means that for all positive t closeenough to 0,

(1.77) inf(x′,α′)∈epi f

{‖x + tJ∗(x∗) − x′‖2 + (α′ − α)2

}= ‖tJ∗(x∗)‖2.

Choose any δ > 0 with δ < α− f (x). Fix any (x′, α′) ∈ epi f ∩ B((x, f (x)), δ). We haveα′ − f (x) < δ < α − f (x), hence α′ < α so (x′, α) ∈ epi f . From (1.77) we obtain

(1.78) ‖x + tJ∗(x∗) − x′‖2 + (α − α)2 ≥ ‖tJ∗(x∗)‖2,

soinf

(x′,α′)∈epi f∩B((x, f (x)),δ)

{‖x + tJ∗(x∗) − x′‖2 + (α′ − f (x))2

}≥

inf(x′,α′)∈epi f∩B((x, f (x)),δ)

{‖x + tJ∗(x∗) − x′‖2

}≥ ‖tJ∗(x∗)‖2,

the last inequality being due to (1.78). Since the concept of proximal normal is local(see (1.57)), we conclude that (x∗, 0) ∈ NP

epi f (x, f (x)). �


Proof of Proposition 1.4.9. By definition of J-plr function (see Definition 1.4.4),there exist Θ, ε, r > 0 such that for any t ≥ Θ, any x ∈ B(x, ε) and x∗ ∈ ∂p f (x) with‖x∗‖ ≤ rt,

(1.79) f (y) ≥ f (x) + 〈J[J∗(x∗) − t(y − x)], y − x〉 for all y ∈ B(x, ε).

Take (x, α) ∈ epi f and (x∗,−λ) ∈ NPepi f (x, α) where ‖x − x‖ < ε, |α − f (x)| < ε and

‖(x∗,−λ)‖ ≤ 1. Fix any (x′, α′) ∈ epi f with x′ ∈ B(x, ε).If λ > 0, then α = f (x) and λ−1x∗ ∈ ∂p f (x) with ‖λ−1x∗‖ ≤ 1/λ ≤ rt for everyt ≥ max(Θ, 1/(λr)). Hence by (1.79)

α′ ≥ f (x′) ≥ f (x) + 〈J[J∗(λ−1x∗) − t(x′ − x)], x′ − x〉

which entails

0 ≥ 〈J[J∗(x∗) − λt(x′ − x)], x′ − x〉 − λ(α′ − α) − λt(α′ − α)2.

For any t′ ≥ max(Θ, 1/r), since t′ ≥ max(λΘ, 1/r) because 1 ≥ λ, the latter inequalitywith t = t′/λ, according to (1.56), implies

(1.80) 0 ≥ 〈JX×R[J∗X∗×R(x∗,−λ) − t′((x′, α′) − (x, α))], (x′, α′) − (x, α)〉.

If λ = 0, we get by Lemma 1.4.10 that (x∗, 0) ∈ NPepi f (x, f (x)). Since NP(·) ⊂ NF(·), by

the approximation result in Ioffe [89, p. 190] (see also Mordukhovich [124, Lemma2.37]) there exist sequences (un, f (un)) → (x, f (x)), (u∗n,−λn) ∈ NF

epi f (un, f (un)) suchthat λn > 0 and ‖(u∗n,−λn) − (x∗, 0)‖ → 0. By Proposition 1.4.5, for each integer nchoose (xn, αn) ∈ epi f and then (y∗n,−µn) ∈ NP

epi f (xn, αn) such that

‖(un, f (un)) − (xn, αn)‖ < λn/2 and ‖(y∗n,−µn) − (u∗n,−λn)‖ < λn/2.

The latter inequality ensures in particular µn > (λn/2) > 0 and hence αn = f (xn).Consequently for x∗n := µ−1

n y∗n we have x∗n ∈ ∂P f (xn), (xn, f (xn)) → (x, f (x)), µn ↓ 0,and ‖µnx∗n− x∗‖ → 0. For n large enough, say n ≥ N, we have ‖xn− x‖ < ε and µn < 1.Suppose for a moment that x∗ , 0. Putting

tn := max{

1µn

Θ,1‖x∗‖‖x∗n‖

r

}for n ≥ N, we see that tn ≥ max{Θ, ‖x∗n‖/r} since 1/µn > 1 and 1/‖x∗‖ ≥ 1, and henceapplying (1.79) with y = x′ we have

f (x′) ≥ f (xn) + 〈J[J∗(x∗n) − tn(x′ − xn)], x′ − xn〉.

Multiplying this inequality by µn and taking the limit we obtain

0 ≥ 〈J[J∗(x∗) − ρ(x′ − x)], x′ − x〉


for ρ := max{Θ, 1/r}. This yields in particular

0 ≥ 〈J[J∗(x∗) − ρ(x′ − x)], x′ − x〉 − ρ(α′ − α)2,

i.e.,

(1.81) 0 ≥ 〈JX×R[J∗X∗×R(x∗, 0) − ρ((x′, α′) − (x, α))], (x′, α′) − (x, α)〉,

and it is obvious that the inequality continues to hold for x∗ = 0. Hence both (1.80)and (1.81) hold with t′ = ρ, which entails that the truncated set-valued mapping NP 1

epi fis J-hypomonotone of degree ρ on B(x, ε′) and hence by Theorem 1.3.25, the setepi f is prox-regular at (x, f (x)). �

Remembering that the Clarke subdifferential operator ∂C and the Mordukhovichlimiting subdifferential ∂L satisfy, for any lower semicontinuous function f (seeClarke [44], Mordukhovich [124])

∂C f (x) = {x∗ : (x∗,−1) ∈ NClepi f (x, f (x))} and ∂L f (x) = {x∗ : (x∗,−1) ∈ NL

epi f (x, f (x))},

from Theorem 1.4.6 we have the following

Corollary 1.4.11. Assume that the moduli of uniform convexity and smoothnessof the norm ‖ · ‖ of X are of power type. If f is J-plr at x, then there exists aneighbourhood U of x such that for β = 2 one has

∂β f (x) ⊂ ∂p f (x) = ∂F f (x) = ∂L f (x) = ∂C f (x) for all x ∈ U.

By considering the following property of f being uniformly J-plr, we obtainsimilarly the uniform prox-regularity of epi f . The function f is uniformly J-plr onX provided there exist positive constants Θ, r such that for any t ≥ Θ, any x ∈ X andx∗ ∈ ∂p f (x) with ‖x∗‖ ≤ rt,

f (y) ≥ f (x) + 〈J[J∗(x∗) − t(y − x)], y − x〉 for all y ∈ X.

Proposition 1.4.12. Assume that the moduli of uniform convexity and smoothness ofthe norm ‖ · ‖ of X are of power type. If f is uniformly J-plr on X with parametersr, Θ, then epi f is uniformly r′-prox-regular for some r′ ≥ 1

2 min{1/Θ, r}.

Proof. According to the proof of Proposition 1.4.9, the truncated set-valued mappingNP 1

epi f is J-hypomonotone of degree ρ = max{Θ, 1/r}. With δ = ρ−1, that fact isequivalent to NP δ

epi f being J-hypomonotone of degree 1 which by Proposition 1.3.31allows us to conclude. �

That proposition allows us to establish the J-plr property of the basic functionequal to the opposite of the square of the norm whenever the latter has moduli ofconvexity and smoothness of power type.


Proposition 1.4.13. Assume that the moduli of uniform convexity and smoothness ofthe norm ‖ · ‖ of X are of power type. Then the opposite function of the square ofthe norm, say −‖ · ‖2, is uniformly J-plr on X and hence in particular epi (−‖ · ‖2) isuniformly r-prox-regular for some r ≥ 1

4 .

Proof. For any x ∈ X, y = x + h ∈ X, x∗ = ∂(−‖ · ‖2)(x) = −2J(x), by the convexity of‖ · ‖2, we may write

‖x‖2 = ‖y − h‖2 ≥ ‖y‖2 + 〈2J(y),−h〉.

So, we have

−‖y‖2 ≥ −‖x‖2 + 〈−2J(y), h〉

which yields, according to the equality −2y = J∗(x∗) − 2(y − x), that

−‖y‖2 ≥ −‖x‖2 + 〈J[J∗(x∗) − 2(y − x)], y − x〉.

This entails the J-hypomonotonicity of degree 2 of NP 1C where C := epi (−‖.‖2), hence

the conclusion follows from Proposition 1.3.31 and Proposition 1.4.12.

In order to compare J-plr functions with primal lower-nice functions, we supposein Proposition 1.4.14 and Corollary 1.4.15 below that the modulus of uniform con-vexity of the norm is of power type q = 2. Let us recall the definition of primal lowernice functions, introduced by Poliquin in [136] in Rn is studied in the Hilbert settingwith further developments by Levi, Poliquin and Thibault in [111].

Below we will consider an presubdifferential operator ∂ that associates with eachfunction f : X → R∪ {+∞} a multivalued mapping ∂ f : X ⇒ X∗ and satisfies variousproperties commonly fulfilled by the usual subdifferentials on appropriate spaces (seeDefinition 2.1.1 and the discussion after it). Here we will just assume that for anyfunction f from X into R ∪ {+∞}, one has

(1.82) ∂p f (x) ⊂ ∂ f (x) for all x ∈ X.

Let us recall that a lower semicontinuous function f is ∂-pln at x ∈ dom f (seeDefinition 1.1.4) if there are ε, c,Θ > 0 such that whenever t ≥ Θ, x∗ ∈ ∂ f (x) with‖x − x‖ < ε and ‖x∗‖ ≤ ct, one has

f (x′) ≥ f (x) + 〈x∗, x′ − x〉 −t2‖x′ − x‖2 for all x′ ∈ B(x, ε).

Obviously, for β = 2 one has ∂ f (·) ∩ ctB∗ ⊂ ∂β f (·) on B(x, ε) for such a function f .Examples of subdifferential operators that contain the proximal subdifferential

operator ∂p are given by ∂p itself, by ∂F , and hence also by any of the many subdif-ferential operators that contain ∂F .


Proposition 1.4.14. Assume that the modulus of uniform convexity of the norm ‖ · ‖of X is of power type q = 2. If f is ∂-pln at x for ∂(·) ⊃ ∂p(·), then f is J-plr at x.

Proof. Under the assumptions, there exist positive numbers ε, r,Θ such that for any(x, x∗) ∈ gph ∂p f with ‖x − x‖ < ε, ‖x∗‖ ≤ rt and t ≥ Θ,

(1.83) f (x′) ≥ f (x) + 〈x∗, x′ − x〉 −t2‖x′ − x‖2 for all x′ ∈ B(x, ε).

Further, for any x∗ ∈ X∗, h ∈ X, t′ > 0 we may write

〈x∗, h〉 = 〈J[J∗(x∗)] − J[J∗(x∗) − t′h], h〉 + 〈J[J∗(x∗) − t′h], h〉

≥Kt′‖t′h‖2 + 〈J[J∗(x∗) − t′h], h〉,

the inequality being due to (1.52) with q = 2. Hence taking h = x′ − x, we have forany t′ > 0,

〈x∗, x′ − x〉 − Kt′‖x′ − x‖2 ≥ 〈J[J∗(x∗) − t′(x′ − x)], x′ − x〉.

Combining the latter inequality and (1.83), we see that for any t′ ≥ Θ/(2K), (x, x∗) ∈gph ∂p f with ‖x − x‖ < ε and ‖x∗‖ ≤ (2Kr)t′, we have

f (x′) ≥ f (x) + 〈J[J∗(x∗) − t′(x′ − x)], x′ − x〉 for all x′ ∈ B(x, ε),

that is, f is J-plr with parameters ε, 2rK,Θ/(2K). �

Corollary 1.4.15. Assume that the moduli of uniform convexity and smoothness ofthe norm ‖ · ‖ of X are of power type. Under the inclusion ∂ f (·) ⊂ ∂C f (·) and theassumption of Proposition 1.4.14, the lower semicontinuous function f is ∂-pln at xif and only if f is ∂p-pln at x and then for β = 2 one has

∂β f (x) = ∂p f (x) = ∂ f (x) = ∂F f (x) = ∂L f (x) = ∂C f (x)

for all x in a neighbourhood of x.

Proof. Suppose that f is ∂-pln at x. On the one hand, by (1.82), it is ∂p-pln at x. Onthe other hand, by definition of ∂-pln property the inclusion ∂ f (x) ⊂ ∂β f (x) holds forβ = 2 and for x in a neighbourhood of x, and by Proposition 1.4.14, the function f isalso J-plr at x. Therefore using (1.82) and Corollary 1.4.11 we obtain the equalitiesof the corollary.

Conversely, if f is ∂p-pln at x, then Proposition 1.4.14 again entails that f is J-plrat x and hence by Proposition 1.4.9 and Theorem 1.4.6 we have ∂p f (x) = ∂C f (x) forall x in some neighbourhood of x. Combining the latter equality with (1.82) and theinclusion assumption ∂ f (·) ⊂ ∂C f (·) we obtain that ∂ f (x) = ∂p f (x) for all x near x.So f is ∂-pln at x. �


We have a symmetrical result when the modulus of smoothness is s = 2.

Proposition 1.4.16. Assume that the moduli of uniform convexity and uniformsmoothness of the norm ‖ · ‖ are of power type and that the power type of smoothnessis s = 2. If f is J-plr at x, then f is ∂-pln at x for any subdifferential ∂ satisfying theinclusions ∂p f (·) ⊂ ∂ f (·) ⊂ ∂C f (·).

Proof. By Corollary 1.4.11, for all x near x we have ∂C f (x) = ∂p f (x) and hence∂p f (x) = ∂ f (x). The J-plr assumption then entails that there exist positive numbersε, r,Θ such that for any t ≥ Θ, (x, x∗) ∈ gph ∂p f with ‖x − x‖ < ε, ‖x∗‖ ≤ rt,

(1.84) f (x′) ≥ f (x) + 〈J[J∗(x∗) − t(x′ − x)], x′ − x〉 for all x′ ∈ B(x, ε).

Fix now any t ≥ Θ, x′, x ∈ B(x, ε) and x∗ ∈ ∂ f (x) with ‖x∗‖ ≤ rt. By (1.53) sinces = 2 (the power type of smoothness of the norm), we have

〈J[J∗(x∗) − t(x′ − x)] − J[J∗(x∗)],−t(x′ − x)〉 ≤ L‖t(x′ − x)‖2,

which is equivalent to

〈J[J∗(x∗) − t(x′ − x)], x′ − x〉 ≥ 〈x∗, x′ − x〉 − tL‖x′ − x‖2.

Then according to (1.84) we obtain

f (x′) − f (x) ≥ 〈x∗, x′ − x〉 − tL‖x′ − x‖2.

So f is ∂-pln at x. �

Corollary 1.4.17. Assume that the moduli of uniform convexity and smoothness ofthe norm ‖ · ‖ of X are of power type and that the power type of smoothness is s = 2.Under the assumption of Proposition 1.4.16, if the lower semicontinuous function fis J-plr at x then for β = 2

∂β f (x) = ∂p f (x) = ∂ f (x) = ∂F f (x) = ∂L f (x) = ∂C f (x)

for all x in a neighbourhood of x.

Proof. This a consequence of Corollary 1.4.11 and Proposition 1.4.16. �

Uniformly ∂ primal lower-nice functions are defined analogously to uniformlyJ-plr functions, i.e., f is uniformly ∂-pln if there are c,Θ > 0 such that whenevert ≥ Θ, x∗ ∈ ∂ f (x) with ‖x∗‖ ≤ ct,

f (x′) ≥ f (x) + 〈x∗, x′ − x〉 −t2‖x′ − x‖2 for all x′ ∈ X.

The two previous results above proved for ∂-pln functions at a point x of X also holdfor uniformly ∂-pln functions: just replace “∂-pln (resp. J-plr) at x ” with “uniformly∂-pln (resp. uniformly J-plr)”.


1.4.3 Prox-regularity and N-hyporegularity

Let N(·) be a given normal cone concept (e.g., NF(·), NP(·), etc.) associated witha subdifferential operator ∂; i.e., for any closed set C with indicator function ψC, onehas NC = ∂ψC and ∂ f (x) = {x∗ ∈ X∗ : (x∗,−1) ∈ Nepi f (x, f (x))}.

The normal cone N(·) is assumed to satisfy, for any closed set C, the inclusionNP

C(·) ⊂ NC(·).Following Poliquin and Rockafellar [137] and the adaptation in Bernard and

Thibault [14], the set C ⊂ X is N-hyporegular at x ∈ C if there exist ε, r > 0such that for any x∗ ∈ NC(x) with ‖x − x‖ < ε, ‖x∗‖ ≤ 1,

(1.85) 0 ≥ 〈x∗, x′ − x〉 −12r‖x′ − x‖2 for all x′ ∈ B(x, ε) ∩C.

Of course, (1.85) holds if and only if the truncated normal cone NC(·) ∩ B∗ is hy-pomonotone near x in the usual sense, that is, there exist some ε, r > 0 such that forall xi ∈ C ∩ B(x, ε) and x∗i ∈ NC(xi) ∩ B∗, i = 1, 2 one has

(1.86) 〈x∗1 − x∗2, x1 − x2〉 ≥ −1r‖x1 − x2‖

2.

In Bernard and Thibault [14] a set C satisfying (1.85) with N(·) has been calledprox-regular with respect to the normal cone N(·). Here we prefer to use the nameof N-hyporegular set because of the characterization (1.86) as the hypomonotonicityof NC(·) ∩ B∗ (which has nothing to do with the metric projection mapping) and toreserve the name of prox-regularity set merely to translate the regularity property ofthe metric projection mapping in Definition 1.4.2 or in (c) of Theorem 1.3.25.

We observe by (1.85) that C is N-hyporegular at x if and only if its indicatorfunction ψC is ∂-pln at x. Remember (see Theorem 1.3.25 (l)) that C is prox-regularat x if and only if ψC is J-plr at x. Similarly, the uniform N-hyporegularity means thatthe inequality (1.85) holds for all x, x′ ∈ C and x∗ ∈ NC(x)∩B∗ and this corresponds tothe uniform ∂-pln property for the indicator function ψC. In the same way, the uniformprox-regularity of C is equivalent to the uniform J-plr property of the function ψC.Remember that q and s are the moduli of convexity and of smoothness respectively,and that they satisfy 1 < s ≤ 2 ≤ q. The case q = 2 corresponds for instance to Lp

spaces with p ∈]1, 2] and s = 2 to Lp spaces with p ∈ [2,+∞[.In the case q = 2, we have from results in the previous subsection the following

result.

Corollary 1.4.18. Assume that the moduli of uniform convexity and smoothness ofthe norm ‖ · ‖ of X are of power type. When the power type of the modulus ofconvexity is q = 2, a closed set C ⊂ X is N-hyporegular at x ∈ C (resp. uniformlyN-hyporegular) for a normal cone N which satisfies NP(·) ⊂ N(·) ⊂ NCl(·) if and


only if it is NP-hyporegular at x (resp. uniformly NP-hyporegular), and then it isalso prox-regular at x (resp. uniformly prox-regular) and for β = 2 one has Nβ

C(x) =

NPC(x) = NC(x) = NCl

C (x) for any x in some neighbourhood of x (resp. any x ∈ C).

Proof. It is a direct consequence of Proposition 1.4.14 and Corollary 1.4.15. �

In the case s = 2 we have the reverse implication. If the moduli of uniformconvexity and smoothness are of power type with the power type of smoothnesss = 2, then its proof directly follows from Proposition 1.4.16 and Corollary 1.4.17. Infact the proof below shows that the result holds without requiring that the modulusof uniform convexity be of power type.

Proposition 1.4.19. Assume that the modulus of smoothness of the norm ‖ · ‖ of X isof power type s = 2. If a closed set C ⊂ X is prox-regular at x ∈ C (resp. uniformlyprox-regular), then it is N-hyporegular at x (resp. uniformly N-hyporegular) for anynormal cone N which satisfies NP(·) ⊂ N(·) ⊂ NCl(·), and further for β = 2 one has

NβC(x) = NP

C(x) = NC(x) = NClC (x)

for any x in some neighbourhood of x (resp. any x ∈ C).

Proof. We prove the result when we have local prox-regularity at a point. The prox-regularity assumption entails by Proposition 1.3.4 that there exist positive numbersε, r such that for any x ∈ B(x, ε) ∩C and for any x∗ ∈ NP

C(x) with ‖x∗‖ ≤ 1,

(1.87) 0 ≥ 〈J[J∗(x∗) − r−1(x′ − x)], x′ − x〉, ∀x′ ∈ C with ‖x′ − x‖ < ε.

Fix now any x′, x ∈ C ∩ B(x, ε) and x∗ ∈ NPC(x) with ‖x∗‖ ≤ 1. By (1.53) since s = 2

(the power type of smoothness of the norm), we have

〈J[J∗(x∗) − r−1(x′ − x)] − J[J∗(x∗)],−r−1(x′ − x)〉 ≤ L‖r−1(x′ − x)‖2,


〈J[J∗(x∗) − r−1(x′ − x)], x′ − x〉 ≥ 〈x∗, x′ − x〉 − Lr−1‖x′ − x‖2.

Then according to (1.87) we obtain

0 ≥ 〈x∗, x′ − x〉 − Lr−1‖x′ − x‖2.

So the set C is also NP-hyporegular at x.Further, putting β = 2, on the one hand the NP-hyporegularity ensures us that for

an appropriate neighbourhood U of x, for any x ∈ U∩C, the inclusion NPC(x) ⊂ Nβ

C(x)holds, and on the other hand by Theorem 1.4.6, we know that Nβ

C(x) ⊂ NPC(x) = NCl

C (x).Since by assumption we have the inclusions NP

C(·) ⊂ NC(·) ⊂ NClC (·), we obtain that

for β = 2Nβ

C(x) = NPC(x) = NC(x) = NCl

C (x) for all x ∈ U ∩C

and consequently, C is also N-hyporegular. �


For a family of norms (‖ · ‖i)i∈I , let us denote by N?i the normal cone N? (e.g., NF ,

NCl, NP, etc.) obtained by using the norm ‖ · ‖i in its definition and computation. TheNP

i (·) cones for a closed set C will be denoted by NPi (C; .). When the norms ‖ · ‖i and

‖ · ‖ j are equivalent, some normal cone concepts N? yield N?i = N?

j , as in the casesN? = NF ,Nβ,NL,s,NL,NCl for instance. We will speak of (‖ · ‖i,N)-hyporegularity ofa set C when the property (1.85) is satisfied with the norm ‖ · ‖ = ‖ · ‖i and with thecone N = N?

i . When C is (‖ · ‖i,N?)-hyporegular for any i ∈ I, we will say that Cis N?-hyporegular relatively to the family (‖ · ‖i)i∈I . Similarly, the concept of prox-regularity in Definition 1.4.2 is a priori norm dependent, so for a given norm ‖ · ‖ ithas been said in that definition that the set C is ‖ · ‖-prox-regular.

From now on in the remaining of this subsection we address the problem ofcomparing the (‖ · ‖i,N?)-hyporegularities and ‖ · ‖i-prox-regularities of a set C fornorms in a given family (‖ · ‖i)i∈I , and of comparing accordingly the normal conesNP

i (C; .) for such a set C.Note first that by passing to the limit in (1.85) and by using the comments pre-

ceding and following Proposition 1.4.5, it is easily seen, for a norm ‖ · ‖ which is bothuniformly convex and uniformly smooth, that whenever C is (‖ · ‖,N?)-hyporegular ata point x with N? = NP or N? = NF , then it is (‖ · ‖,NL,s)-hyporegular at x with thesame parameters. Further for some neighbourhood U of x and for β = 2 the inclusionNL

C(x) ⊂ NβC(x) holds for all x ∈ U and hence, Nβ

C(x) = NFC (x) = NL

C(x) = NClC (x) and

C is (‖ · ‖,N?)-hyporegular at x for any N? with N?(·) ⊂ NCl(·).Suppose that (‖ · ‖i)i∈I is a family of equivalent norms which are Frechet dif-

ferentiable outside zero and such that for some given i0 ∈ I, the norm ‖ · ‖i0 isuniformly smooth and uniformly convex. From the previous remark we know that ifC is (‖ · ‖i0 ,N

P)-hyporegular at x ∈ C there exists some γ > 0 and some open neigh-bourhood U of x such that for all x, x′ ∈ U ∩ C and x∗ ∈ NL,s

i0(C; x) with ‖x∗‖i0 ≤ 1

(denoting the dual norm of ‖ · ‖i0 in the same way)

0 ≥ 〈x∗, x′ − x〉 − γ‖x′ − x0‖2i0 .

Then for the same open neighbourhood U of x, for each i ∈ I there exists some γi > 0such that for all x, x′ ∈ U ∩C and x∗ ∈ NF(C; x) with ‖x∗‖i ≤ 1 one has

0 ≥ 〈x∗, x′ − x〉 − γi‖x′ − x‖2i .

Therefore for any i ∈ I the set C is (‖ · ‖i,NP)-hyporegular at any x ∈ U ∩ C and forβ = 2 one has

(1.88) NPi (C; x) ⊂ NF

C (x) = NβC(x) = NCl

C (x) for all x ∈ U ∩C.

By specializing to certain families (‖ · ‖i)i∈I we have the next results of the sub-section. Here is the first one.


Proposition 1.4.20. Suppose that (‖ · ‖i)i∈I is a family of equivalent norms which areuniformly smooth and uniformly convex with the moduli of convexity δ‖·‖i of powertype q = 2 for all i ∈ I.

If the closed set C is (‖ · ‖i0 ,NP)-hyporegular at x ∈ C for some i0 ∈ I, then the

following hold:(a) the set C is (‖ · ‖i,N)-hyporegular at x for any i ∈ I, and for any normal cone Nsuch that N(·) ⊂ NCl(·);(b) the set C is ‖ · ‖i-prox-regular at x for any i ∈ I;(c) there exists some neighbourhood U of x such that for any x ∈ U ∩ C and anyi ∈ I and for β = 2 the equalities Nβ

C(x) = NPi (C; x) = NCl

C (x) hold, and C is (‖ · ‖i,N)-hyporegular at x for any normal cone NC(·) with Nβ

C(·) ⊂ NC(·) ⊂ NClC (·).

Proof. The first point has been seen above. The second and third ones come fromthe first one, from (1.88), and from Corollary 1.4.18. �

We specialize further, to the case of Rn endowed with the family (‖ · ‖p)p>1 ofclassical lp-norms, say ‖x‖p = (

∑pk=1 |xk|

p)1/p.

Proposition 1.4.21. Suppose that X = Rn and (‖ · ‖p)p>1 is the family of lp-norms withp > 1.1) If for some p0 > 1 the set C is (‖ · ‖p0 ,N

P)-hyporegular at x ∈ C, then(a) the set C is (‖ · ‖p,N)-hyporegular at x for any p > 1 and any normal cone N suchthat N(·) ⊂ NCl(·);(b) the set C is ‖ · ‖p-prox-regular at x for any p ∈]1, 2];(c) there is some neighbourhood U of x such that for all p ∈]1, 2] and p′ > 2 and forβ = 2 one has

NPp′(C; x) ⊂ NP

p (C; x) = NβC(x) = NCl

C (x) for any x ∈ U.

2) If for some p0 ≥ 2 the set C is ‖ · ‖p0-prox-regular at x ∈ C, then(a′) the set C is (‖ · ‖p,N)-hyporegular at x for any p > 1 and any normal cone Nsuch that N(·) ⊂ NCl(·);(b′) the set C is ‖ · ‖p-prox-regular at x for any p ∈]1, 2];(c′) there is some neighbourhood U of x such that for all p ∈]1, 2] and p′ > 2 and forβ = 2 one has

NPp′(C; x) ⊂ NP

p (C; x) = NPp0

(C; x) = NβC(x) = NCl

C (x) for any x ∈ U.

Proof. Under the assumption of the first case, the property (a) like (a) in Proposi-tion 1.4.20 has been seen ahead this latter. The second property (b) is a consequenceof (b) in Proposition 1.4.20 since for any p ∈]1, 2], the norm ‖ · ‖p has modulus ofconvexity of power type q = 2. Taking the open neighbourhood U given by (c) of


Proposition 1.4.20 for the family of norms (‖ · ‖p)p∈]1,2], only the first inclusion of thethird property (c) remains to be argued. It is a consequence of C being (‖ · ‖p′ ,NP)-hyporegular at any x ∈ U ∩ C for any p′ > 2 according to the property (a), whichentails for β = 2 the inclusion NP

p′(C; x) ⊂ NβC(x) for β = 2 and for any x ∈ U ∩C.

Now concerning the second case, suppose that for some p0 ≥ 2 the set C is‖ · ‖p0-prox-regular at x ∈ C. We first observe that Definition 1.4.2 furnishes someopen neighbourhood U of x such that C is ‖ · ‖p0-prox-regular at any point x ∈ C ∩U.Since the norm ‖ · ‖p0 is uniformly smooth with modulus of smoothness s = 2,Proposition 1.4.19 yields that C is NF-hyporegular at any x ∈ C ∩ U relatively to thefamily (‖ · ‖p)p>1 and that NP

p0(C; x) = NF

C (x) for any x ∈ U. Then for any p > 1 theset C is (‖ · ‖p,NP)-hyporegular at any x ∈ C ∩ U and then from the first case andwhat precedes, we have the second and third properties (b′) and (c′). �

1.4.4 Comparison of normal cones

The proposition of this subsection compares, for any closed subset C of X, thecone NP

C(·) of proximal normal functionals with the normal cone NβC(·) when β is the

power type of the modulus of uniform convexity or of smoothness of the norm ‖ · ‖of X.

Proposition 1.4.22. Let C be a closed subset of X and let x ∈ C.(a) If the modulus of convexity of the norm ‖ · ‖ of X is of power type q, then

NqC(x) ⊂ NP

C(x).

(b) Similarly, if the modulus of smoothness of the norm ‖ · ‖ of X is of power type s,then

NPC(x) ⊂ N s

C(x).

Proof. Suppose that the modulus of convexity of ‖ · ‖ is of power type q. From (1.50),there exists some constant K > 0 such that for all x, y ∈ X,

‖x + y‖q ≥ ‖x‖q + q〈Jq(x), y〉 + K‖y‖q.

Take x∗ ∈ NqC(x), x∗ , 0. There is some γ, ε > 0 such that for any x′ ∈ B(x, ε) ∩ C,

〈x∗, x′ − x〉 ≤ γ‖x′ − x‖q. For any x′ ∈ B(x, ε) ∩C we then have for each t > 0,

‖x + tJ−1q (x∗) − x′‖q ≥ ‖tJ−1

q (x∗)‖q + q〈Jq(tJ−1q (x∗)), x − x′〉 + K‖x − x′‖q,

that is, by the equality in (1.47)

‖x + tJ−1q (x∗) − x′‖q ≥ ‖tJ−1

q (x∗)‖q + tq−1q〈Jq(J−1q (x∗)), x − x′〉 + K‖x − x′‖q,


hence‖x + tJ−1

q (x∗) − x′‖q ≥ ‖tJ−1q (x∗)‖q − tq−1qγ‖x′ − x‖q + K‖x − x′‖q

so, whenever t ≤ (K/(ργ))1/(q−1), we obtain

‖x + tJ−1q (x∗) − x′‖q ≥ ‖tJ−1

q (x∗)‖q.

Because of the local character (1.57) of primal normal vector, the latter equalityentails that J−1

q (x∗) ∈ PNC(x), that is, J(J−1q (x∗)) ∈ NP

C(x). As J(J−1q (x∗)) ∈ R+x∗, the

first inclusion follows.To prove now (b) fix any x∗ ∈ NP

C(x) with x∗ , 0. By definition of NPC(·), there

exists δ > 0 such that for any t ∈]0, δ] and any x′ ∈ C

‖x + tJ−1(x∗) − x′‖s ≥ ‖tJ−1(x∗)‖s.

From (1.51), there exists some constant L > 0 such that, for every x, y ∈ X,

‖x + y‖s ≤ ‖x‖s + s〈Js(x), y〉 + L‖y‖s.

From the two previous estimations we derive for any x′ ∈ C

s〈Js(δJ−1(x∗)), x − x′〉 + L‖x − x′‖s ≥ 0,

that is, according to the equality in (1.47)

sδs−1〈Js(J−1(x∗)), x − x′〉 + L‖x − x′‖s ≥ 0.

As Js(J−1(x∗)) ∈ R+x∗, we deduce that there exists some σ > 0 such that

〈x∗, x′ − x〉 ≤ σ‖x′ − x‖s for all x′ ∈ C.

This entails that x∗ ∈ N sC(x) and the inclusion of (b) is proved. �

1.4.5 Preservation of hyporegularity and prox-regularity

This subsection is devoted to the study of the preservation of prox-regularity forthe intersection of finitely many sets and for the inverse image. The characterizationsof prox-regularity in Theorem 1.3.25 by the hypomonotonicity of the proximal normalcone as well as by local single valuedness and by continuity of the metric projectionmapping have been crucial in the study of Edmond and Thibault [68] of differentialinclusions of sweeping process type governed by nonconvex prox-regular sets. Theregularization of such differential inclusions in Hilbert space in Thibault [159] uses theproperty (f) of the same theorem. Below we will take advantage of the property (1.58)


Fig. 1. Intersection of prox-regular sets

to investigate the stability of metrical prox-regularity under the above set-operations.In fact we will start with the stability of NP-hyporegularity through the property(1.85) which is in the line of property (1.58).

To see first that the intersection of finitely many prox-regular sets may not beprox-regular, consider the example in R2 with its euclidian norm illustrated in Fig. 1,where the set C1 is the whole line (x1x2), {xi}i being a sequence of points on C1 thatconverges to some x ∈ C1, and the set C2 is the hatched surface delimited on one side

by the closed arc _x1x of the half circle with radius R = ‖x1 − x‖/2 and on its other

side by the arcs of the circles (all with the same radius r < R) _xixi+1 , i ∈ N, so thatx belongs also to C2. It is easily seen that while C1 and C2 are both prox-regular at x(they are even uniformly prox-regular), their intersection C1 ∩ C2 is not prox-regularat the point x.

To provide general sufficient conditions under which the prox-regularity of inter-section or inverse image is preserved, we need first to recall the concept of calmness.Translating the concept of calmness of a multivalued mapping in Rockafellar andWets [152] we say that the intersection of a finite family of sets {Ck}

mk=1 is metrically

calm at a point x ∈m∩

k=1Ck provided there exist a constant γ > 0 and a neighbourhood

U of x such that

(1.89) d(x,

m∩

k=1Ck

)≤ γ ( d(x,C1) + · · · + d(x,Cm)) for all x ∈ U.

Let now F : X → Y be a mapping from X into another uniformly convex Banachspace Y and let D be a subset of Y and x ∈ F−1(D). As above we say that the mapping


F is metrically calm at x relatively to the set D when there exist a constant γ > 0and a neighbourhood U of x such that

(1.90) d(x, F−1(D)) ≤ γd(F(x),D) for all x ∈ U,

where d(x, A) is the distance from x to the set A.

Proposition 1.4.23. Let {Ck}mk=1 be a finite family of closed sets of X and let D be a

closed set of Y .(a) If all sets Ck are NP-hyporegular at a point x of all the sets Ck and if the

intersection is metrically calm at x, then this intersection setm∩

k=1Ck is NP-hyporegular

at x too.(b) If a mapping F : X → Y is of class C1,1 around a point x ∈ F−1(D) and metricallycalm at x relatively to D and if D is NP-hyporegular at F(x), then the set F−1(D) isNP-hyporegular at x.

Proof. Assume that each set Ck is NP-hyporegular at x and put C :=m∩

k=1Ck. By

definition of NP-hyporegularity (see 1.85) it is easily seen that there exist some realnumber r > 0 and some open neighbourhood U of x such that for each k = 1, . . . ,mwe have for all x ∈ Ck ∩ U and u∗ ∈ NP

Ck(x) ∩ B∗

(1.91) 〈u∗, x′ − x〉 ≤ (2r)−1‖x′ − x‖2 for all x′ ∈ Ck ∩ U.

Restricting the neighbourhood U if necessary, we may suppose that the inequality(1.89) holds upon U. Fix any x ∈ C ∩ U and take any x∗ ∈ NP

C(x) with ‖x∗‖ ≤ 1.We have that x∗ ∈ NF

C (x) and hence x∗ ∈ ∂FdC(x) since one knows that ∂FdC(x) =

NFC (x)∩B∗ (see, e.g., Mordukhovich [124, Corollary 1.96]). This inequality (1.89) and

the definition of Frechet subgradient easily yields that x∗ ∈ γ∂F(dC1 + · · ·+ dCm)(x) andhence in particular x∗ ∈ γ∂L(dC1 + · · · + dCm)(x). The functions dCk being Lipschitzian,the formula of the Mordukhovich subdifferential of a finite sum of locally Lipschitzfunctions (see, e.g., Mordukhovich [124, Theorem 3.36]) ensures us that there existu∗k ∈ ∂LdCk(x) such that γ−1x∗ = u∗1 + · · · + u∗m. Restricting again the neighbouhoodU, by Theorem 1.4.6 we have u∗k ∈ ∂pdCk(x) = NP

C(x) ∩ B∗ which by (1.91) gives〈u∗k, x

′ − x〉 ≤ (2r)−1‖x′ − x‖2 for every x′ ∈ Ck ∩ U. Consequently for any x′ ∈ C ∩ Uwe obtain

〈γ−1x∗, x′ − x〉 ≤ m(2r)−1‖x′ − x‖2

and this obviously implies that the intersection set C is NP-hyporegular at x, that is,(a) is proven.

Let us now establish (b). Fix some open convex neighbourhood U of x overwhich (1.90) holds and over which the mapping F as well as its derivative DF(·)are Lipschitz with Lipschitz constants K and K1 respectively. For S := F−1(D) fix


any x ∈ U ∩ S and x∗ ∈ NPS (x) with ‖x∗‖ ≤ 1. As above we have x∗ ∈ ∂FdS (x) and

this entails by (1.90) that x∗ ∈ γ∂F(dD ◦ F)(x) and hence x∗ ∈ γ∂L(dD ◦ F)(x). Thesubdifferential chain rule (see, e.g., Mordukhovich [124, Corollary 3.43]) gives somey∗ ∈ ∂LdD(F(x)) such that γ−1x∗ = y∗◦DF(x). Fix by (1.85) some open neighbourhoodV of F(x) and some constant r > 0 such that

(1.92) 〈v∗, y′−y〉 ≤ (2r)−1‖y′−y‖2 for all y ∈ V∩D, y′ ∈ V∩D and v∗ ∈ NPD(y)∩B∗,

and such that the equality concerning the subdifferentials of the distance functionin Theorem 1.4.6 holds with dD at all points of V ∩ D. Restricting the open con-vex neighbourhood U of x if necessary, we may suppose that F(U) ⊂ V and that∂LdD(F(u)) = ∂pdD(F(u)) for all u ∈ U ∩ S according to Theorem 1.4.6. Then (1.92)yields

(1.93) 〈y∗, y′ − F(x)〉 ≤ (2r)−1‖y′ − F(x)‖2 for all y′ ∈ V ∩ D.

Fix any x′ ∈ S ∩ U and write

F(x′) − F(x) = DF(x)(x′ − x) +

∫ 1

0

(DF(x + t(x′ − x)) − DF(x)

)(x′ − x) dt

and then

〈γ−1x∗, x′ − x〉 = 〈y∗, F(x′) − F(x)〉 −∫ 1

0〈y∗ ◦

(DF(x + t(x′ − x)) − DF(x)

), x′ − x〉 dt.

Using (1.93) we obtain

〈x∗, x′ − x〉 ≤ γ(2r)−1‖F(x′) − F(x)‖2 + γK1‖x′ − x‖2 ≤ γ(K2(2r)−1 + K1)‖x′ − x‖2.

The latter being true for all x, x′ ∈ S ∩ U and x∗ ∈ NPS (x) ∩ B∗, we conclude that the

set S is NP-hyporegular at x. �

For the convenience of the reader we made the choice to prove first the NP-hyporegularity of the intersection and then to give the additional arguments yielding tothe NP-hyporegularity of the inverse image. However the case of the intersection canbe derived from that of the inverse image. Indeed putting Y = Xm, F(x) = (x, . . . , x)for all x ∈ X, and D = C1 × · · · ×Cm we see that F−1(D) =

m∩

k=1Ck and then some direct

arguments and computation allow us to obtain through (b) the result of (a).

We state now the following corollary which is a direct consequence of Corol-lary 1.4.18.


Corollary 1.4.24. Assume that the moduli of convexity and smoothness of the normof X are of power type and the power type of convexity is q = 2. Then under the

assumptions of Proposition 1.4.23 the setsm∩

k=1Ck and F−1(D) are prox-regular at x.

When the spaces (X, ‖ · ‖) and (Y, ‖ · ‖) are Hilbert spaces, Proposition 1.4.23translates, according to Corollary 1.4.18 and Proposition 1.4.19, the preservation ofprox-regularity.

Corollary 1.4.25. Assume that X and Y are Hilbert spaces for their respective norms.Let (Ck)m

k=1 be a finite family of closed sets of X and let D be a closed set of Y .(a) If all sets Ck are prox-regular at a point x of all the sets Ck and if the intersection

is metrically calm at x, then this intersection setm∩

k=1Ck is prox-regular at the point x.

(b) If a mapping F : X → Y is of class C1,1 around a point x ∈ F−1(D) and metricallycalm at x relatively to D and if D is prox-regular at F(x), then the set F−1(D) isprox-regular at x.

An analysis of the proof of Proposition 1.4.23 reveals that a uniform version alsoholds. We state it only in the case of an intersection and let the case of the inverseimage to the reader.

Proposition 1.4.26. Assume that all the closed sets Ck, k = 1, . . . ,m, are r-uniformlyNP-hyporegular (resp. r-uniformly prox-regular and (X, ‖ · ‖) is a Hilbert space) andthat their intersection is calm at anyone of its points with the same modulus γ of

metric calmness in (1.89). Then the setm∩

k=1Ck is r′-uniformly NP-hyporegular (resp.

r′-uniformly metrically prox-regular) with r′ := r/(mγ).

In the literature there are conditions with normal cones easy to handle ensuringthe metric calmness inequalities (1.89) and (1.90). For example such a condition for(1.89) is known (see, e.g., Ioffe [90, p. 548-549]) under the name of general positioncondition for intersection of finitely many sets. In the context of our uniformly convexBanach space, the concept can be translated by saying that the closed sets C1, . . . ,Cm

are (relative to the Frechet normal cone) in sequential general position at x ∈m∩

k=1Ck

whenever for any sequence of tuples (x1,n, · · · , xm,n, x∗1,n, · · · , x∗m,n)n such that xk,n ∈ Ck,

xk,n → x, x∗k,n ∈ NFCk

(xk,n) ∩ B∗ and such that∥∥∥∥∥ m∑

k=1x∗k,n

∥∥∥∥∥→ 0, one has ‖x∗k,n‖ → 0 for any

k = 1, . . . ,m.Another condition which is much easier to handle involves as above a property

with normal cones but at the fixed point x. It probably appears for the first time asone of the assumptions of Theorem 4.10 of Federer’s seminal paper [73] for sets ofRn which are not submanifolds. The sets Ck, k = 1, . . . ,m, are (relative to the limiting


normal cone) in pointbased general position at x provided the equality x∗1+· · ·+x∗m = 0with x∗k ∈ NL

Ck(xk) entails x∗1 = · · · = x∗m = 0.

Recall now that a closed set C of X is compactly epi-Lipschitzian at x ∈ C(a concept due to Borwein and Strojwas [30]) if there exist a compact set Q, aneighbourhood V of zero, a neighbourhood U of x in X, and a positive real numberε such that

C ∩ U + tV ⊂ C + tQ for all t ∈ ]0, ε].

Of course any closed subset of X is compactly epi-Lipschitzian at any of its pointswhenever the space X is finite dimensional.

Corollary 1.4.27. Let {Ck}mk=1 be a finite family of closed sets of X and x ∈

m∩

k=1Ck.

Then the following hold:(a) If the sets Ck are in sequential general position at x and if each set Ck is NP-hyporegular at x (resp. prox-regular at x and (X, ‖ · ‖) is a Hilbert space), then the

intersectionm∩

k=1Ck is N p-hyporegular (resp. prox-regular) at x.

(b) If all the sets Ck except at most one are compactly epi-Lipschitzian at x, then thesequential general position in (a) may be replaced by the corresponding pointbasedgeneral position at x.(c) If the space X is finite dimensional, again the sequential general position in (a)may be replaced by the corresponding pointbased general position at x.

Proof. (a) Since the uniformly convex Banach space X is an Asplund space, thesequential general position is known to entail the metric calmness property (1.89)(and even more) according to Proposition 6 in p. 548 and Theorem 2 in p. 545 ofIoffe [89] (for example). Then the conclusion of (a) follows from Proposition 1.4.23(resp. Corollary 1.4.25).(b) The result comes from the fact that the assumption of compactly epi-Lipschitzianproperty combined with the pointbased general position at x ensures (see, e.g., Jouraniand Thibault [98, Theorem 3.4]) that the intersection is metrically regular and hencemetrically calm at x.(c) is a direct consequence of (b). �

The result in (c) was previously established in (5) of Theorem 4.10 in Federer [73].

1.4.6 Conical derivative of the mapping PC

The following result is well-known for convex sets of Hilbert space (see Zaran-tonello [171, p. 300]). The term conical derivative has been coined in page 301 ofZarantonello [171]. The result has been independently extended by Canino [37] to


closed p-convex sets of Hilbert space and by Shapiro [153] to closed sets with 2-order tangential property of Hilbert space too. It is actually known (see Poliquin,Rockafellar and Thibault [138]) that, in the context of Hilbert space, the conceptof p-convex set is equivalent to uniform prox-regularity and the 2-order tangentialproperty is equivalent to the local prox-regularity. The related result in Shapiro [153]may then be translated for prox-regular sets. Note also that the proof in Canino [37]still holds for sets which are prox-regular at the considered point of the Hilbert space.Our proposition below deals with the context of uniformly convex space.

Proposition 1.4.28. Assume that the moduli of uniform convexity and smoothnessof the norm ‖ · ‖ of X are of power type. Assume also that the closed set C of X isprox-regular at x ∈ C. Then there exists an open neighbourhood U of x such that PC

is single valued and continuous on U and such that for any x ∈ U ∩ C and y ∈ X for

which the mapping t 7→1t[PC(x + ty)− x] is bounded on some interval ]0, t0], one has

limt↓0

1t[PC(x + ty) − x] = PKC(x)(y)

and the directional derivative d′C(x; y) := limt↓0

1t[dC(x + ty) − dC(x)] of dC at x in the

direction y exists andd′C(x; y) = d(y,KC(x)),

which translates in the terminology of Zarantonello [171] that d′C(x; ·) is a conicalderivative.

Proof. By the inequality (1.74) in the proof of Theorem 1.4.6, there exist ε, r > 0with ε < 1/2 such that PC is a single-valued continuous mapping on B(x, ε) and

(1.94) 〈x∗ − u∗, u − x〉 ≤2r‖x − u‖ωr+1(‖x − u‖)

for all x, u ∈ B(x, ε) and all x∗ ∈ NPC(x), u∗ ∈ NP

C(u) with ‖x∗‖ ≤ 1 and ‖u∗‖ ≤ 1,where ωr+1 is the modulus of uniform continuity of the duality mapping J over thebounded set (r + 1)B. Let x ∈ B(x, ε) ∩ C, y ∈ X, and t0 > 0 such that the mapping

t 7→1t[PC(x + ty)− x] is bounded on ]0, t0], say by β > 0. We may suppose that for all

t ∈]0, t0] we have x + ty ∈ B(x, ε) and PC(x + ty) ∈ B(x, ε). For any t ∈]0, t0] putting

h′t :=1t[PC(x + ty) − x], we see that

y − h′t =1t[x + ty − PC(x + ty)]

and hence y − h′t ∈ PNC(PC(x + ty)), i.e., J(y − h′t) ∈ NPC(PC(x + ty)).


Now take any sequence (tn)n of ]0, t0] converging to 0. The sequence (J(y − h′tn))n

being bounded, there exists an increasing function σ : N→ N such that the sequence(J(y − h′tσ(n)

))n converges weakly star to J(y − h) for some vector h ∈ X. As NPC(x) =

NLC(x) (see Theorem 1.4.6), we have J(y − h) ∈ NP

C(x). Put hn := h′tσ(n). By (1.94) there

exists some constant γ > 0 independent of n such that for λn := tσ(n)

〈J(y − hn) − J(y − h), x − PC(x + λny)〉 ≤γ

r‖PC(x + λny) − x‖ωr+1(‖PC(x + λny) − x‖).

The latter inequality is equivalent to

〈J(y − hn) − J(y − h),−λnhn〉 ≤γ

r‖PC(x + λny) − x‖ωr+1(‖PC(x + λny) − x‖),

that is,

〈J(y − hn) − J(y − h), (y − hn) − (y − h)〉 − 〈J(y − hn) − J(y − h), h〉 ≤

γ

r

∥∥∥∥∥ 1λn

[PC(x + λny) − x]∥∥∥∥∥ ωr+1(‖PC(x + λny) − x‖).

If max{‖y − hn‖, ‖y − h‖} = 0 for infinitely many n, then for all these integers n wehave y = hn = h and hence by definition of hn we obtain h = 1

λn[PC(x + λny) − x],

which yields h ∈ KC(x). Suppose now that max{‖y− hn‖, ‖y− h‖} > 0 for all n not lessthan some n0. According to (1.48) for some constant K2 > 0 we have for all n ≥ n0

K2(max{‖y − hn‖, ‖y − h‖})2δ‖·‖

(‖hn − h‖

2(max{‖y − hn‖, ‖y − h‖})

)≤

〈J(y − hn) − J(y − h), h〉 +γ

r

∥∥∥∥∥ 1λn

[PC(x + λny) − x]∥∥∥∥∥ ωr+1(‖PC(x + λny) − x‖),

and since the first expression of the second member tends to 0 and the modulus ofconvexity δ‖·‖ is an increasing function with δ‖·‖(t)−→

t↓00, it is not difficult to see that

‖hn − h‖ → 0 and by definition of hn we obtain h ∈ KC(x). So in any case, we haveh ∈ KC(x) and the sequence (hn)n strongly converges to h.

Now take any v ∈ KC(x). Since KC(x) = TC(x) according to Theorem 1.4.6, thereis a sequence (zn)n converging to v such that x + λnzn ∈ C for all n. This entails thatfor all n

‖x + λny − (x + λnzn)‖ ≥ ‖x + λny − PC(x + λny)‖,


‖y − zn‖ ≥ ‖y +1λn

[x − PC(x + λny)]‖.


Then passing to the limit with n → ∞, we obtain ‖y − v‖ ≥ ‖y − h‖ for all v ∈KC(x), which means that h = PKC(x)(y). So for any sequence of positive numbers (tn)n

converging to 0 there exists a subsequence (tσ(n))n such that

limn→∞

1tσ(n)

[PC(x + tσ(n)y) − x] = PKC(x)(y).

Consequently

(1.95) limt↓0

1t[PC(x + ty) − x] = PKC(x)(y).

Concerning the equality of the directional derivative of the distance function tothe set C, observe first that for any positive number t small enough

1t[dC(x + ty) − dC(x)] =

1tdC(x + ty) = 1

t ‖PC(x + ty) − (x + ty)‖

=∥∥∥1

t [PC(x + ty) − x] − y∥∥∥ .

Then taking (1.95) into account and passing to the limit when t ↓ 0 give that thedirectional derivative d′C(x; y) exists and

d′C(x; y) = ‖PKC(x)(y) − y‖ = d(y,KC(x)). �

In the case of a Hilbert space X the mapping PC is Lipschitz near x (see Poliquin,Rockafellar and Thibault [138]) and hence the boundedness assumption of the propo-sition above is fulfilled and one recovers in Corollary 1.4.29 below the result ofProposition 2.12 in Canino [37] and Theorem 3.1 in Shapiro [153]. The result isproved in Shapiro [153] through some properties of solutions of perturbed optimiza-tion problems satisfying some appropriate conditions. The proof in Canino [37] isbased on the duality in Hilbert space between the tangent cone KC(x) and the primalnormal cone PNC(x) for prox-regular sets C. Such a duality property is not availablein the non Hilbert setting because of the nonlinearity of the duality mapping J. In-stead, our proof of Proposition 1.4.28 above is related to the tangential regularity ofC established in Theorem 1.4.6 and to the sequential characterization of the Clarketangent cone (recalled in the beginning of Subsection 1.4.1).

Corollary 1.4.29 (Canino [37, Proposition 2.12] and Shapiro [153, Theorem3.1]). Assume that X is a Hilbert space and that C is prox-regular at x ∈ C. Then forsome neighbourhood U of x one has that for all x ∈ U ∩ C and y ∈ X the directionalderivatives of PC and dC at x in the direction y exist and

limt↓0

1t[PC(x + ty) − x] = PKC(x)(y) and d′C(x; y) = d(y,KC(x)).


1.4.7 Convergence

In this subsection we are interested in the behaviour of the projection mappingunder convergence of prox-regular sets.

Let us begin by recalling some properties of the projection mapping of a prox-regular set and by providing some additional facts. Consider a uniformly r-prox-regular closed set C. We see from the proof of Theorem 1.3.27 that the function d2

Cis Frechet differentiable on the r-open neighborhood OC(r) := {x ∈ X : dC(x) < r} ofC and that PC is a single-valued and norm-to-norm continuous mapping from OC(r)into C. Moreover, from the proof of Proposition 1.3.30, for any x ∈ OC(r)

(1.96) ∇F

(12

d2C

)(x) = J(x − PC(x)).

Concerning the continuity of PC, the proof of Theorem 1.3.27 says more, in the sensethat PC is locally Holder continuous on OC(r). The following two propositions furnishsome further points that we will need in Theorem 1.4.33 below. Recall that q (resp. s)denotes the power type of the modulus of convexity (resp. smoothness) of the normof X.

Proposition 1.4.30. Assume that the moduli of uniform convexity and smoothness ofthe norm ‖ · ‖ of X are of power type q and s respectively. Let r, r′ be two positivenumbers with r′ < r

2 and let a number ρ > 0. Then there exists some constant γ ≥ 0depending only on r, r′ and ρ such that, for any uniformly r-prox-regular closedsubset C of X, one has

(1.97) ‖PC(x1) − PC(x2)‖ ≤ γ‖x1 − x2‖1q and ‖∇Fd2

C(x1) − ∇Fd2C(x2)‖ ≤ γ‖x1 − x2‖

s−1q

for all x1, x2 ∈ OC(r′) ∩ ρB.

Proof. Take α ∈ ]0, 1/2[ such that r′ = αr. By Step 1 of the proof of Theorem 1.3.27the truncated normal cone set-valued mapping NP r

C (·) is J-hypomonotone of degree 1and hence so is the set-valued mapping NPαr

C (·). Referring to the proof of Lemma 1.3.8with T (·) := NPαr

C (·) and r := 1, since α−1 > 2r the development there reveals that,fixing λ satisfying 2α < λ < 1, there exists some constant γ′ > 0 depending onlyon α, ρ, and r such that Q := (I + αJ∗ ◦ NP r

C )−1 is on ρB ∩ Dom Q a single valuedmapping for which

(1.98) ‖Q(x1) − Q(x2)‖ ≤ γ′‖x1 − x2‖1q for all x1, x2 ∈ ρB ∩ Dom Q.

Observe now that for x ∈ OC(αr) we have J(x − PC(x)) ∈ NPC(PC(x)) (by definition

of proximal normal functional) and ‖J(x − PC(x))‖ < αr and that those two facts


imply J(x− PC(x)) ∈ αNP rC (PC(x)), that is, PC(x) = (I +αJ∗ ◦NP r

C )−1(x). We may thenreformulate (1.98) in the form

‖PC(x1) − PC(x2)‖ ≤ γ′‖x1 − x2‖1q for all x1, x2 ∈ OC(r′) ∩ ρB.

Concerning the square distance function d2C, we know that it is (see 1.96 ) Frechet

differentiable on OC(r) with ∇Fd2C(x) = 2J(x − PC(x)) for all x ∈ OC(r). So using

(1.55) and the latter inequality above, we have for all x1, x2 ∈ OC(r′) ∩ ρB

‖∇Fd2C(x1) − ∇Fd2

C(x2)‖ ≤ 2Lρ‖x1 − PC(x1) − x2 + PC(x2)‖s−1

≤ 2Lρ(‖x1 − x2‖ + γ′‖x1 − x2‖1q )s−1

≤ 2Lρ(γ′ + ‖x1 − x2‖1− 1

q )s−1‖x1 − x2‖s−1q

≤ 2Lρ(γ′ + (2ρ)1− 1q )s−1‖x1 − x2‖

s−1q .

It suffices to take γ = max{γ′, 2Lρ(γ′ + (2ρ)1− 1

q )s−1}

to obtain both inequalities in thestatement of the proposition. �

In the proposition above, under the restriction r′ < r/2, the power of Holdercontinuity of PC over OC(r′) ∩ ρB is the constant 1/q (the inverse of the power typeof uniform convexity of ‖ · ‖), i.e., only the modulus γ of Holder continuity varieswith r′ and ρ. Relaxing the restriction r′ < r/2 into r′ < r and letting the power ofHolder continuity depending also on r′ and ρ, we can prove the following result ofHolder continuity of PC on OC(r′) ∩ ρB but this time with any r′ < r.

Before stating the result, recall that for any r > 0 the (closed) r-enlargement C(r)of C is given by C(r) := {x ∈ X : d(x,C) ≤ r}.

Proposition 1.4.31. Assume that the moduli of uniform convexity and smoothnessof the norm ‖ · ‖ of X are of power type q and s respectively. Let ρ, r, r′ be positivereal numbers with r′ < r. Then there exist some positive constants γ and θ ≤ 1 bothdepending only on r′, r and ρ, such that for any uniformly r-prox-regular closed setC of X one has

(1.99) ‖PC(x1)− PC(x2)‖ ≤ γ‖x1 − x2‖θ and ‖∇Fd2

C(x1)−∇Fd2C(x2)‖ ≤ γ‖x1 − x2‖

θ(s−1)

for all x1, x2 ∈ OC(r′) ∩ ρB.More precisely, defining (αn)n by α1 = 1

2 and αn+1 = 1+αn2 , then for any n ∈ N,

ρ > 0, α′ ∈ ]0, αn[, there exists some positive real number k (depending only on ρ,α′ and n) such that, for any uniformly r-prox-regular closed set C of X, the metricprojection mapping PC is Holder continuous on OC(α′r)∩ ρB with power θ := 1

qn andwith modulus k.


Proof. Let us call P(n) the property above in the second part of the proposition forPC of any uniformly r-prox-regular closed set C. Note that P(1) is Proposition 1.4.30.Suppose that P(n) is fulfilled, and fix any uniformly r-prox regular closed set C andany ρ > 0, α′ ∈ ]0, αn[ . We may suppose that OC(r) ∩ ρB , ∅. By the proofs ofTheorem 1.3.27 and Lemma 1.3.29, the closed set C(α′r) is uniformly (1−α′)r-prox-regular. Then applying now Proposition 1.4.30 to the set C(α′r) yields that for anyλ ∈ ]0, 1[ ,

(1.100) PC(α′r) is Holder continuous on OC(α′r)

(λ

1 − α′

2r)∩ ρB with power 1/q

and with modulus depending only on ρ, λ, α′, and r. On the other hand, we havethat for any t > 0, OC(α′r + t) = OC(α′r)(t). Indeed, from Lemma 1.3.28 (a) in(see also Bounkhel and Thibault [33]), for any τ > 0 and any u < C(τ) one hasdC(u) = τ + dC(τ)(u). Hence, for any u ∈ X, one has the equivalence dC(u) < α′r + t ⇔dC(α′r)(u) < t, that is, the desired equality holds. This entails that

(1.101) OC(α′r)

(λ

1 − α′

2r)

= OC

(α′r + λ

1 − α′

2r).

By (1.100) and (1.101) there exists some constant K1 > 0 (depending only on ρ, λ,α′, and r) such that

(1.102) ‖PC(α′r)(u1) − PC(α′r)(u2)‖ ≤ K1‖u1 − u2‖1/q for all ui ∈ OC

(α′r + λ

1 − α′

2r).

Fixing bρ ∈ OC(r) ∩ ρB , ∅, we find some aρ ∈ C with ‖aρ‖ ≤ r + ρ. Combining thiswith (1.102), we obtain that there is some ρ′ > 0 (depending only on ρ, λ, α′, and r)such that

(1.103) ‖PC(α′r)(u)‖ ≤ ρ′ for all u ∈ (ρB) ∩ OC

(α′r + λ

1 − α′

2r)

(take u1 = u and u2 = aρ in (1.102)). The equality (1.101) also implies, according toTheorem 1.3.27 that for any u ∈ OC

(α′r + λ 1−α′

2 r)\ C(α′r), the points y := PC(α′r)(u)

and z := PC(y) exist. By (c) of Lemma 1.3.28, for any such u we have conse-quently z = PC(u) and so, PC(u) = PC ◦ PC(α′r)(u). From this, if u1, u2 ∈ (ρB) ∩(OC

(α′r + λ1−α′

2 r)\C(α′r)

), then for any α′′ ∈]α′, αn[ we have

‖PC(u1) − PC(u2)‖ = ‖PC(PC(α′r)(u1)) − PC(PC(α′r)(u2))‖

≤ K2‖PC(α′r)(u1) − PC(α′r)(u2)‖1/qn

for some constant K2 > 0 (depending only on ρ′, α′′, and n), according to P(n) andto the fact that, by (1.103), for i = 1, 2, PC(α′r)(ui) ∈ ρ′B ∩ OC(α′′r). Taking (1.102)


into account we obtain that there exists some constant K3 (depending only on ρ, λ,α′, r and n) such that for all u1, u2 ∈ (ρB) ∩

(OC

(α′r + λ 1−α′

2 r)\C(α′r)

)(1.104) ‖PC(u1) − PC(u2)‖ ≤ K3‖u1 − u2‖

1/qn+1

and this inequality is still true for all ui ∈ ρB satisfying α′r ≤ d(ui,C) < α′r + λ 1−α′2 r.

On ρB ∩ C(α′r), P(n) ensures that PC also has the Holder continuity property withpower 1/qn and hence also with power 1/qn+1 (because of the boundedness of ρB ∩C(α′r)). Then there exists some positive constant K4 (depending only on ρ, λ, α′, rand n) such that

‖PC(u1) − PC(u2)‖ ≤ (K4/2)‖u1 − u2‖1/qn+1

for all u1, u2 ∈ (ρB) ∩ OC

(α′r + λ 1−α′

2 r)

satisfying either d(ui,C) ≥ α′r for i = 1and i = 2 or d(ui,C) ≤ α′r for i = 1 and i = 2. In the remaining case whereu1, u2 ∈ (ρB)∩OC

(α′r + λ 1−α′

2 r)

with d(u1,C) < α′r and d(u2,C) > α′r, the continuityof d(·,C) over the line segment [u1, u2] yields some point u0 ∈]u1, u2[ such thatd(u0,C) = α′r; then

‖PC(u1) − PC(u2)‖ ≤ (K4/2)(‖u1 − u0‖

1/qn+1+ ‖u0 − u2‖

1/qn+1)≤ K4‖u1 − u2‖

1/qn+1,

the second inequality being due to the inequalities ‖ui − u0‖ ≤ ‖u1 − u2‖. The mappingPC is then Holder continuous on ρB∩OC

(α′r + λ 1−α′r

2 r)

with modulus K4 and power1/qn+1.

We have shown that, for any λ ∈]0, 1[ , α′ ∈]0, αn[ , and ρ > 0, there existssome positive real number k (depending only on ρ, α′, λ, and n) such that, forany uniformly r-prox-regular set C, the metric projection mapping PC is Holdercontinuous on ρB ∩ OC

(α′r + λ 1−α′

2 r)

with power 1/qn+1 and with modulus k. Thismeans that for any α′ ∈]0, αn[ there is some positive k (depending only on ρ, α′, andn) such that, for any uniformly r-prox-regular set C of X, the mapping PC is Holdercontinuous on ρB∩OC

(α′r + 1−α′

2 r)

with power 1/qn+1 and modulus k. Consequently

P(n + 1) holds true since α′r + 1−α′2 r = 1+α′

2 r.The Holder continuity of the function ∇Fd2

C(·) is obtained like in the proof ofProposition 1.4.30. �

Remark 1.4.32. When (X, ‖ · ‖) is a Hilbert space, where ‖ · ‖ is the norm associatedwith the inner product, the behaviour of PC(·) and ∇Fd2

C(·) is distinctly better. Indeed,by Poliquin, Rockafellar and Thibault [138] for any real positive numbers r, r′ withr′ < r and any uniformly r-prox-regular closed set C of X one has

‖PC(x1) − PC(x2)‖ ≤r

r − r′‖x1 − x2‖

for all x1, x2 ∈ OC(r′), and hence

‖∇Fd2C(x1) − ∇Fd2

C(x2)‖ ≤ 2(1 +

rr − r′

)‖x1 − x2‖


according to (1.96) and to the Lipschitz property of J here with Lipschitz modulus 1.So, not only the Lipschitz continuity (instead of the Holder one) is available for themetric projection but also no restriction to bounded subsets of OC(r′) is required.

We can now study convergence properties of families of uniformly prox-regularsets. Let T ∪ {t0} be a topological space with t0 as a cluster point of T . Recall that afamily (Ct)t∈T of non empty closed subsets of X converges in the sense of Attouch-Wets (see Attouch and Wets [2], Beer [13], Rockafellar and Wets [152]) to a closedsubset C of X when t goes to t0 provided for each positive real number ρ one has

(1.105) supx∈ρB|dCt(x) − dC(x)| −→

t→t00.

Theorem 1.4.33. Assume that the moduli of uniform convexity and smoothness ofthe norm ‖ · ‖ of X are of power type. Let (Ct)t∈T be a family of closed uniformlyr-prox-regular sets of X which converges in the sense of Attouch-Wets to a closedset C of X. Then C is uniformly r-prox-regular and for each x0 ∈ X satisfyingd(x0,C) < r the mapping t 7→ PCt(x0) and the function t 7→ ∇Fd2

Ct(x0) are defined on

a neighbourhood of t0 and

(1.106) PCt(x0)−→t→t0

PC(x0) and ∇Fd2Ct

(x0)−→t→t0∇Fd2

C(x0).

Proof. Fix a positive number r′ with d(x0,C) < r′ < r and choose by (1.105) someneighbourhood T ′0 of t0 and some β > 0 such that d(x,Ct) < r′ for all t ∈ T ′0 \ {t0} andx ∈ x0 + 2βB. According to Proposition 1.4.31, there exist some positive constants γand σ ≤ 1 (both depending only x0, β, r′ and r) such that for all t ∈ T ′0 with t , t0

and x1, x2 ∈ x0 + 2ρB the derivatives ∇Fd2Ct

(xi), i = 1, 2, exist and

(1.107) ‖∇Fd2Ct

(x1) − ∇Fd2Ct

(x2)‖ ≤ γ‖x1 − x2‖σ.

Fix another neighbourhood T′′

0 of t0 with T′′

0 ⊂ T ′0 and such that, according to (1.105),the function v given for ρ := 1 + 2β + ‖x0‖ by

v(t) := supx∈ρB|dCt(x) − dC(x)|

is bounded on T0 := T′′

0 \ {t0}. Writing

dCt(x) ≤ dCt(x0) + ‖x − x0‖ ≤ dC(x0) + ‖x − x0‖ + v(t) for all x ∈ X and t ∈ T,

we see that there exists some constant M > 0 such that

(1.108) dCt(x) ≤ M/2 for all (t, x) ∈ T0 × ρB.


Fix now any positive number ε < β. Choose by (1.105) a neighbourhood T ′ε ⊂ T′′

0 oft0 such that for Tε := T ′ε \ {t0}

(1.109) |dCt(x) − dC(x)| ≤εσ+1

2for all t ∈ Tε and x ∈ ρB.

Fix any x ∈ B(x0, β), any w ∈ X with ‖w‖ = ε, and any t, τ ∈ Tε. We have

d2Ct

(x + w) − d2C(x) = 〈∇Fd2

Ct(x),w〉 +

∫ 1

0〈∇Fd2

Ct(x + sw) − ∇Fd2

Ct(x),w〉 ds

and a similar equality with d2Cτ

in place of d2Ct

, hence

〈∇Fd2Ct

(x) − ∇Fd2Cτ

(x),w〉

= [d2Ct

(x + w) − d2Cτ

(x + w)] − [d2Ct

(x) − d2Cτ

(x)]

−

∫ 1

0〈∇Fd2


Ct(x),w〉 +

∫ 1

0〈∇Fd2

Cτ(x + sw) − ∇Fd2

Cτ(x),w〉.

Further, on the one hand by (1.107) we have∣∣∣∣∣∣∫ 1

0〈∇Fd2


Ct(x),w〉 ds

∣∣∣∣∣∣ ≤∫ 1

0γsσ‖w‖σ+1 ds =

1σ + 1

γ‖w‖σ+1 ≤ γεσ+1

and a similar inequality with d2Cτ

in place of d2Ct

. On the other hand by (1.108) and(1.109), with y := x + w or y := x we also have

|d2Ct

(y) − d2Cτ

(y)| = (dCt(y) + dCτ(y))|dCt(y) − dCτ

(y)| ≤ Mεσ+1.

Consequently〈∇Fd2

Ct(x) − ∇Fd2

Cτ(x),w〉 ≤ (2M + 2γ)εσ+1

and hence that for every u in X with ‖u‖ = 1

〈∇Fd2Ct

(x) − ∇Fd2Cτ

(x), u〉 ≤ (2M + 2γ)εσ,

which ensures that‖∇Fd2

Ct(x) − ∇Fd2

Cτ(x)‖ ≤ (2M + 2γ)εσ.

This uniform Cauchy property says that the family (∇Fd2Ct

)t converges uniformly tosome mapping from B(x0, β) into X∗. This is known to entail that the function d2

C is


Frechet differentiable on B(x0, β) and hence, in particular, Frechet differentiable at x0

with

(1.110) ∇Fd2Ct

(x0)−→t→t0∇Fd2

C(x0).

Then the function dC is Frechet differentiable on the open tube UC(r) and this differ-entiability property translates through Theorem 1.3.27 the uniform r-prox-regularityof C. Further (1.96) and (1.110) yield

PCt(x0)−→t→t0

PC(x0).

This completes the proof of the proposition. �

Chapter 2

Integrability of subdifferentials offunctions

The purpose of studying integration of subdifferentials is to answer the questionwhether or not the condition that the subdifferential of g contains the subdifferentialof f implies that f and g differ by a constant. It is accepted instead of integrability ofsubdifferentials of functions to speak of the integrability of functions, in both casesbeing understood one and the same.

The famous Moreau-Rockafellar’s integration result (see Rockafellar [147]) givesa positive answer when f and g are lower semicontinuous convex functions definedon a Banach space. The answer is also affirmative for some classes of locally Lips-chitz functions (see Borwein and Moors [27], Borwein, Moors and Wang [28], Correaand Jofre [50], Correa and Thibault [54]). The first significant extension outside theconvex and locally Lipschitz case is due to Poliquin [135] who shows that the in-tegration result holds in a finite dimensional setting for functions f and g that areprimal lower nice. Later, Thibault and Zagrodny [160] extend the result of Poliquin tothe class of convexly subdifferentially similar functions defined on a Banach space.This class includes primal lower nice functions defined on a Hilbert space, as well asthe differences of convex functions. Ivanov and Zlateva [93] established the integra-tion result for the class of semi-convex functions. New insight on this topic can befound in the work of Thibault and Zagrodny [161], where a more general inclusionof subdifferentials is investigated.

In Section 2.1 the famous Moreau-Rockafellar result is proved by using regular-ization (and approximation) techniques which was the initial idea of Moreau. Theresult is published by Zlateva in [172].

Using a quantitative version of subdifferential characterization of a class of direc-

99

100 Chapter 2. Integrability of subdifferentials of functions

tionally Lipschitz functions in Section 2.2 we establish the integrability of subdiffer-entials of such functions. The result is published by Thibault and Zlateva in [163].

In Section 2.3 we study integrability properties of bivariate functions defined ona product of Banach spaces. Progress in this direction was already made by Correaand Thibault [54], and by Wu and Ye [168]. The problem in this setting is interestingfrom the point of view that the product structure allows us to introduce differentconcepts of continuity and regularity of the function and study how they rely to theintegrability. Special attention here will be given to directionally Lipschitz bivariatefunctions in order to develop the results for directionally Lipschitz functions of onevariable from Section 2.2. The results are published by Thibault and Zlateva in [162].

In Section 2.4 we study on a product Banach space the properties of a class ofsaddle functions called partially ball weakly inf-compact. For such a function weprove that the domain of the subdifferential is non-empty, that the operator naturallyassociated with the subdifferential is maximal monotone, and that the subdifferentialof the function is integrable. For a function in a large subclass of that class we provethe density of the domain of the subdifferential in the domain of the function. Theresults are published by Thibault and Zlateva in [164].

2.1 Integrability of the subdifferential of a convexfunction through infimal regularization

As usual, X is a Banach space with dual X∗. A function f : X → R ∪ {+∞} isconvex, whenever

f (λx + (1 − λ)y) ≤ λ f (x) + (1 − λ) f (y), ∀x, y ∈ X, and ∀λ ∈ [0, 1],

or, which is equivalent, epi f is a convex set in X × R.Subgradient of a convex function f at point x is any x∗ ∈ X∗ such that

f (x′) ≥ f (x) + 〈x′ − x, x∗〉, ∀x′ ∈ X.

The set of all subgradients of f at x (which could be empty) is denoted by

∂ f (x) = {p ∈ X∗; 〈p, y − x〉 ≤ f (y) − f (x), ∀y ∈ X},

and the multivalued map ∂ f : X → 2X∗ is called convex subdifferential (or Fenchelsubdifferential) of f whose domain dom ∂ f ⊂ dom f if f is proper.

2.1. Integrability of the subdifferential of a convex function through infimal . . . 101

For a proper, convex and lower semicontinuous function f : X → R ∪ {+∞}the subdifferential of f at x ∈ dom f in the sense of convex analysis (or Fenchelsubdifferential) is denoted by ∂ f (x), that is

∂ f (x) = {p ∈ X∗; 〈p, y − x〉 ≤ f (y) − f (x), ∀y ∈ X},

and ∂ f = ∅ on X \ dom f .The famous integration theorem of Moreau and Rockafellar, see Moreau [128]

and Rockafellar [144, 147] states that each proper, convex and lower semicontinuousfunction is determined up to an additive constant by its subdifferential.

This result was firstly stated and proved by Moreau in [128] on a Hilbert space H.The proof uses infimal regularizations and works also in reflexive Banach space asmentioned of Moreau at [129, p.87].

Below we present a brief sketch of Moreau proof (see Moreau [128]). When H isa Hilbert space, for all z ∈ H the function

u→ f (u) +12‖u − z‖2

possesses a strict minimum called prox f z and z can be represented as the sum z = x+yof x such that x = prox f z and y such that y ∈ ∂ f (x). Moreover, the decomposition ofz as a sum of x and y such that y ∈ ∂ f (x) is unique and holds only for x = prox f z(see Moreau [128, Proposition 4a]).

Consider the function ϕ : H → R

ϕ(z) = infu∈H

{f (u) +

12‖u − z‖2

},

which is convex and Frechet differentiable with Frechet derivative ϕ′(z) = prox f z− z.Similarly, if g : X → R ∪ {+∞} is a proper, convex and lower semicontinuous

function, we consider

ψ(z) = infu∈H

{g(u) +

12‖u − z‖2

},

which is convex and Frechet differentiable with Frechet derivative ψ′(z) = proxgz − zand for all z ∈ H the unique decomposition of the form z = x + y where y ∈ ∂g(x)holds for x = prox gz.

If the inclusion ∂ f (x) ⊂ ∂g(x) holds for all x ∈ H, then prox gz = prox f z forall z ∈ H, which implies that ϕ′(z) = ψ′(z) for all z ∈ H. This yields that for someconstant c, ϕ(z) = ψ(z) + c for all z ∈ H, which for their conjugate functions yieldsϕ∗(z) = ψ(z)∗ − c for all z ∈ H. As it is well known that ϕ∗ = f ∗ + 1

2‖ · ‖2, where f ∗


is the conjugate of f and that ψ∗ = g∗ + 12‖ · ‖

2, where g∗ is the conjugate of g. So itholds that f ∗ = g∗ − c, and then f = g + c (see Moreau [128, Proposition 8a]).

It is clear that the idea of Moreau was to replace f and g with more regu-lar functions ϕ and ψ generated by them. Rockafellar’s proof in Banach space iscompletely different and uses duality arguments. Our purpose here is to prove theMoreau-Rockafellar result in Banach space by using regularization (and approxima-tion) techniques which was the initial idea of Moreau.

Given a proper, convex and lower semicontinuous function f : X → R ∪ {+∞}one defines the multi-valued operator ∂ f : X ⇒ X∗ which to any x ∈ X assigns the(probably empty) set ∂ f (x) of the convex subdifferential of f at x.

Recall that an operator T : X ⇒ X∗ is monotone if for all x∗ ∈ T (x) and y∗ ∈ T (y)it holds 〈x∗ − y∗, x − y〉 ≥ 0.

We consider proper, convex and lower semicontinuous functions f , g : X →R ∪ {+∞} such that

(2.1) ∂ f (x) ⊂ ∂g(x), ∀x ∈ X.

At this place it is worth noting that the formally weaker assumption: it is given aproper lower semicontinuous function f : X → R ∪ {+∞} and a proper, convex andlower semicontinuous function g : X → R∪ {+∞} such that (2.1) holds for ∂ f , where∂ : X ⇒ X∗ is some arbitrary presubdifferential in fact entails the convexity of f .

Following Thibault and Zagrodny [160, 161], Correa, Jofre and Thibault [53] wedefine

Definition 2.1.1. Presubdifferential operator is an operator ∂ which associates to anyfunction f : X → R ∪ {+∞} and any point x ∈ X a (possibly empty) subset ∂ f (x) ofX∗ that we will call a subdifferential of f at x such that:

Property 1. ∂ f (x) = ∅ if x < dom g;Property 2. ∂ f (x) = ∂g(x) whenever f and g coincide on a neighbourhood of x;Property 3. ∂ f (x) is equal to the subdifferential in the sense of convex analysis

whenever f is convex;Property 4. For g lower semicontinuous near x and f convex and continuous on

a neighbourhood of x, whenever x ∈ dom( f + g) is a local minimum point of f + g,

0 ∈ ∂ f (x) + lim supy→g x

∂g(y),

where lim sup denotes the weak star sequential upper limit and y →g x means thaty→ x and g(y)→ g(x).


It is easy to see that any abstract subdifferential operator according to Defini-tion 1.1.1 is a presubdifferential operator according to Definition 2.1.1. Further, mostof the widely used subdifferentials over appropriate Banach spaces are subdiffer-entials in the sense of the above definition (see Thibault and Zagrodny [160]) –the Clarke-Rockafellar subdifferential [44], the Michel-Penot subdifferential [120],the Mordukhovich subdifferential [121], the Kruger-Mordukhovich subdifferential[106, 107], the Ioffe A-subdifferential [87, 88]), the limiting Frechet subdifferen-tial, the limiting proximal subdifferential (see for example Ioffe [89], Kruger [106]),as well as the Frechet subdifferential, the proximal subdifferential, etc.

It is well-known that if g : X → R∪ {+∞} is a proper, convex and lower semicon-tinuous function, then the convex subdifferential ∂g is a monotone operator. Inclusion(2.1) for a lower semicontinuous function f and ∂ f where ∂ is arbitrary subdifferentialentails that ∂ f is a monotone operator too. But the monotonicity of a subdifferential∂ f yields the convexity of the function f (see Correa, Jofre and Thibault [53, Theo-rem 2.4] that generalizes Correa, Jofre and Thibault [52, Theorem 3]). So, any lowersemicontinuous function f satisfying (2.1) with arbitrary subdifferential ∂ has to beconvex and thanks to Property 3 in Definition 2.1.1 the inclusion (2.1) holds for itsconvex subdifferential ∂ f .

Going back to the case of a Hilbert space H let us recall that the function

(2.2) ϕ(z) := minu∈H

{f (u) +

12‖u − z‖2

}= f (prox f z) +

12‖prox f z − z‖2

is convex with continuous Frechet derivative. Moreover, z = x + y, where x = prox f zand it is easy to derive that

(2.3) ϕ′(z) ∈ ∂ f (x) ∩(12‖ · ‖2

)′(y) ⊂ ∂g(x) ∩

(12‖ · ‖2

)′(y) = ψ′(z)

In the case when X is not a Hilbert space, the function

u→ f (u) +12‖u − z‖2

may not attains its minimum and a weaker version of the equality (2.2) involvingε-minima has to be considered, as well as, a relevant generalization of (2.3).

We prefer to perturb the convex function f (u) with functions of the type n‖u − z‖instead of the square of the norm. In other words, we use Hausdorff regularizationinstead of Moreau-Yosida regularization. Hence, we will consider the functions

u→ f (u) + n‖u − z‖


and we will study the relations between the intersection of ε-subdifferentials of theconvex functions f (·) and n‖ · −z‖ at the points of ε-minima of the function u →f (u) + n‖u − z‖ and the ε-subdifferential of the approximating function.

Let us recall that for ε ≥ 0, the ε–subdifferential of a proper, convex and lowersemicontinuous function f : X → R ∪ {+∞} at x ∈ dom f is the set

(2.4) ∂ε f (x) = {p ∈ X∗ : −ε + 〈p, y − x〉 ≤ f (y) − f (x), ∀y ∈ X}.

Of course, for ε = 0, ∂0 f (x) ≡ ∂ f (x). But while ∂ f (x) could be empty, for anyx ∈ dom f and ε > 0, the sets ∂ε f (x) are non-empty. Moreover, for any real numbersε1 and ε2 such that 0 < ε1 ≤ ε2 one has ∂ε1 f (x) ⊂ ∂ε2 f (x) and ∂ f (x) =

⋂ε>0

∂ε f (x).

The result of Brøndsted and Rockafellar saying that the graph of ∂ε f is close tothe graph of ∂ f is well known:

Brøndsted-Rockafellar Lemma ([36]). Let f : X → R ∪ {+∞}, be a proper, convexand lower semicontinuous function, let ε > 0 and p ∈ ∂ε f (x). Then there existsq ∈ ∂ f (z) such that

‖z − x‖ ≤√ε, and ‖q − p‖ ≤

√ε.

Proof of the integration result for convex function

Let f : X → R ∪ {+∞} be a proper, lower semicontinuous and convex function.For n ∈ N define the inf-convolutions { fn} by

(2.5) fn(x) := infy∈X{ f (y) + n‖x − y‖}.

(The approximating sequence { fn} was originally introduced by Hausdorff [82] for anylower bounded lower semicontinuous function f of a real variable.) It is clear thatfor sufficiently large n the function fn is finite valued and we will always considerthis case even if it is not stated explicitly. Some well-known properties of these inf-convolutions of f (see, for instance, Laurent [109], Hiriart-Urruty [84], Fitzpatrickand Phelps [76]) are listed in next

Lemma 2.1.2. For a proper, lower semicontinuous and convex function f : X →R ∪ {+∞} and n large enough,

(i) fn is convex and n-Lipschitzian;(ii) fn(x) ≤ fn+1(x) ≤ f (x) for all x ∈ X and all n ∈ N;(iii) fn(x)→ f (x) as n→ ∞ for all x ∈ X.


Let us denote the set of all ε-minima of the function f (·) + n‖x − ·‖ over X by

Mnε( f ; x) := {y ∈ X : f (y) + n‖x − y‖ ≤ fn(x) + ε}.

It is clear that for ε > 0 the sets Mnε( f ; x) are non-empty.

Since fn are convex continuous functions, then ∂ fn(x) is a nonempty set for allx ∈ X. We will show that if p is a subgradient of fn at x, and y is an arbitraryε-minimum point of f (·) + n‖x − ·‖, then actually p is an ε-subgradient of f at y, aswell as, it is an ε-subgradient of the function n‖ · ‖ at x − y:

Lemma 2.1.3. For any ε ≥ 0, and any y ∈ Mnε( f ; x) it holds that

(2.6) ∂ fn(x) ⊂ ∂ε f (y) ∩ ∂εn‖ · ‖(x − y).

Proof. Take p ∈ ∂ fn(x) and y ∈ Mnε( f ; x). By the definition of subgradient, for all

x′ ∈ X,

(2.7) 〈p, x′ − x〉 ≤ fn(x′) − fn(x).

We estimate from above the right hand side of (2.7) by using that fn(x′) ≤ f (x′) (seeLemma 2.1.2 (ii)) and f (y) + n‖x − y‖ ≤ fn(x) + ε (because y ∈ Mn

ε( f ; x)) and we get

(2.8) fn(x′) − fn(x) ≤ f (x′) − f (y) − n‖x − y‖ + ε.

We estimate from below the left hand side of (2.7) using that ‖p‖ ≤ n(see Lemma 2.1.2 (i)), thus

(2.9) 〈p, x′ − x〉 ≥ 〈p, x′ − y〉 − ‖p‖‖x − y‖ ≥ 〈p, x′ − y〉 − n‖x − y‖.

Combining (2.8) and (2.9) with (2.7) we get for all x′ ∈ X that

〈p, x′ − y〉 − n‖x − y‖ ≤ f (x′) − f (y) − n‖x − y‖ + ε,

or〈p, x′ − y〉 ≤ f (x′) − f (y) + ε,

which means that p ∈ ∂ε f (y).Other obvious estimation of the right hand side of (2.7) gives us for all x′ ∈ X

that〈p, x′ − x〉 ≤ fn(x′) − fn(x) ≤

≤ f (y) + n‖x′ − y‖ − f (y) − n‖x − y‖ + ε =

n‖x′ − y‖ − n‖x − y‖ + ε,

which means that p ∈ ∂εn‖ · ‖(x − y). �


We recall that the proper, convex and lower semicontinuous functions f , g : X →R ∪ {+∞} are such that

∂ f (x) ⊂ ∂g(x), ∀x ∈ X.

Since dom ∂ f is dense in dom f and the latter is non-empty because f is assumedto be proper, then there exist x ∈ dom ∂ f and p ∈ ∂ f (x).

Define the function f (x) := f (x + x) − 〈p, x〉 − f (x).The function f : X → R ∪ {+∞} is proper, convex and lower semicontinuous,

f (0) = 0 and ∂ f (x) = ∂ f (x + x) − p. Hence, 0 ∈ ∂ f (0).Analogously, define g(x) := g(x + x) − 〈p, x〉 − g(x), and observe that g : X →

R∪ {+∞} is proper, convex and lower semicontinuous function, g(0) = 0 and ∂g(x) =

∂g(x + x) − p. Moreover, 0 ∈ ∂g(0), since p ∈ ∂ f (x) ⊂ ∂g(x).It is clear also that ∂ f (x) ⊂ ∂g(x) for all x ∈ X.

Lemma 2.1.4. Let f : X → R ∪ {+∞} be a proper, convex and lower semicontinuousfunction such that f (0) = 0 and 0 ∈ ∂ f (0). Let s > 0. Then for x ∈ B(0, s), Mn

ε( f , x) ⊂B[0, 3s] for n ≥ 1/s and ε ≤ 1.

Proof. From f (0) = 0 and 0 ∈ ∂ f (0) it follows that f (y) ≥ 0 for any y ∈ X. Then

f (y) + n‖x − y‖ ≥ n‖x − y‖ ≥ n‖y‖ − n‖x‖,

and if ‖x‖ < s and ‖y‖ > 3s then

f (y) + n‖x − y‖ ≥ n‖y‖ − n‖x‖ > 3ns − ns = 2ns = ns + ns >

n‖x‖ + 1 ≥ infy{ f (y) + n‖x − y‖} + 1 = f n(x) + 1,

and no such y can be in Mnε( f , x) for ε ≤ 1. �

Lemma 2.1.5. Let f , g : X → R∪ {+∞} be proper, convex and lower semicontinuousfunctions such that f (0) = g(0) = 0, 0 ∈ ∂ f (0) ∩ ∂g(0), and ∂ f (x) ⊂ ∂g(x) for allx ∈ X. Let s > 0. Then for n ≥ 1/s, ∂ f n(x) ⊂ ∂gn(x) for all x ∈ B(0, s).

Proof. Fix ε ∈ (0, 1] and n ≥1s.

Take arbitrary x ∈ B(0, s) and let p ∈ ∂ f n(x). Obviously, ‖p‖ ≤ n.By Lemma 2.1.3, p ∈ ∂ε f (y) ∩ ∂εn‖x − y‖ for all y ∈ Mn

ε( f , x). Fix y ∈ Mnε( f , x)

and note that y ∈ B[0, 3s] by Lemma 2.1.4.Since p ∈ ∂ε f (y) by Brøndsted-Rockafellar lemma there exists q ∈ ∂ f (z) for some

z such that ‖z − y‖ ≤√ε and ‖p − q‖ ≤

√ε.


Since ∂ f (z) ⊂ ∂g(z) it follows that q ∈ ∂g(z), and

(2.10) g(x′) − g(z) ≥ 〈q, x′ − z〉, ∀x′ ∈ X.

Since p ∈ ∂εn‖x − y‖ this means that

n‖w − x′‖ − n‖x − y‖ ≥ 〈p,w − x′ + y − x〉 − ε, ∀w, x′ ∈ X.

Hence, for all w, x′ ∈ X we have

(2.11)

n‖w − x′‖ − n‖x − z‖ ≥ n‖x − y‖ − n‖x − z‖ + 〈p,w − x′ + z − x〉+〈p, y − z〉 − ε

≥ 〈p,w − x′ + z − x〉 − 2n‖y − z‖ − ε≥ 〈p,w − x′ + z − x〉 − 2n

√ε − ε.

By (2.10) and (2.11) for all w, x′ ∈ X we get

g(x′) + n‖w − x′‖ ≥ g(z) + n‖x − z‖ + 〈p,w − x〉+〈q − p, x′ − z〉 − 2n

√ε − ε

≥ gn(x) + 〈p,w − x〉 −√ε‖x′ − z‖ − 2n

√ε − ε.

Applying Lemma 2.1.4 for w ∈ B(0, s) and the function g we get

gn(w) = infx′∈B[0,3s]

{g(x′) + n‖w − x′‖},

andgn(w) = inf

x′∈B[0,3s]{g(x′) + n‖w − x′‖}

≥ gn(x) + 〈p,w − x〉 −√ε sup

x′∈B[0,3s]‖x′ − z‖ − 2n

√ε − ε

≥ gn(x) + 〈p,w − x〉 −√ε(6s +

√ε) − 2n

√ε − ε.

Letting ε to zero we obtain that

gn(w) ≥ gn(x) + 〈p,w − x〉, ∀w ∈ B(0, s),

which means that p ∈ ∂gn(x).Finally, for any n ≥ 1/s, ∂ f n(x) ⊂ ∂gn(x) for all x ∈ B(0, s). �

For continuous convex functions – such as f n and gn – the inclusion of subdiffe-rentials easily yields that they differ by a finite constant (see Rockafellar [147]).

As an immediate consequence of this and Lemma 2.1.5 one deduces

Corollary 2.1.6. Let f , g : X → R∪{+∞} be proper, convex and lower semicontinuousfunctions such that f (0) = g(0) = 0, 0 ∈ ∂ f (0) ∩ ∂g(0), and ∂ f (x) ⊂ ∂g(x) for allx ∈ X. Let s > 0. For n ≥ 1/s there exists a finite constant cn such that

(2.12) f n(x) = gn(x) + cn, for all x ∈ B(0, s).


Now it is easy to finish the reasoning.Indeed, for 0 ∈ B(0, s) by Lemma 2.1.2 (iii) we have that f n(0) → f (0) = 0 and

gn(0) → g(0) = 0 as n → ∞. Passing to limit in (2.12) yields f (0) = g(0) + limn→∞

cn,

and then limn→∞

cn = 0.

By Lemma 2.1.2 (iii) for all x ∈ B(0, s), f (x) = limn→∞

f n(x) and g(x) = limn→∞

gn(x),hence passing to limit in (2.12) yields

f (x) − g(x) = limn→∞

( f n(x) − gn(x)) = limn→∞

cn = 0,

andf (x) = g(x), ∀x ∈ B(0, s).

Having in mind that s was arbitrary, this means that

f (x) = g(x), ∀x ∈ X.

The latter is equivalent to say that for all x ∈ X,

f (x + x) − 〈p, x〉 − f (x) = g(x + x) − 〈p, x〉 − g(x),

f (x + x) − g(x + x) = f (x) − g(x),

or f (x) = g(x) + c for all x ∈ X with c = f (x) − g(x).This completes the proof.

The regularization technique presented in this section can be used for establishingintegrability of convex composite function on Asplund spaces but this will be asubject of future work.

2.2 Integrability of subdifferentials of directionally Lip-schitz functions

We will use a quantitative version of subdifferential characterization of direction-ally Lipschitz functions to study the integrability of subdifferentials of such functionsover arbitrary Banach space.

We prove results concerning the subdifferential characterization of directionallyLipschitz property of a given function. We finish by establishing the local integrabilityof subdifferentials of strictly directionally Lipschitz regular functions, continuous ontheir domains (Theorem 2.2.6).

2.2. Integrability of subdifferentials of directionally Lipschitz functions 109

Let us begin by giving some necessary definitions and preliminaries.We work in a real Banach space X with open unit ball B◦ and topological dual X∗.We will work also with a general presubdifferential operator ∂ that associates

with each function g : X → R ∪ {+∞} and with each point x ∈ X a subset ∂g(x) ofX∗, that we will call a subdifferential of g at x, and for which the properties listed inDefinition 2.1.1 hold.

Recall that the lower semicontinuous function g : X → R ∪ {+∞} is said to bedirectionally Lipschitz at x0 ∈ dom g with respect to a vector h0 ∈ X (see Rockafel-lar [150]), if there exist constants K ∈ R, ε > 0, δ > 0 such that

(2.13)t−1[g(x + th) − g(x)] ≤ K,

∀t ∈]0, ε], ∀h ∈ h0 + δB◦, ∀x ∈ x0 + δB◦ with |g(x) − g(x0)| ≤ ε.

It is clear that g is Lipschitz around x0 exactly when it is directionally Lipschitzat x0 with respect to h0 = 0. Also, it is easy to see that in the case when the lowersemicontinuous function g considered in the definition above is convex, or continuousrelative to its domain (i.e. for any x0 ∈ dom g and any γ > 0 there exists η > 0 suchthat |g(x) − g(x0)| ≤ γ for all x ∈ dom g ∩ (x0 + ηB◦)), then (2.13) is equivalent to theexistence of constants K ∈ R, ε > 0, δ > 0 such that

(2.14)t−1[g(x + th) − g(x)] ≤ K,

∀t ∈]0, ε], ∀h ∈ h0 + δB◦, and ∀x ∈ (x0 + δB◦) ∩ dom g.

The lower semicontinuous function g : X → R∪ {+∞} is said to be (see Jofre andThibault [97]) strictly directionally Lipschitz at x0 ∈ dom g with respect to h0 ∈ Xif it is directionally Lipschitz at x0 with respect to h0 with some constants K, ε, δsatisfying (2.13) and, moreover,

(2.15)g is locally Lipschitz on any set x+]0, ε](h0 + δB◦),

where x ∈ x0 + δB◦ and |g(x) − g(x0)| ≤ ε.

If the lower semicontinuous function g in the latter definition is also supposed tobe continuous relative to its domain, or convex, then (2.15) can be replaced by:

(2.16)g is locally Lipschitz on any set x+]0, ε](h0 + δB◦),

where x ∈ (x0 + δB◦) ∩ dom g.

Let us also recall that the lower semicontinuous function g : X → R ∪ {+∞} issaid to be regular at x0 ∈ dom g if lim inf

h′→ht↓0

t−1[g(x0 + th′) − g(x0)] = g↑(x0; h) for any


h ∈ X, where

d↑ϕ(x0; h) = supη>0

lim supx→ϕ x0

t↓0

(inf

h′∈h+ηB◦Xt−1[ϕ(x + th′) − ϕ(x)]

).

is the Clarke generalized derivative and it is said to be regular if it is regular at anypoint of its domain.

We wish to remind the Mean Value Theorem, established by Zagrodny in [170](see also Thibault and Zagrodny [161]), as it will be essentially used in what follows.

Mean Value Theorem of Zagrodny [170, 161]. Let f : X → R ∪ {+∞} be a lowersemicontinuous function, let a, b ∈ dom f with a , b and let ∂ be any presubdiffer-ential operator. Then there exist xn → f c ∈ [a, b[:= {(1 − t)a + tb : t ∈ [0, 1[}, andx∗n ∈ ∂ f (xn) such that:

(i) f (b) − f (a) ≤ limn→∞〈x∗n, b − a〉;

(ii)‖b − c‖‖b − a‖

( f (b) − f (a)) ≤ limn→∞〈x∗n, b − xn〉;

(iii) ‖b − a‖( f (c) − f (a)) ≤ ‖c − a‖( f (b) − f (a)).

2.2.1 Subdifferential properties of directionally Lipschitz func-tions

In this subsection we study how the directionally Lipschitz property of the functionrefers to the properties of its subdifferential. Work in this direction with the Clarkesubdifferential is that of Treiman [165], which is strongly based on the technique ofBishop and Phelps and on the result (see Treiman [165]) stating that lim inf

S3y→xK(S ; y) ⊂

T (S ; x) (here K(S ; .) and T (S ; .) denote respectively the Bouligand contingent coneand the Clarke tangent cone of a closed subset S ⊂ X). Our results are given interms of arbitrary subdifferential and their proofs are merely based on the MeanValue Theorem. They also may be considered as a quantitative version of a resultof Treiman [165, Theorem 6]. This quantitative version will be needed further toestablish Lemma 2.2.4 that is a key step in our development.

In all the sequel ∂ stands for any presubdifferential operator such that the corre-sponding subdifferential is included in the Clarke subdifferential, i.e., ∂g(x) ⊂ ∂Cg(x)for any function g : X → R ∪ {+∞} and any x ∈ X.


Lemma 2.2.1. Assume that the lower semicontinuous function g : X → R ∪ {+∞} isdirectionally Lipschitz at x0 with respect to h0 with constants K, ε, δ satisfying (2.13)(resp. (2.14)). Then

(2.17) 〈x∗, h0〉 + δ‖x∗‖ ≤ K, ∀x∗ ∈ ∂g(x), ∀x ∈ x0 + δB◦ with |g(x) − g(x0)| ≤ ε,

(resp.

(2.18) 〈x∗, h0〉 + δ‖x∗‖ ≤ K, ∀x∗ ∈ ∂g(x), ∀x ∈ x0 + δB◦).

Proof. Let g : X → R ∪ {+∞} be directionally Lipschitz at x0 with respect to h0

with constants K, ε, δ satisfying (2.14) (resp. (2.13)). Then the condition (2.14)(resp. (2.13)) obviously entails

g↑(x; h) ≤ K, ∀h ∈ h0 + δB◦, ∀x ∈ x0 + δB◦ (with |g(x) − g(x0)| ≤ ε).

Hence, if ∂g(x) , ∅ for some x ∈ x0 + δB◦ (with |g(x) − g(x0)| ≤ ε), then as ∂g(x) ⊂∂Cg(x) we have

〈x∗, h〉 ≤ g↑(x; h) ≤ K, ∀h ∈ h0 + δB◦, ∀x∗ ∈ ∂g(x) (with |g(x) − g(x0)| ≤ ε), i.e.,

〈x∗, h0〉 + δ‖x∗‖ ≤ K, ∀x∗ ∈ ∂g(x), x ∈ x0 + δB◦ (with |g(x) − g(x0)| ≤ ε). �

We proceed to show the reverse implication, i.e., that (2.18) yields the prop-erty (2.14). As it can be seen below, that case is more simple than the one establishingthat (2.17) ensures the property (2.13). So, we made the choice of separating the twoproofs.

Lemma 2.2.2. Let g : X → R ∪ {+∞} be a lower semicontinuous function, letx0 ∈ dom g and h0 ∈ X. Assume that there exist constants K ∈ R and δ > 0 suchthat (2.18) holds.

Then for any positive numbers ε0 and δ0 such that ε0(‖h0‖ + δ0) + δ0 < δ, theproperty (2.14) holds for the function g with K, ε0, δ0.

Proof. Fix ε0 > 0 and δ0 > 0 such that ε0(‖h0‖ + δ0) + δ0 < δ and take any x ∈(x0 + δ0B◦) ∩ dom g, any h ∈ h0 + δ0B◦ with h , 0, and any t ∈]0, ε0]. Considerg(x + th) and fix arbitrary real number r ∈ R with r < g(x + th). Define the lowersemicontinuous function

gr(y) :=

g(y), if y , x + th,

r, if y = x + th.


Apply the Mean Value Theorem of Zagrodny to estimate

(2.19) gr(x + th) − gr(x) = r − g(x) ≤ limn→∞〈x∗n, th〉

with x∗n ∈ ∂g(xn), xn −→n→∞

z ∈ [x, x + th[, g(xn) −→n→∞

g(z). Then for n large enough,

‖xn − x0‖ ≤ ‖xn − z‖ + ‖z − x‖ + ‖x − x0‖ ≤ ‖xn − z‖ + t‖h‖ + δ0

≤ ‖xn − z‖ + ε0(‖h0‖ + δ0) + δ0 < δ.

Hence, xn ∈ x0 + δB◦ and we can use (2.18) to get

〈x∗n, th〉 = 〈x∗n, th0〉 + t〈x∗n, h − h0〉 ≤ t[〈x∗n, h0〉 + 〈x∗n, h − h0〉]

≤ t[〈x∗n, h0〉 + δ0‖x∗n‖] ≤ tK.

This implies because of (2.19)r − g(x) ≤ tK,

which yields, on the one hand, that g(x + th) is finite (i.e. x + th ∈ dom g) and, on theother hand, that

g(x + th) − g(x) ≤ tK.

Observe that the inequality also holds for h = 0, in the case when 0 ∈ h0 + δ0B◦ sinceK ≥ 0 in that case because of (2.18). Therefore, the property (2.14) holds for g at x0

with respect to h0 with K, ε0, δ0 as above. �

Theorem 2.2.3 (subdifferential characterization of directionally Lipschitz prop-erty). Let g : X → R ∪ {+∞} be a lower semicontinuous function, let x0 ∈ dom g andh0 ∈ X. Then g is directionally Lipschitz at x0 with respect to h0 with constants K,ε′, δ′ satisfying (2.13), if and only if, there exist constants ε > 0 and δ > 0 such that(2.17) holds with K, ε, δ.

Proof. The implication =⇒ follows from Lemma 2.2.1.Let us prove the converse.Without loss of generality we may suppose that ε ∈]0, 1[ and δ ∈]0, ε[ , and that

they are such that

(2.20) g(x0) ≤ g(x) + ε, ∀x ∈ x0 + δB◦.

Now take ε′ > 0 such that ε′(‖h0‖+ |K|+ 2) < δ, and δ′ ∈]0, ε′[. Fix any x ∈ x0 + δ′B◦

with |g(x) − g(x0)| ≤ ε′, any h ∈ h0 + δ′B◦ with h , 0, and any t ∈]0, ε′]. Considerg(x + th) and suppose g(x + th) > g(x) + tK. We may choose and fix a real number


µ ∈]0, 1[ such that g(x + th) > g(x) + t(K + µ). Set r := g(x) + t(K + µ) and define thelower semicontinuous function

gr(y) :=

g(y), if y , x + th,

r, if y = x + th.

Apply the Mean Value Theorem of Zagrodny to estimate

(2.21) gr(x + th) − gr(x) = r − g(x) ≤ limn→∞〈x∗n, th〉

with some sequences x∗n ∈ ∂g(xn), xn −→n→∞

z ∈ [x, x + th[, g(xn) −→n→∞

g(z). Then, for nlarge enough, ‖xn − x0‖ ≤ ‖xn − z‖ + ε′(‖h0‖ + δ′) + δ′ < δ.

Case I. z = x.Then g(xn) −→

n→∞g(x) and for n large enough

|g(xn) − g(x0)| ≤ |g(xn) − g(x)| + |g(x) − g(x0)| ≤ |g(xn) − g(x)| + ε′ < ε.

Case II. z , x.Then Mean Value Theorem of Zagrodny and in particular (iii) ensures that

t‖h‖(g(z) − g(x)) ≤ ‖z − x‖(r − g(x)).

We rewrite the left hand side

t‖h‖(g(z) − g(x0)) ≤ ‖z − x‖(r − g(x)) + t‖h‖(g(x) − g(x0))

≤ ‖z − x‖(r − g(x)) + t‖h‖ε′

≤ t‖h‖(r − g(x) + ε′),

from where

g(z) − g(x0) ≤ r − g(x) + ε′ = t(K + µ) + ε′

≤ t(|K| + µ) + ε′ ≤ ε′(|K| + µ + 1) ≤ ε′(|K| + 2).

Hence, for sufficiently large n we have

g(xn) − g(x0) = g(xn) − g(z) + g(z) − g(x0) ≤ |g(xn) − g(z)| + ε′(|K| + 2) < ε.

Obviously, for n large enough, the points xn ∈ x0 + δB◦ and hence by (2.20) we alsohave that g(x0) − g(xn) ≤ ε. Therefore, we obtain |g(xn) − g(x0)| ≤ ε.


In both cases one has for n large enough, xn ∈ x0 + δB◦ and |g(xn) − g(x0)| ≤ ε,then by (2.17) one has 〈x∗n, h〉 ≤ K. So, by (2.21)

r − g(x) ≤ tK, i.e., t(K + µ) ≤ tK,

and the latter yields µ ≤ 0, which is a contradiction, since µ > 0. We conclude thatg(x + th) ≤ g(x) + tK, and further, if 0 ∈ h0 + δ′B◦, the inequality still holds for h = 0because, according to (2.17), K ≥ 0 in that case. The proof is then complete. �

2.2.2 Local integrability

In this subsection we use the results proved in the previous one to establish thelocal integrability of any subdifferential of a class of directionally Lipschitz functions.

We begin by showing how the property (2.14) is entailed by inclusion of subdif-ferentials for lower semicontinuous functions.

Lemma 2.2.4. Let g : X → R ∪ {+∞} be a lower semicontinuous function whichsatisfies property (2.14) at x0 ∈ dom g with respect to h0 ∈ X with some constantsK, ε, δ. Let f : X → R ∪ {+∞} be a lower semicontinuous function such that∂ f (x) ⊂ ∂g(x) for any x ∈ x0 + δB◦.

Then for any constants δ0 > 0 and ε0 > 0 such that δ0 + ε0(‖h0‖ + δ0) < δ withdom f ∩(x0+δ0B◦) , ∅, one has that the property (2.14) holds for f with the constantsK, ε0, δ0, i.e.,

t−1[ f (x + th) − f (x)] ≤ K, ∀t ∈]0, ε0], ∀h ∈ h0 + δ0B◦, and ∀x ∈ (x0 + δ0B◦) ∩ dom f .

In particular, x + [0, ε0](h0 + δ0B◦) ⊂ dom f , for all x ∈ dom f ∩ (x0 + δ0B◦). Further,f is directionally Lipschitz at x0 with respect to h0, whenever x0 ∈ dom f .

Proof. Suppose that g satisfies (2.14) with constants K, ε, δ. By Lemma 2.2.1, forany x ∈ dom g∩(x0 +δB◦) and any x∗ ∈ ∂g(x) we have 〈x∗, h0〉+δ‖x∗‖ ≤ K. Because ofthe assumption ∂ f (x) ⊂ ∂g(x) for any x ∈ x0 + δB◦, that inequality holds in particularfor all x ∈ dom ∂ f ∩ (x0 + δB◦) and x∗ ∈ ∂ f (x).

Fix any δ0 > 0 and ε0 > 0 such that δ0 + ε0(‖h0‖ + δ0) < δ. Lemma 2.2.2ensures that for all x ∈ dom f ∩ (x0 + δ0B◦), t ∈]0, ε0], and h ∈ h0 + δ0B◦ one hast−1[ f (x + th) − f (x)] ≤ K. The proof is then complete. �

We will also need the following second lemma of general interest. It concerns thegraphical density of the domain of any presubdifferential.


Lemma 2.2.5. Let f : X → R ∪ {+∞} be a lower semicontinuous function withdom f , ∅. Then dom ∂ f is f -graphically dense in dom f , i.e., for any a ∈ dom fthere exists a sequence xn ∈ dom ∂ f with xn −→

n→∞a and f (xn) −→

n→∞f (a).

Proof. Fix a ∈ dom f and ε > 0. Choose by the lower semicontinuity of f somepositive number r < ε such that f (x) > f (a) − ε for all x ∈ a + rB◦. If for anyb ∈ a + rB◦ one has f (b) ≥ f (a), then a is a local minimum point of f and byProperty 4 we have that

0 ∈ lim supxn→ f a

∂ f (xn),

i.e., there exist xn ∈ dom ∂ f such that xn −→n→∞

a and f (xn) −→n→∞

f (a). Otherwise, there

exists some b ∈ a + rB◦ with f (b) < f (a) and the Mean Value Theorem of Zagrodnyyields xn −→

n→∞c ∈ [a, b[ with f (xn) −→

n→∞f (c) and ∂ f (xn) , ∅, and such that the

conclusion (iii) of that theorem holds. The latter gives limn→∞ f (xn) = f (c) ≤ f (a).We deduce the existence of some N such that ‖xn − a‖ < r < ε and | f (xn) − f (a)| < εfor all n ≥ N, and hence the proof is complete. �

We establish now the integrability result.

Theorem 2.2.6 (integrability of regular directionally Lipschitz functions). Letg : X → R ∪ {+∞} be a lower semicontinuous regular function, continuous relative toits domain, and strictly directionally Lipschitz at x0 ∈ dom g.

Then there exist constants α > 0 and β ∈]0, α[ such that for any lower semi-continuous function f : X → R ∪ {+∞} with dom f ∩ (x0 + βB◦) , ∅ the inclusion∂ f (x) ⊂ ∂g(x) for all x ∈ x0 + αB◦ implies

f = g + const on x0 + βB◦.

If the strict directionally Lipschitz property for g at x0 with respect to h0 holds withconstants K, ε, δ satisfying (2.14), one may take α = δ and β = min

{δ2

4(‖h0‖+2δ) ,εδ2

}.

Proof. Let K, ε, and δ be given by the strictly directionally Lipschitz property of gat x0 with respect to h0, i.e., properties (2.14) and (2.16) hold for them.

Set δ0 := δ2 and ε0 := min

{δ

2(‖h0‖+2δ) , ε}, and observe that ε0 < 1. It is easy to see

that the conclusion of Lemma 2.2.4 holds for such δ0 and ε0.Put now α := δ and β := ε0δ0 = min

{δ2

4(‖h0‖+2δ) ,εδ2

}.

We claim that for arbitrary x ∈ x0 + βB◦ there exists hx ∈ h0 + δ2 B◦ such that

(2.22) x + ε0hx = x0 + ε0h0.


Indeed, for x ∈ x0 + βB◦ we may set hx := x0−xε0

+ h0 and since ‖x−x0‖

ε0< β

ε0= ε0δ0

ε0= δ

2 ,we obtain that hx ∈ h0 + δ

2 B◦.Moreover, the inclusion hx ∈ h0 + δ

2 B◦ ensures that

(2.23) x+]0, ε0](hx +

δ

2B◦

)⊂ x+]0, ε](h0 + δB◦), ∀x ∈ x0 + βB◦.

Now, fix arbitrary v ∈ dom f ∩ (x0 +βB◦) which is a non-empty set by assumption.As the set dom ∂ f is f -graphically dense in dom f by Lemma 2.2.5, we obtain asequence {xn} ⊂ dom ∂ f ⊂ dom ∂g ⊂ dom g, such that xn −→

n→∞v and f (xn) −→

n→∞f (v).

Writing‖xn − x0‖ ≤ ‖xn − v‖ + ‖v − x0‖ < ‖xn − v‖ + β,

we see that for sufficiently large n (for example n ≥ N1) we have

(2.24) ‖xn − x0‖ < β.

For n ≥ N1 denote by Cn the open convex set Cn := xn+]0, ε0](hxn + δ

2 B◦). Recall

that for n ≥ N1 we have xn ∈ (x0 + βB◦) ∩ dom g and observe also, by what precedes,that hxn ∈

(h0 + δ

2 B◦). So, by the definition of strictly directionally Lipschitz property

of g and by (2.23) it follows that g is locally Lipschitz and regular on Cn. FromLemma 2.2.4 it is clear that dom f ∩Cn , ∅ for n ≥ N1, since xn+]0, ε0]hxn ⊂ dom f .

It remains to observe that the inclusion of subdifferentials holds on Cn for n ≥ N1,since Cn ⊂ x0 + αB◦. Indeed, for any x ∈ Cn we may write x := xn + t(hxn + p) forsome t ∈]0, ε0] and some p ∈ δ

2 B◦ which ensures by (2.24) that

‖x − x0‖ = ‖xn + t(hxn + p) − x0‖ ≤ ‖xn − x0‖ + t‖hxn + p‖

< β + ε0

(‖h0‖ +

δ

2+δ

2

)= ε0

δ

2+ ε0(‖h0‖ + δ)

= ε0

(‖h0‖ +

3δ2

)≤

δ

2(‖h0‖ + 2δ)

(‖h0‖ +

3δ2

)< δ = α.

Hence, we can apply the integrability result for locally Lipschitz regular functions,see Correa and Jofre [50] and Theorem 4.1 in Thibault and Zagrodny [161], to obtainthat there exists some real constant cn such that

f (x) = g(x) + cn, ∀x ∈ Cn, ∀n ≥ N1.

From (2.22) we have that xn + ε0hxn = x0 + ε0h0 ∈ Cn, for any n ≥ N1, hence

f (x0 + ε0h0) = g(x0 + ε0h0) + cn, ∀n ≥ N1,


and, in particular because g(x0 + ε0h0) is finite according to (2.14), the value of cn

does not depend on n for n ≥ N1, say cn = c = f (x0 + ε0h0) − g(x0 + ε0h0), for alln ≥ N1, i.e.,

(2.25) f (x) = g(x) + c, ∀x ∈ Cn, ∀n ≥ N1.

Now we proceed to prove the equality f (v) = g(v) + c.

First, let us show that f (xn) = g(xn) + c for n ≥ N1. Observing by the definitionof Cn that for some vectors h0 one may have xn < Cn, we begin by verifying thatf (xn + thxn) −−→t↓0

f (xn).

As f is lower semicontinuous, we always have that f (xn) ≤ lim inft↓0

f (xn + thxn).

Further, for t ∈]0, ε0], Lemma 2.2.4 gives that xn + thxn ∈ dom f and f (xn + thxn) −f (xn) ≤ Kt. Hence, lim sup

t↓0f (xn + thxn) ≤ f (xn) and the claim is proved.

Analogously, using the strictly directionally Lipschitz property of g at x0 withrespect to h0 along with its lower semicontinuity, it is not difficult to see that g(xn +

thxn) −−→t↓0g(xn), and then, reminding that f (xn) and g(xn) are finite because of xn ∈

dom ∂ f ⊂ dom ∂g, we may conclude that

(2.26) ( f − g)(xn) = limt↓0

( f − g)(xn + thxn) = c, ∀n ≥ N1.

As xn −→n→∞

v, the lower semicontinuity of g implies that

g(v) ≤ lim infn→∞

g(xn) = limn→∞

f (xn) − c = f (v) − c,

in particular, v ∈ dom g, hence v ∈ dom g ∩ (x0 + βB◦). The continuity of g relative toits domain ensures that g(v) = lim

n→∞g(xn). By (2.26) and the fact that f (xn) −→

n→∞f (v)

we obtaing(v) = lim

n→∞g(xn) = lim

n→∞[ f (xn) − c] = f (v) − c,

i.e. f (v) = g(v) + c.Hence we have proved that

f (v) = g(v) + c, ∀v ∈ dom f ∩ (x0 + βB◦),

and in the same time we obtain via that equality

dom f ∩ (x0 + βB◦) ⊂ dom g ∩ (x0 + βB◦).

To finish the proof, it remains only to establish the opposite inclusion of thedomains. Take arbitrary u ∈ dom g∩ (x0 +βB◦) and set C := u+]0, ε0]

(hu + δ

2 B◦). Note


that for any x ∈ dom f ∩ (x0 + βB◦) , ∅, the point x + ε0hx ∈ dom f by Lemma 2.2.4and, moreover

(2.27) x + ε0hx = x0 + ε0h0 = u + ε0hu.

This ensures that dom f ∩ C , ∅. The assumptions of Theorem 4.1 in Thibault andZagrodny [161] hold for f , g and C, and we apply it to conclude that

f (x) = g(x) + c, ∀x ∈ C.

The constant is still c = f (x0 + ε0h0) − g(x0 + ε0h0) because by (2.27) one hasx0 + ε0h0 ∈ C. Observe that for any t ∈]0, ε0] the points u + thu ∈ C ⊂ dom g, wherethe last inclusion holds because of (2.14). Using the lower semicontinuity of f at u,and the continuity of g with respect to its domain, we obtain that

f (u) ≤ lim inft↓0

f (u + thu) = lim inft↓0

g(u + thu) + c = g(u) + c,

hence, u ∈ dom f . The proof is then complete. �

Note that the continuity assumption of the restriction of g on V ∩ dom g (where Vis some neighbourhood of x0) is crucial. It suffices to consider the function g from Rinto R with g(x) = 0 if x ≥ 0 and g(x) = 1 if x < 0 and the function f from R into Rwith f (x) = 0 if x ≥ 0 and f (x) = 2 if x < 0. We have ∂ f (x) ⊂ ∂g(x) for all x ∈ R butthe functions f and g are not equal near 0 up to a constant.

In the same way, it is easily seen that the lower semicontinuity assumption of fis also essential.

2.3 Integrability of subdifferentials of certain bivari-ate functions

The aim of this section is to develop the subject in studying integrability propertiesof bivariate functions defined on a product of two Banach spaces. The problem inthis setting is interesting from the point of view that the product structure allows usto introduce different concepts of continuity and regularity of the function and studyhow they rely to the integrability.

In Subsection 2.3.1 we introduce and discuss two concepts of regularity for abivariate function.

2.3. Integrability of subdifferentials of certain bivariate functions 119

The integrability of subdifferentials of locally Lipschitz bivariate functions forwhich some regularity concept holds is proved in Subsection 2.3.2.

In the following Subsection 2.3.3 we define and study two notions of directionalLipschitzness for a bivariate function.

These notions are used in Subsection 2.3.4 to establish the local integrability ofcertain directionally Lipschitz bivariate functions.

Throughout the section X,Y are real Banach spaces with open unit balls B◦X andB◦Y and topological duals X∗ and Y∗, respectively. The product space Z = X × Yequipped with the norm ‖(x, y)‖ = max{‖x‖, ‖y‖} is a Banach space whose open unitball is denoted by B◦ := B◦X × B◦Y . For a function ϕ : X → R ∪ {−∞,+∞} we denotethe effective domain by domϕ := {x ∈ X : |ϕ(x)| < +∞}.

Recall also that, for a function ϕ : X → R ∪ {−∞,+∞} with x ∈ domϕ, the lowerDini derivative in the direction h ∈ X is defined by

d−ϕ(x; h) := lim infh′→h

t↓0

t−1[ϕ(x + th′) − ϕ(x)].

Similarly one defines d+ϕ(x; h) := −d−(−ϕ)(x; h) and d↓ϕ(x; h) := −d↑(−ϕ)(x; h).Recall that the function ϕ : X → R ∪ {−∞,+∞} is said to be upper (resp. lower)

regular at x in the direction h when

d−ϕ(x; h) = d↑ϕ(x; h) (resp. d+ϕ(x; h) = d↓ϕ(x; h)),

and it is said to be upper (resp. lower) regular at x when it is so in any direction.When ϕ is locally Lipschitz near x, it is easy to see that the upper regularity in

the direction h corresponds to the existence of dϕ(x; h) := limt↓0

t−1[ϕ(x + th)−ϕ(x)] and

to the equality dϕ(x; h) = d↑ϕ(x; h).We will also use the notation

d◦ϕ(x; h) := lim sup(x′ ,α)↓ϕ(x,ϕ(x))

h′→h, t↓0

t−1[ϕ(x′ + th′) − α].

Obviously, when ϕ is Lipschitz near x one has

d↑ϕ(x; h) = d◦ϕ(x; h) = lim supx′→xt↓0

t−1[ϕ(x′ + th) − ϕ(x′)].

It is trivial that all above definitions hold for a bivariate function g : X × Y →R ∪ {−∞,+∞} considered as a function of one variable defined on the Banach spaceZ = X × Y . To simplify the notations, for (x, y) ∈ dom g we set

d−g(x, y; h, k) := d−g((x, y); (h, k)),


as well as,

d−1 g(x, y; h) := d−g(·, y)(x; h) and d−2 g(x, y; k) := d−g(x, ·)(y; k),

where g(·, y) denotes the function x′ → g(x′, y) and g(x, ·) denotes the function y′ →g(x, y′). One defines similarly d↑g(x, y; h, k), d↑1g(x, y; h), d↑2g(x, y; k), etc.

We work with a general presubdifferential operator ∂, that associates with eachfunction g : X → R∪{−∞,+∞} and with each point z = (x, y) ∈ Z a subset ∂g(z) of Z∗,that we will call a joint subdifferential of g at z, and for which the four properties fromDefinition 2.1.1 hold, i.e., a subdifferential of g when it is considered as a function ofone “joint” variable z. Also, we suppose that the joint subdifferential is included in theClarke subdifferential, i.e., ∂g(x, y) ⊂ ∂Cg(x, y) for any bivariate function g : X × Y →R∪{−∞,+∞} and any (x, y) ∈ X×Y . We will also consider the partial subdifferentialsof the function g, namely, ∂1g(x, y) := ∂1g(·, y)(x) and ∂2g(x, y) := ∂2g(x, ·)(y), where∂1ϕ (resp. ∂2ϕ) denotes some subdifferential in the sense of Definition 2.1.1 forfunctions ϕ defined on X (resp. on Y). The subdifferentials ∂1 and ∂2 are also assumedto be included in the Clarke subdifferential.

2.3.1 Concepts of regularity

For a bivariate function g defined on a product Banach space Z = X × Y weconsider two regularity concepts.

Definition 2.3.1. (Correa and Thibault [54]) The function g : X×Y → R∪{−∞,+∞}is said to be upper-upper regular (resp. upper-lower regular) at (x, y) ∈ dom g in thedirection (h, k) ∈ X × Y if(i) d−g(x, y; h, 0) = d↑g(x, y; h, 0) and(ii) d−g(x, y; 0, k) = d↑g(x, y; 0, k) (resp. d+g(x, y; 0, k) = d↓g(x, y; 0, k)).The function g is said to be upper-upper regular (resp. upper-lower regular) at(x, y) ∈ dom g if it is so in any direction (h, k) ∈ X × Y .

It is a direct consequence from the definition that the class of upper-upper reg-ular bivariate functions contains all upper regular bivariate functions (considered asfunctions of one variable (x, y)).

Definition 2.3.2. The function g : X × Y → R ∪ {−∞,+∞} is said to be separatelyupper-upper regular (resp. separately upper-lower regular) at (x, y) ∈ dom g in thedirections h ∈ X and k ∈ Y if


(i) g(·, y) is upper regular at x in the direction h ∈ X (i.e. d−1 g(x, y; h) = d↑1g(x, y; h))and(ii) g(x, ·) is upper regular at y in the direction k ∈ Y (i.e. d−2 g(x, y; k) = d↑2g(x, y; k))(resp. g(x, ·) is lower regular at y in the direction k ∈ Y (i.e. d+

2 g(x, y; k) = d↓2g(x, y; k))).The function g is said to be separately upper-upper regular (resp. separately upper-lower regular) at (x, y) ∈ dom g if it is so in any directions h ∈ X and k ∈ Y .

Below we consider only the upper-upper case of regularity in order to simplifythe presentation. As it is stated in Subsection 2.3.4, the upper-lower case can be dealtin a similar way.

It is not difficult to see that when the function g is locally Lipschitz, the upper-upper regularity implies the separate upper-upper regularity. Indeed, suppose that g islocally Lipschitz and upper-upper regular at (x, y) with respect to (h, k). Then g(·, y)is locally Lipschitz and

d−1 g(x, y; h) = d−g(x, y; h, 0) = d◦g(x, y; h, 0) ≥ d◦1g(x, y; h) ≥ d−1 g(x, y; h).

Hence, d−1 g(x, y; h) = d◦1g(x, y; h) = d↑1g(x, y; h). Analogously, one has d−2 g(x, y; k) =

d◦2g(x, y; k) = d↑2g(x, y; k) and, hence, g is separately upper-upper regular at (x, y) withrespect to h and k.

Any continuous convex-convex function g : X×Y → R is upper-upper regular (seeCorrea and Thibault [54, Proposition 2.2]) and then, the above observation impliesthat continuous convex-convex functions are also separately upper-upper regular.

A locally Lipschitz function may be separately upper-upper regular at some pointwithout being upper-upper regular at that point as the following example demon-strates.

Example 2.3.3. The locally Lipschitz function g : R × R→ R defined by

g(x, y) := min{|x|, |y|}

is separately upper-upper regular at (0, 0) in any directions h and k, but it is not upper-upper regular at (0, 0), since it is not upper-upper regular at (0, 0) in the direction(h, k) = (1, 1).

Outside the case of a locally Lipschitz function, the upper-upper regularity maynot imply the separate upper-upper regularity, as the following example shows.

Example 2.3.4. The function g : R × R→ R defined by

g(x, y) :=

x2 sin 1x , if y = 0, x > 0;

−x2, otherwise


is continuous and upper-upper regular at (0, 0), but it is not separately upper-upperregular at (0, 0), since it is not separately upper-upper regular at (0, 0) in directionsh = 1 and k = 1.

2.3.2 Integrability of subdifferentials of certain locally Lipschitzbivariate functions

In this section we establish the integrability of subdifferentials of locally Lipschitzbivariate functions g : X × Y → R for which the introduced concepts of regularity inthe previous subsection hold.

First, we prove the integrability of the partial subdifferentials of a separate upper-upper regular locally Lipschitz bivariate function.

Theorem 2.3.5. Let g : X×Y → R be a locally Lipschitz function which is separatelyupper-upper regular and let C ⊂ X × Y be an open convex set.

Then for any function f : X × Y → R ∪ {+∞} which is separately lower semicon-tinuous with dom f ∩C , ∅ the condition on partial subdifferentials

(PS) ∂1 f (x, y) ⊂ ∂1g(x, y) and ∂2 f (x, y) ⊂ ∂2g(x, y), ∀(x, y) ∈ C

impliesf (x, y) = g(x, y) + const, ∀(x, y) ∈ C.

Proof. Take any (x0, y0) ∈ C ∩ dom f which is a non-empty set by assumption, andconsider the functions f (·, y0) and g(·, y0) on the set U(y0) := {x ∈ X : (x, y0) ∈ C}.

The function g(·, y0) : X → R is locally Lipschitz and regular and the set U(y0) isopen and convex in X. The function f (·, y0) : X → R∪{+∞} is lower semicontinuous,such that dom f (·, y0)∩U(y0) , ∅ (since x0 ∈ U(y0)) and by (PS) we have ∂1 f (x, y0) ⊂∂1g(x, y0), for all x ∈ U(y0). Hence, the conditions of Theorem 4.1 in Thibault andZagrodny [161] are satisfied and we apply it to f (·, y0), g(·, y0) and U(y0) to deducethat there exists a finite constant c(y0) such that

(2.28)f (x, y0) = g(x, y0) + c(y0), ∀x ∈ U(y0) and

dom f (·, y0) ∩ U(y0) ≡ dom g(·, y0) ∩ U(y0) ≡ U(y0).

Now, consider f (x0, ·) and g(x0, ·) on U(x0) := {y ∈ Y : (x0, y) ∈ C}. The func-tion g(x0, ·) : Y → R is regular and locally Lipschitz and the set U(x0) is openand convex in Y . The function f (x0, ·) : Y → R ∪ {+∞} is lower semicontinu-ous with dom f (x0, ·)∩U(x0),∅ (since y0 ∈ U(x0)). Moreover, from (PS) we have


∂2 f (x0, y) ⊂ ∂2g(x0, y), for all y ∈ U(x0). We apply again Theorem 4.1 in Thibault andZagrodny [161] to f (x0, ·), g(x0, ·) and U(x0) to obtain a finite constant d(x0) suchthat

(2.29)f (x0, y) = g(x0, y) + d(x0), ∀y ∈ U(x0) and

dom f (x0, ·) ∩ U(x0) ≡ dom g(x0, ·) ∩ U(x0) ≡ U(x0).

We claim that for arbitrary (x, y) ∈ C we have

(2.30) ( f − g)(x, y) = ( f − g)(x0, y0),

and succeeding in proving the claim the proof will be complete.To establish the claim, let us fix arbitrary (x, y) ∈ C with (x, y) , (x0, y0).There exists δ > 0 such that the inclusion⋃{

(x′ + δB◦X) × (y′ + δB◦Y) : (x′, y′) ∈ the segment [(x0, y0), (x, y)]}⊂ C

holds, which means in particular that

(2.31) x′ + δB◦X ⊂ U(y′) and y′ + δB◦Y ⊂ U(x′), whenever (x′, y′) ∈ [(x0, y0), (x, y)].

Put u := x − x0, v := y − y0, and w := (u, v) ∈ X × Y . Choose 0 < t < min{δ||w|| , 1

}such that M := t−1 ∈ N, recalling that ||w|| := max{||u||, ||v||}.

First, we will show that

(2.32) ( f − g)(x0, y0) = ( f − g)(x0 + tu, y0 + tv) = ( f − g)((x0, y0) + t(u, v)).

To this end, observe that x0 ∈ U(y0) and x0 + tu ∈ U(y0). The latter is a consequenceof (2.31), because (x0, y0) is in the segment [(x, y), (x0, y0)] and t||u|| < δ. Using (2.28)we obtain that

(2.33) c(y0) = ( f − g)(x0, y0) = ( f − g)(x0 + tu, y0).

Similarly, we have that y0 + tv ∈ U(x0 + tu) and y0 ∈ U(x0 + tu) as a consequence of(2.31), because (x0 + tu, y0 + tv) is in the segment [(x, y), (x0, y0)] and t||v|| < δ. Notealso by (2.33) that (x0 + tu, y0) ∈ C ∩ dom f . Then we may apply (2.29) to obtain that

d(x0 + tu) = ( f − g)(x0 + tu, y0) = ( f − g)(x0 + tu, y0 + tv),

and combining with (2.33) we get (2.32).Repeating the considerations, after M steps we obtain (2.30). The proof is then

complete. �


Theorem 2.3.5 extends to the product of arbitrary Banach spaces a result of Wuand Ye (see [168, Corollary 3.16] where some restrictions on the Banach spaces arerequired).

Investigating the integrability of the joint subdifferential of a locally Lipschitzupper-upper regular bivariate function, one can prove as in Correa and Thibault [54]the following result.

Theorem 2.3.6 (Correa and Thibault [54, Proposition 3.7]). Let g : X × Y → Rbe a locally Lipschitz function which is upper-upper regular and let C ⊂ X × Y be anopen convex set.

Then for any lower semicontinuous function f : X × Y → R∪ {+∞} with dom f ∩C,∅ the condition on the joint subdifferentials

(JS) ∂ f (x, y) ⊂ ∂g(x, y), ∀(x, y) ∈ C

impliesf (x, y) = g(x, y) + const, ∀(x, y) ∈ C.

It is easy to see that, if in Theorem 2.3.6 one replaces the condition (JS) on jointsubdifferentials by the condition (PS) on partial subdifferentials, then its conclusionstill holds. This is because the upper-upper regular locally Lipschitz function g isalso separately upper-upper regular and one may invoke Theorem 2.3.5 to conclude.Hence, for a locally Lipschitz upper-upper regular bivariate function the integrabilityresult holds when the inclusion assumption is made on the partial subdifferentials, aswell as when it is made on the joint subdifferentials.

Let us note that Theorem 2.3.5 and Theorem 2.3.6 hold for convex-convex con-tinuous functions.

2.3.3 Concepts of directional Lipschitzness

In this section we will consider two notions of directional Lipschitzness for abivariate function. First is the classical definition stated for a bivariate function,namely:

Definition 2.3.7 (Rockafellar [150]). The lower semicontinuous function g : X×Y →R ∪ {+∞} is said to be directionally Lipschitz at (x0, y0) ∈ dom g with respect to avector (h0, k0) ∈ X × Y if there exist constants K ∈ R, ε > 0, δ > 0 such that:

(2.34)t−1[g(x + th, y + tk) − g(x, y)] ≤ K,for all t ∈]0, ε], (h, k) ∈ (h0, k0) + δB◦,and (x, y) ∈ (x0, y0) + δB◦ with |g(x, y) − g(x0, y0)| ≤ ε.


It is not difficult to see that g is Lipschitz around (x0, y0) exactly when it isdirectionally Lipschitz at (x0, y0) with respect to (h0, k0) = (0, 0). Also, it is easyto see that in the case when the lower semicontinuous function g considered inthe above definition is convex, or continuous relative to its domain (i.e. for any(x0, y0) ∈ dom g and any η > 0 there exists γ > 0 such that |(g(x, y) − g(x0, y0)| < η

for any (x, y) ∈ ((x0, y0) + γB◦) ∩ dom g), then (2.34) is equivalent to the existence ofconstants K ∈ R, ε > 0, δ > 0 such that

(2.35)t−1[g(x + th, y + tk) − g(x, y)] ≤ K,for all t ∈]0, ε], (h, k) ∈ (h0, k0) + δB◦,and (x, y) ∈ ((x0, y0) + δB◦) ∩ dom g.

The lower semicontinuous function g : X × Y → R ∪ {+∞} is said to be strictlydirectionally Lipschitz at (x0, y0) ∈ dom g with respect to (h0, k0) ∈ X × Y if it isdirectionally Lipschitz at (x0, y0) with respect to (h0, k0) with some constants K, ε, δsatisfying (2.34) and, moreover (see Jofre and Thibault [97])

(2.36)g is locally Lipschitz on any drop (x, y)+]0, ε]((h0, k0) + δB◦),where (x, y) ∈ (x0, y0) + δB with |g(x, y) − g(x0, y0)| ≤ ε.

If the lower semicontinuous function g in the latter definition is also supposed tobe continuous relative to its domain, or convex, then (2.36) can be replaced by:

(2.37)g is locally Lipschitz on any drop (x, y)+]0, ε]((h0, k0) + δB◦),where (x, y) ∈ ((x0, y0) + δB◦) ∩ dom g.

Now we will introduce another concept of directional Lipschitzness for a bivariatefunction which is inspired by the product structure of the space.

Definition 2.3.8. The separately lower semicontinuous function g : X×Y → R∪{+∞}will be said to be separately directionally Lipschitz at (x0, y0) ∈ dom g with respect tovectors h0 ∈ X and k0 ∈ Y , if there exist constants K ∈ R, ε > 0, δ > 0 such that:

(2.38)t−1[g(x + th, y) − g(x, y)] ≤ K, and t−1[g(x, y + tk) − g(x, y)] ≤ K,for all t ∈]0, ε], (h, k) ∈ (h0, k0) + δB◦, and(x, y) ∈ (x0, y0) + δB◦ with |g(x, y) − g(x0, y0)| ≤ ε.

Note that Definition 2.3.8 may be easily adapted to the function that is uppersemicontinuous on some of the variables in assuming that the corresponding inequalityin (2.38) holds for −g instead of g.

It is clear that when the separately lower semicontinuous function g, consideredin the above definition is supposed to be also continuous relative to its domain, then


the separate directionally Lipschitz property (2.38) is equivalent to the existence ofconstants K ∈ R, ε > 0, δ > 0 such that

(2.39)t−1[g(x + th, y) − g(x, y)] ≤ K, and t−1[g(x, y + tk) − g(x, y)] ≤ K,for all t ∈]0, ε], (h, k) ∈ (h0, k0)+δB◦, and(x, y) ∈ ((x0, y0)+δB◦) ∩ dom g.

To compare so introduced definitions of directional Lipschitzness, let us observethat for a lower semicontinuous function g : X × Y → R ∪ {+∞} the directionallyLipschitz property at (x0, y0) with respect to (h0, k0), i.e. (2.34) is equivalent to

d◦g(x, y; h, k) ≤ K,∀(h, k) ∈ (h0, k0) + δB◦, ∀(x, y) ∈ (x0, y0) + δB◦ with |g(x, y) − g(x0, y0)| ≤ ε,

while the separate directionally Lipschitz property at (x0, y0) with respect to h0 andk0, i.e., (2.38) is equivalent to

d◦1g(x, y; h) ≤ K, and d◦2g(x, y; k) ≤ K,∀(h, k) ∈ (h0, k0) + δB◦, ∀(x, y) ∈ (x0, y0) + δB◦ with |g(x, y) − g(x0, y0)| ≤ ε.

The following example (due to Rockafellar [148] and considered there in anothercontext) provides in R × R a lower semicontinuous function which is directionallyLipschitz at the origin with respect to (h0, k0) but it is not separately directionallyLipschitz with respect to h0 and k0.

Example 2.3.9. The lower semicontinuous function

f (x, y) :=

x2

y − y, if y > 0,0, if x = y = 0,

+∞, otherwise

is directionally Lipschitz at (0, 0) with respect to (h0, k0) = (1, 1) but it is not separatelydirectionally Lipschitz with respect to h0 = 1 and k0 = 1, as it is infinite on the positivex-ray.

Further, the separate directionally Lipschitz property is stronger than the non-separate one in such a way that, for a lower semicontinuous function, it implies itsdirectional Lipschitzness. Here we give the proof for a lower semicontinuous function,which is continuous relative to its domain, as this is the case that we will need later.

Lemma 2.3.10. Let g : X × Y → R ∪ {+∞} be a lower semicontinuous function,which is continuous relative to its domain. Assume that g is separately directionally


Lipschitz at (x0, y0) ∈ dom g with respect to h0 ∈ X and k0 ∈ Y with constants K, ε, δsatisfying (2.39) and fix any m ≥ max{‖h0‖, ‖k0‖}. Then g is directionally Lipschitz at(x0, y0) with respect to (h0, k0) with constants 2K, ε, δ satisfying (2.35) for any positiveconstants ε and δ such that δ + ε(m + δ) < δ and ε ≤ ε.

Proof. Take any K, ε, δ satisfying (2.39) and any positive ε and δ such thatδ + ε(m + δ) < δ and ε ≤ ε. Then, for any (x, y) ∈ ((x0, y0) + δB◦) ∩ dom g, anyt ∈]0, ε[ , and any k ∈ k0 + δB◦Y one has from (2.39) that

(2.40) t−1[g(x, y + tk) − g(x, y)] ≤ K.

Hence,

(2.41) (x, y + tk) ∈ dom g ∩ ((x0, y0) + (δ + ε(m + δ))B◦) ⊂ dom g ∩ ((x0, y0) + δB◦)

by the choice of δ and ε.Again from (2.39) we have for any h ∈ h0 + δB◦X that

(2.42) t−1[g(x + th, y + tk) − g(x, y + tk)] ≤ K.

Combining (2.40) and (2.42) one gets

t−1[g(x + th, y + tk) − g(x, y)] ≤ 2K,for all t ∈]0, ε], (h, k) ∈ (h0, k0) + δB◦, and (x, y) ∈ ((x0, y0) + δB◦) ∩ dom g.

So, according to the definition, g is directionally Lipschitz at (x0, y0) with respect to(h0, k0) with constants 2K, ε, and δ. �

Further, we show that the inclusion of partial subdifferentials entails the separatedirectional Lipschitzness. The proof of this result follows the main steps of the proofof Lemma 2.2.4 and for this reason it is omitted.

Lemma 2.3.11. Let g : X × Y → R ∪ {+∞} be a separately lower semicontinuousfunction, which is continuous relative to its domain. Suppose that g is separatelydirectionally Lipschitz at (x0, y0) ∈ dom g with respect to h0 ∈ X and k0 ∈ Y withconstants K, ε, δ satisfying (2.39) and fix any m ≥ max{‖h0‖, ‖k0‖}.

Let f : X × Y → R ∪ {+∞} be a separately lower semicontinuous function suchthat

(PS) ∂1 f (x, y) ⊂ ∂1g(x, y) and ∂2 f (x, y) ⊂ ∂2g(x, y), ∀(x, y) ∈ (x0, y0) + δB◦.

Then for any positive constants δ and ε such that δ + ε(m + δ) < δ and ε ≤ ε withdom f ∩ ((x0, y0) + δB◦) , ∅ one has

(2.43)t−1[ f (x + th, y) − f (x, y)] ≤ K, and t−1[ f (x, y + tk) − f (x, y)] ≤ K,for all t ∈]0, ε], (h, k) ∈ (h0, k0)+δB◦, and (x, y) ∈ ((x0, y0)+δB◦) ∩ dom f .


Now we put together Lemma 2.3.10 and Lemma 2.3.11 to obtain

Proposition 2.3.12. Let g : X × Y → R ∪ {+∞} be a separately lower semicontinuousfunction, which is continuous relative to its domain. Suppose that g is separatelydirectionally Lipschitz at (x0, y0) ∈ dom g with respect to h0 ∈ X and k0 ∈ Y withconstants K, ε, δ satisfying (2.39) and fix any m ≥ max{‖h0‖, ‖k0‖}. Let f : X × Y →R ∪ {+∞} be a separately lower semicontinuous function such that

(PS) ∂1 f (x, y) ⊂ ∂1g(x, y) and ∂2 f (x, y) ⊂ ∂2g(x, y), ∀(x, y) ∈ (x0, y0) + δB◦.

Then for any positive constants δ0 and ε0 such that δ0 + ε0(m + δ0) < δ/2 andε0 ≤ min

{ε, δ

2(m+δ)

}with dom f ∩ ((x0, y0) + δ0B◦) , ∅ one has

(2.44)t−1[ f (x + th, y + tk) − f (x, y)] ≤ 2K,for all t ∈]0, ε0], (h, k)∈(h0, k0)+δ0B◦, and(x, y)∈((x0, y0)+δ0B◦) ∩ dom f .

In particular, (x, y) + [0, ε0]((h0, k0) + δ0B◦) ⊂ dom f , whenever(x, y) ∈ dom f ∩ ((x0, y0) + δ0B◦).

Proof. We have from Lemma 2.3.11 that, for δ = δ/2 and ε = min{ε, δ

m+2δ

}, property

(2.43) holds for f . Further, exactly as in the proof of Lemma 2.3.10, but now workingwith f and with δ and ε in the place of δ and ε, respectively, we can take any positiveε0 and δ0 such that δ0 + ε0(m + δ0) < δ = δ/2 and ε0 ≤ ε = min

{ε, δ

2(m+δ)

}(for example

δ0 = δ/4 and ε0 = min{ε, δ

4(m+δ/2)

}) for which (2.44) holds. �

2.3.4 Local integrability of subdifferentials of certain direction-ally Lipschitz bivariate functions

The first result concerns the local integrability of the joint subdifferential ofa bivariate function for which non-separate regularity and non-separate directionalLipschitzness hold.

Theorem 2.3.13. Let g : X × Y → R ∪ {+∞} be a lower semicontinuous function,which is continuous relative to its domain. Suppose that g is upper-upper regular andstrictly directionally Lipschitz at (x0, y0) ∈ dom g with respect to (h0, k0) ∈ X × Y .

Then there exist some constants α > 0 and β ∈]0, α[ such that for any lowersemicontinuous function f : X × Y → R ∪ {+∞} with dom f ∩ ((x0, y0) + βB◦) , ∅ thecondition

(JS) ∂ f (x, y) ⊂ ∂g(x, y), ∀(x, y) ∈ (x0, y0) + αB◦


implies

f (x, y) = g(x, y) + const, ∀(x, y) ∈ (x0, y0) + βB◦.

Proof. The proof may be derived by following the steps of the proof of Theorem 2.2.6concerning the case of a directionally Lipschitz function of one variable, so we madethe choice to omit it. We need only to observe first, that Lemma 2.2.4 (that isessentially used in the above mentioned proof of Theorem 2.2.6) holds in the productBanach space Z = X × Y , and second, that we can apply Theorem 2.3.6 working onany drop that appears in the proof of Theorem 2.2.6. �

The second result establishes the local integrability of the partial subdifferentialsof a bivariate function for which the separate regularity and the separate directionalLipschitzness hold.

We need first to introduce the strict version of the separately directionally Lips-chitz property as we have already done for the directionally Lipschitz property.

Definition 2.3.14. Let g : X × Y → R ∪ {+∞} be a separately lower semicontinuousfunction that is continuous relative to its domain. We say that g is strictly separatelydirectionally Lipschitz at (x0, y0) ∈ dom g with respect to h0 ∈ X and k0 ∈ Y if itis separately directionally Lipschitz at (x0, y0) with respect to h0 and k0 with someconstants K, ε, δ satisfying (2.39) and, moreover, for each (x, y) ∈ ((x0, y0) + δB◦) ∩dom g the function g(·, y) is locally Lipschitz on the drop x+]0, ε](h0 + δB◦X), and thefunction g(x, ·) is locally Lipschitz on the drop y+]0, ε](k0 + δB◦Y).

We can now prove the result.

Theorem 2.3.15. Let g : X × Y → R ∪ {+∞} be a separately lower semicontinuousfunction which is continuous relative to its domain. Suppose also that g is separatelyupper-upper regular and strictly separately directionally Lipschitz at (x0, y0) ∈ dom gwith respect to (h0, k0) ∈ X × Y .

Then there exist some constants α > 0 and β ∈]0, α[ such that for any functionf : X×Y → R∪{+∞} that is separately lower semicontinuous with dom f ∩ ((x0, y0) +

βB◦) , ∅ the condition

(PS) ∂1 f (x, y) ⊂ ∂1g(x, y) and ∂2 f (x, y) ⊂ ∂2g(x, y), ∀(x, y) ∈ (x0, y0) + αB◦

implies

f (x, y) = g(x, y) + const, ∀(x, y) ∈ (x0, y0) + βB◦.


Proof. Let g be strictly separately directionally Lipschitz at (x0, y0) ∈ dom g withrespect to h0 ∈ X and k0 ∈ Y with constants K, ε and δ satisfying (2.39). Fix anym > max{||h0||, ||k0||}. Further, fix δ ∈]0, δ/4[ such that

δ

m + 2δ< min

{ε,

δ

2(m + δ)

}and

0 < 4δ < min{

(δ − 2δ)2

4(m + 2(δ − 2δ)),ε(δ − 2δ)

2

}.

Set α := δ, β := δ2

m+2δ and β := min{

(δ−2δ)2

4(m+2(δ−2δ)) ,ε(δ−2δ)

2

}. Note that β < δ and 4δ < β.

Let f be any function satisfying the assumptions of the theorem with α and β. Weneed several steps.

I. Set ∆ := ((x0, y0) + 2δB◦) ∩ dom f , ∅. Fix arbitrary y′ ∈ proj Y∆. By densityof dom ∂1 f (., y′) in dom f (., y′) (see Lemma 2.2.5), one may find x with (x, y′) ∈((x0, y0) + 2δB◦) such that (x, y′) ∈ dom ∂1 f ⊂ dom ∂1g ⊂ dom g.

The function g(·, y′) is lower semicontinuous, regular, continuous relative to itsdomain and according to (2.39) and Definition 2.3.14 it is also strictly directionallyLipschitz at x with respect to h0 with constants K, ε and δ − 2δ.

The function f (·, y′) is lower semicontinuous and such thatdom f (·, y′) ∩ (x+βB◦X),∅ (because it contains x), and ∂1 f (x, y′) ⊂ ∂1g(x, y′), for allx ∈ x + (δ − 2δ)B◦X ⊂ x0 + αB◦X.

Hence, according to Theorem 2.2.6 applied with α = α and β, one has that thereexists a finite constant c(y′) such that

f (x, y′) − g(x, y′) = c(y′), ∀x ∈ (x + βB◦X) ∩ dom f (·, y′), anddom f (·, y′) ∩ (x + βB◦X) ≡ dom g(·, y′) ∩ (x + βB◦X).

Further, since x + βB◦X ⊃ x0 + 2δB◦X, it follows that

(2.45)f (x, y′) − g(x, y′) = c(y′), ∀x ∈ (x0 + 2δB◦X) ∩ dom f (·, y′), and

dom f (·, y′) ∩ (x0 + 2δB◦X) ≡ dom g(·, y′) ∩ (x0 + 2δB◦X).

II. Consider now x′ ∈ proj X∆. One may find y with (x′, y) ∈ ((x0, y0) + 2δB◦) suchthat (x′, y) ∈ dom ∂2 f ⊂ dom ∂2g ⊂ dom g.

The function g(x′, ·) is lower semicontinuous, regular, continuous relative to itsdomain and according to (2.39) and Definition 2.3.14 it is also strictly directionallyLipschitz at y with respect to k0 with constants K, ε and δ − 2δ.


The function f (x′, ·) is lower semicontinuous and such thatdom f (x′, ·) ∩ (y + βB◦Y) , ∅ (because it contains y), and ∂2 f (x′, y) ⊂ ∂2g(x′, y), for ally ∈ y + (δ − 2δ)B◦Y ⊂ y0 + αB◦Y .

Again applying Theorem 2.2.6 with the same α and β, one obtains the existenceof a finite constant d(x′) such that

f (x′, y) − g(x′, y) = d(x′), ∀y ∈ (y + βB◦X) ∩ dom f (x′, ·), anddom f (x′, ·) ∩ (y + βB◦Y) ≡ dom g(x′, ·) ∩ (y + βB◦Y).

Since y + βB◦Y ⊃ y0 + 2δB◦Y , it follows that

(2.46)f (x′, y) − g(x′, y) = d(x′), ∀y ∈ (y0 + 2δB◦Y) ∩ dom f (x′, ·), and

dom f (x′, ·) ∩ (y0 + 2δB◦Y) ≡ dom g(x′, ·) ∩ (y0 + 2δB◦Y).

Combining (2.45) and (2.46) we obtain for any (x′, y′) ∈ dom f ∩ ((x0, y0) + 2δB◦)that

(2.47) c(y′) = f (x′, y′) − g(x′, y′) = d(x′).

III. We proceed to showing that c(y′) does not depend on y′, as well as d(x′) doesnot depend on x′, and that they are equal. Let us mention that the latter is not a directconsequence of (2.47) because the set dom f ∩ ((x0, y0) + 2δB◦) may not contain asubset that is a cartesian product.

Observe first that for any (x, y) ∈ (x0, y0)+βB◦ one may find (hx, ky) ∈ (h0, k0)+ δB◦

such that

(x, y) + ε(hx, ky) = (x0, y0) + ε(h0, k0), where ε :=β

δ=

δ

m + 2δ.

Indeed, to obtain hx we set hx := h0 + x0−xε

. It is easy to see that hx ∈ h0 + δB◦X, since

||hx − h0|| =||x0 − x||

ε<β

ε= δ.

To obtain ky we proceed analogously.Put u0 := x0+εh0 and v0 := y0+εk0. Further, observe that for δ and ε the conclusion

of Lemma 2.3.10 holds, since its conditions δ + ε(m + δ) ≤ δ +δ(m+δ)m+2δ < 2δ < δ and

ε ≤ ε are satisfied. Hence, from the proof of Lemma 2.3.10 we have that for any(x, y) ∈ dom g ∩ ((x0, y0) + δB) and any (h, k) ∈ (h0, k0) + δB the points (x, y + εk) and(x + εh, y) are in dom g ∩ ((x0, y0) + 2δB).

Having in mind that β < δ we combine with the preceding to obtain

(x, y) ∈ dom g∩((x0, y0)+βB◦)⇒ (x, y+εky), (x+εhx, y) ∈ dom g∩((x0, y0)+2δB◦), i.e.,


(2.48) (x, y) ∈ dom g ∩ ((x0, y0) + βB◦)⇒ (x, v0), (u0, y) ∈ dom g ∩ ((x0, y0) + 2δB◦).

Now, take any point y′ ∈ proj Y[dom f ∩ ((x0, y0) + βB◦)] ⊂ proj Y∆. Then thereexists x′ ∈ x0 + βB◦X such that (x′, y′) ∈ dom f ∩ ((x0, y0) + βB◦) and hence (x′, y′) ∈dom g ∩ ((x0, y0) + βB◦) where the latter holds according to (2.45). From (2.48) thepoint (u0, y′) ∈ dom g ∩ ((x0, y0) + 2δB◦) and hence (u0, y′) ∈ dom f ∩ ((x0, y0) + 2δB◦),the latter again by (2.45).

Since we have established that u0 ∈ dom f (·, y′), we use (2.45) to obtain

(2.49) f (u0, y′) − g(u0, y′) = c(y′).

Further, from the fact that u0 ∈ proj X∆ and from (2.46) because y′ ∈ dom f (u0, ·), wehave that

(2.50) f (u0, y′) − g(u0, y′) = d(u0).

Combining (2.49) and (2.50) we obtain

c(y′) = d(u0) for any y′ ∈ proj Y[dom f ∩ ((x0, y0) + βB◦)].

Analogously, one gets

d(x′) = c(v0) for any x′ ∈ proj X[dom f ∩ ((x0, y0) + βB◦)].

Since for arbitrary (x′, y′) ∈ dom f ∩ ((x0, y0) +βB◦) from (2.47) we have c(y′) = d(x′),those two constants are equal, i.e., d(u0) = c(v0) =: c. Finally, we obtain that thereexists a finite constant c such that

(2.51)f (x, y) − g(x, y) = c, ∀(x, y) ∈ dom f ∩ ((x0, y0) + βB◦), and

dom f ∩ ((x0, y0) + βB◦) ⊂ dom g ∩ ((x0, y0) + βB◦).

IV. To finish the proof it remains only to establish the opposite inclusion of thedomains, i.e., dom g ∩ ((x0, y0) + βB◦) ⊂ dom f .

To this end, let us take arbitrary (u, v) ∈ dom g ∩ ((x0, y0) + βB◦). First we willshow that (u, v0) ∈ dom f . Consider the drop U := (u, v0)+]0, ε]

({h0 + δB◦X} × {0}

)=:

U1 × {v0} and observe that u0 ∈ U1 because

u0 = x0 + εh0 = u + ε(h0 +

x0 − uε

)and‖x0 − u‖

ε<β

ε= δ.

We claim that U ⊂ ((x0, y0) + αB◦). Indeed, one has

β + ε(m + δ) = εδ + ε(m + δ) = ε(m + 2δ) =δ

m + 2δ(m + 2δ) = δ < α,


and if we take any (x, y) ∈ U we use the above estimation to obtain

||x − x0|| ≤ ||x − u|| + ||u − x0|| ≤ ε(||h0|| + δ) + β < ε(m + δ) + β < α,

||y − y0|| = ||v0 − y0|| = ε||k0|| < εm < α

and the claim is proved. Both last inequalities allow us to invoke (PS) which implies

(2.52) ∂1 f (x, v0) ⊂ ∂1g(x, v0), ∀x ∈ U1.

We claim that (u0, v0) ∈ dom f . This is a consequence of Proposition 2.3.12.Indeed, as δ + ε(m + δ) < 2δ < δ/2 and ε ≤ min

{ε, δ

2(m+δ)

}, we may take δ0 = δ and

ε0 = ε in Proposition 2.3.12. Using the inequality β < δ and fixing any (x′, y′) ∈dom f ∩ ((x0, y0) + βB◦) we then get

(2.53) (x′ + εhx′ , y′ + εky′) = (u0, v0) ∈ dom f ,

since (hx′ , ky′) ∈ (h0, k0) + δ0B◦.We observe now that we have (u, v0) ∈ dom g because of (2.48). So, the choice

of δ and ε together with (2.39) imply that U1 ⊂ dom g(·, v0). Moreover, from thestrict directionally Lipschitz property, g(·, v0) is locally Lipschitz and regular on U1.As u0 ∈ U1 (see the beginning of the present step), we have by (2.53) that u0 ∈

U1 ∩ dom f (·, v0), hence the latter is non-empty. Taking all those facts and (2.52) intoaccount, we can apply Theorem 4.1 in Thibault and Zagrodny [161] with f (·, v0),g(·, v0) and U1 to obtain that there exists a finite constant c1 such that

(2.54) f (x, v0) = g(x, v0) + c1, ∀x ∈ U1.

In fact one can show that c1 = c. For t ∈]0, ε] we have that the point (u + th0, v0) ∈U ∩ dom g. We use the lower semicontinuity of f at (u, v0) with respect to the firstvariable and the continuity of g relative to its domain, as well as (2.54), to get

f (u, v0) ≤ lim inft↓0

f (u + th0, v0) = lim inft↓0

g(u + th0, v0) + c1 = g(u, v0) + c1.

Hence, (u, v0) ∈ dom f .Now, consider the drop V := (u, v)+]0, ε]

({0} × {k0 + δB◦Y}

)=: {u} × V2. We have

that V ⊂ ((x0, y0) + αB◦) exactly by the same reasons as above, so (PS) implies that∂2 f (u, y) ⊂ ∂2g(u, y) for all y ∈ V2. We have already showed that v0 ∈ dom f (u, ·)∩V2.Again the choice of δ and ε and (2.39) ensure that V2 ⊂ dom g(u, ·). Moreover,g(u, ·) is locally Lipschitz and regular on V2. We apply Theorem 4.1 in Thibault and


Zagrodny [161] with f (u, .), g(u, .) and V2 to obtain that there exists a finite constantc2 such that

f (u, y) = g(u, y) + c2, ∀y ∈ V2.

For any t ∈]0, ε] the point v + tk0 ∈ V2, the function f is lower semicontinuous at(u, v) with respect to the second variable, and the function g is continuous relative toits domain, hence

f (u, v) ≤ lim inft↓0

f (u, v + tk0) = lim inft↓0

g(u, v + tk0) + c2 = g(u, v) + c2.

This means that (u, v) ∈ dom f and the proof is then complete. �

Theorem 2.3.15 implies for example the local integrability of any extended realvalued convex-convex function g, which is lower semicontinuous on each variableand continuous relative to its domain, in a neighbourhood of a point (x0, y0) such thatint [dom g(·, y0) × dom g(x0, ·)] , ∅.

In order to deal together with the separate regularity and the joint (non-separate)directionally Lipschitz property, we prove the following result.

Theorem 2.3.16. Let g : X × Y → R ∪ {+∞} be a separately lower semicontinuousfunction which is continuous relative to its domain. Suppose also that g is separatelyupper-upper regular and strictly directionally Lipschitz at (x0, y0)∈ dom g with respectto (h0, 0) ∈ X × Y .

Then there exist some constants α > 0 and β ∈]0, α[ such that for any functionf : X × Y → R ∪ {+∞} that is separately lower semicontinuous withdom f ∩ ((x0, y0) + βB◦) , ∅ the condition

(PS) ∂1 f (x, y) ⊂ ∂1g(x, y), and ∂2 f (x, y) ⊂ ∂2g(x, y), ∀(x, y) ∈ (x0, y0) + αB◦

impliesf (x, y) = g(x, y) + const, ∀(x, y) ∈ (x0, y0) + βB◦.

Proof. We will give only a sketch of the proof because most of the techniques appliedcan be found in the proof of Theorem 2.2.6.

Fix constants K, ε, δ such that properties (2.35) and (2.37) hold for g with respectto (h0, 0). Set δ0 := δ/2, ε0 := min

{ε, δ

2(||h0 ||+2δ)

}and put α := δ, β := ε0δ0.

Let f be any function satisfying the assumptions of the theorem with such α andβ. According to Lemma 2.2.4, it is not difficult to see that the properties of g and thecondition (PS) ensure that if (x, y) ∈ dom f ∩ ((x0, y0) + δ0B◦) then

(2.55) (x, y) + [0, ε0](h, 0) ⊂ dom f , ∀h ∈ h0 + δ0B◦X.


For any (x, y) ∈ X × Y we will put hx = h0 + x0−xε0

and ky =y0−yε0. Fix any (u, v) ∈

dom f ∩ ((x0, y0)+βB◦). Take (see Lemma 2.2.5) for each integer n some xn in X suchthat

(xn, v) ∈ ((x0, y0) + βB◦) ∩ dom ∂1 f ⊂ dom ∂1g ⊂ dom g,

and such that xn → u and f (xn, v)→ f (x, v). The drop

Cn := (xn, v)+]0, ε0]({hxn + δ0B◦X} × {kv + δ0B◦Y}

)contains the point (x0+ε0h0, y0)=(xn+ε0hxn , v+ε0kv) and, since ‖kv‖<δ0, it also containsthe half open segment (xn, v)+]0, ε0](hxn , 0). Further, since β<δ0 by (2.55) one has

(xn, v)+]0, ε0](hxn , 0) ⊂ dom f .

On the open convex set Cn ⊂ X × Y the function g is locally Lipschitz andseparately upper-upper regular. Thus, Theorem 2.3.5 entails that there exists a finiteconstant c := ( f − g)(x0 + ε0h0, y0) such that

f (x, y) = g(x, y) + c, ∀(x, y) ∈ Cn.

Now applying the techniques from the proof of Theorem 2.2.6 to the function f (., v)we obtain that

f (u, v) = g(u, v) + c.

Therefore,

f (x, y) = g(x, y) + c, ∀(x, y) ∈ dom f ∩ ((x0, y0) + βB◦), anddom f ∩ ((x0, y0) + βB◦) ⊂ dom g ∩ ((x0, y0) + βB◦).

To establish the opposite inclusion of the domains it is enough to see that for any(u, v) ∈ dom g∩ ((x0, y0) +βB◦), the drop C := (u, v)+]0, ε0]

((hu + δ0B◦X) × (kv + δ0B◦Y)

)contains the point (x0 + ε0h0, y0) and the latter belongs to dom f , according to thedefinition of c. Hence, applying again Theorem 2.3.5 we obtain that f (x, y) = g(x, y)+cfor all (x, y) ∈ C, in particular for all t ∈]0, ε0]

f (u + thu, v) = g(u + thu, v) + c.

Exploring the lower semicontinuity of f with respect to the first variable and continu-ity of g on its domain, we obtain that (u, v) ∈ dom f . The proof is then complete. �

Theorem 2.3.16 in particular implies the local integrability of any extended realvalued convex-convex function g, which is lower semicontinuous with respect to each


variable and continuous relative to its domain, around a point (x0, y0) such that thereexists some x ∈ X for which (x, y0) ∈ int dom g.

To conclude, let us note that all integration results obtained in this section canbe easily adapted for continuous bivariate functions, for which a type of upper-lowerregularity assumption holds (for example, continuous convex-concave functions).

2.4 Partially ball weakly inf-compact saddle functions

In this section we study saddle functions K : X × Y → [−∞,+∞] defined on aproduct of Banach spaces X and Y . Such functions are closely related to minimaxproblems. The main contribution in the study of saddle functions with values in[−∞,+∞] is due to Rockafellar. In the case when X and Y are finite dimensional,their properties are investigated in detail in a number of works of Rockafellar (cf.,e.g., [142, 145, 148]), McLinden [117, 118, 119], etc. Most of those properties aregeneralized to the case when X and Y are Banach spaces, one of which is reflexive,in subsequent papers of Rockafellar [146, 149], Gossez [79], etc. Our intention hereis to extend their results to a class of saddle functions defined on a product of twoarbitrary Banach spaces X and Y .

The properties of saddle functions on the Banach space X × Y are laid out inSubsection 2.4.1.

Subsection 2.4.2 is devoted to the study of subdifferentiability properties of sad-dle functions, and more precisely, the subdifferential properties of a proper closedsaddle function K : X ×Y → [−∞,+∞] that we call partially ball weakly inf-compact(Definition 2.4.8) and write for short pbwc. We establish that the domain of the sub-differential ∂K is non-empty (Theorem 2.4.11). Moreover, in this setting the operatorTK associated with ∂K is maximal monotone (Theorem 2.4.14). For proper closedsaddle functions K in a large subclass of pbwc saddle functions we show that thedomain of ∂K is dense in the domain of K (Theorem 2.4.12).

In the final Subsection 2.4.3 we prove that for proper closed pbwc saddle functionsK the subdifferential ∂K is integrable (Theorem 2.4.15 and Theorem 2.4.16).

We can conclude that there exist clear parallels between basic properties of properclosed convex functions defined on a Banach space X and those of proper closedpbwc saddle functions defined on a product Banach space X × Y .

We work in a real Banach space (X, ‖ · ‖) with topological dual space X∗. Thedual of the Banach space X∗ is called the bidual of X and it is denoted by X∗∗. Any

2.4. Partially ball weakly inf-compact saddle functions 137

element of X “gives” an element of X∗∗ via the canonical embedding : X ↪→ X∗∗

defined by

〈x∗, x〉 = 〈x, x∗〉 for x ∈ X and x∗ ∈ X∗.

Let us recall that two normed linear spaces are called congruent (relation denoted by�) if there exists a norm preserving isomorphism (called a congruence) between them.It is well-known (see Holmes [86]) that the canonical embedding is a congruencebetween X and its image X in X∗∗, i.e., X � X. The image X of X is a norm closedsubspace of X∗∗ and ‖x‖ = ‖x‖. When X coincides with X∗∗, the Banach space X isreflexive. Analogously, the Banach space X∗ is congruent to X∗, i.e., X∗ � X∗ via thecanonical embedding : X∗ ↪→ X∗∗∗ defined by

〈x∗∗, x∗〉 = 〈x∗, x∗∗〉 for x∗ ∈ X∗ and x∗∗ ∈ X∗∗.

Moreover,(X)∗� X∗ (see Holmes [86, p. 123]).

A convex function f on X is an everywhere defined function with values in theextended real interval [−∞,+∞] whose epigraph epi f := {(x, r) ∈ X × R : f (x) ≤ r}is a convex set in X × R. The effective domain of f is defined by dom f := {x ∈X : f (x) < +∞}. If f (x) > −∞ for all x and f (x) < +∞ for at least one x, then f issaid to be proper. Otherwise, f is said to be improper. The convex function f is saidto be closed if it is proper and lower semicontinuous, or else, if it is identically +∞

or −∞. Through the subsection, unless otherwise be specified, the closure and lowersemicontinuity operations will be taken with respect to the norm topology. Given anyconvex function f on X, there exists a greatest closed convex function majorized byf . This function is called the closure of f and is denoted by cl f . It is clear thatcl f ≤ f and f is closed exactly when f = cl f . When f does not take the value −∞,then

(2.56) cl f (x) = lim infx′→x

f (x′), for all x ∈ X.

For any convex function f on X, the function f ∗ : X∗ → [−∞,+∞] defined by

f ∗(x∗) = supx∈X{〈x, x∗〉 − f (x)}

is called the conjugate of f . The conjugate of a proper lower semicontinuous convexfunction f on X is a proper convex function on X∗ which is lower semicontinuouswith respect to the weak-star topology w(X∗, X), as well as, to the norm topology ofX∗. One defines the biconjugate of a convex function f on X as the conjugate of its


conjugate function, i.e., it is the function f ∗∗ : X∗∗ → [−∞,+∞] on the bidual spaceX∗∗ defined by

f ∗∗(x∗∗) = supx∗∈X∗{〈x∗, x∗∗〉 − f ∗(x∗)}.

The biconjugate of a proper convex lower semicontinuous function f is a properconvex function on X∗∗ which is lower semicontinuous with respect to the w(X∗∗, X∗)topology, as well as, to the norm topology of X∗∗. By Fenchel’s duality result, for aconvex function f one has

(2.57) f ∗∗(x) = cl f (x) for all x ∈ X.

The reader interested in the theory of conjugate convex functions could consult forexample Brøndsted [35], Fenchel [74], Moreau [127, 129], Rockafellar [143, 148].

When f is convex and proper, and ∂ f is its convex subdifferential, dom ∂ f ⊂dom f . We will often use the following well-known result of Brøndsted and Rock-afellar (see [36]):

(2.58)if the convex function f : X → R ∪ {+∞} is properand lower semicontinuous, then dom ∂ f is dense in dom f .

Recall that the range of ∂ f is the subset of X∗ given by Rge ∂ f :=⋃x∈X

∂ f (x) while

the graph of ∂ f is the set gph ∂ f := {(x, x∗) ∈ X × X∗ : x∗ ∈ ∂ f (x)}.It is well known (see for example Aubin and Ekeland [8], Moreau [129] and

Rockafellar [148]) that for a proper convex function f the following are equivalent

x∗ ∈ ∂ f (x) ⇐⇒ f (x) + f ∗(x∗) = 〈x, x∗〉,

and any of those implies that x ∈ dom f . If, in addition f is lower semicontinuous atx, then

(2.59) x∗ ∈ ∂ f (x) ⇐⇒ f (x) + f ∗(x∗) = 〈x, x∗〉 ⇐⇒ x ∈ ∂ f ∗(x∗).

Following Rockafellar [148, Corollary 23.5.2], one derives that

(2.60)if a proper convex function f is subdifferentiable at x thencl f (x) = f (x) and ∂(cl f )(x) = ∂ f (x).

If X is reflexive and f is a proper lower semicontinuous convex function, from(2.59) it is clear that one can identify f ∗∗ with f and that ∂ f ∗ is just the “inverse” of∂ f . In other words, x ∈ ∂ f ∗(x∗), if and only if, x∗ ∈ ∂ f (x). If X is not reflexive, therelationship between ∂ f ∗ and ∂ f is more complicated, but ∂ f ∗ and ∂ f still completelydetermine each other, according to the following result due to Rockafellar [147].


Theorem 2.4.1 (Rockafellar [147, Proposition 1]). Let f : X → R ∪ {+∞} bea proper lower semicontinuous convex function and let x∗ ∈ X∗, x∗∗ ∈ X∗∗. Thenx∗∗ ∈ ∂ f ∗(x∗), if and only if, there exists a net {x∗α}α∈A in X∗ converging to x∗ in thenorm topology and a bounded net {xα}α∈A in X (with the same partially ordered indexset A) converging to x∗∗ in the w(X∗∗, X∗) topology such that x∗α ∈ ∂ f (xα) for everyα ∈ A.

For any r ∈ R, the r-sublevel set of the convex function f is the (possibly empty)set { f ≤ r} := {x ∈ X : f (x) ≤ r}. Obviously, it is a convex set and when f is lowersemicontinuous, it is closed in the norm topology as well as in the weak topology ofX. Let BX denote the closed unit ball of the Banach space X.

We say that the function f : X → R ∪ {+∞} is ball weakly inf-compact (bwc forshort) if for any r ∈ R the sets Levr,n( f ) := { f ≤ r} ∩ nBX are w(X, X∗) compact forany n ∈ N. We make the convention that the empty set is weakly compact. Recallthat the notion of inf-compactness was introduced by Moreau (see [129]).

From Theorem 2.4.1 we derive the following characterizations of a proper closedbwc convex function.

Theorem 2.4.2. Let f : X → R ∪ {+∞} be a proper lower semicontinuous convexfunction. Then the following are equivalent:(a) f is bwc;(b) the range of ∂ f ∗ is a non-empty subset of X;(c) dom f ∗∗ ⊂ X.

Proof. (a) ⇒ (b). Since f ∗ is a proper closed convex function, the range of ∂ f ∗ isnon-empty according to (2.58). Take any x∗∗ ∈ ∂ f ∗(x∗). Then by Theorem 2.4.1 wehave that there exist a net {x∗α}α∈A in X∗ converging to x∗ in the norm topology anda bounded net {xα}α∈A in X converging to x∗∗ in the w(X∗∗, X∗) topology such thatx∗α ∈ ∂ f (xα) for every α ∈ A. As f is proper we take any x0 ∈ dom f and by thedefinition of the subdifferential we have

f (xα) ≤ f (x0) + 〈xα − x0, x∗α〉, ∀α ∈ A.

If we set r := 1 + f (x0) + 〈x∗, x∗∗ − x0〉, there exists some α0 ∈ A such that f (xα) ≤ rfor all α ≥ α0. Since the net {xα}α∈A is norm bounded, for sufficiently large n ∈ N,the points xα ∈ { f ≤ r} ∩ nBX for all α ≥ α0. The w(X, X∗) compactness of the latterset ensures that the w(X∗∗, X∗) closure of its embedding in X∗∗ lies in X (cf., e.g.,Holmes [86, p. 149]). Hence, there exists some x ∈ X such that x∗∗ = x.


(b) ⇒ (c). Suppose the contrary, i.e., there exists some x∗∗ ∈ dom f ∗∗ such thatx∗∗ ∈ X∗∗ \ X. Then there exists a norm open neighbourhood U of x∗∗ in X∗∗ suchthat U ∩ X = ∅. By the norm density of the domain of the subdifferential of a properclosed convex function in its effective domain (see (2.58)) it follows the existenceof some x∗∗0 ∈ U ∩ dom ∂ f ∗∗. Let x∗∗∗0 ∈ ∂ f ∗∗(x∗∗0 ). By Theorem 2.4.1 we have thatthere exist a net {x∗∗α }α∈A in X∗∗ converging to x∗∗0 in the norm topology of X∗∗ and abounded net {x∗α}α∈A in X∗ converging to x∗∗∗0 in the w(X∗∗∗, X∗∗) topology such thatx∗∗α ∈ ∂ f ∗(x∗α) for every α ∈ A. From the norm convergence of {x∗∗α } to x∗∗0 we havethat x∗∗α are eventually in U, in particular x∗∗α < X. But from (b) we have that x∗∗α ∈X,which yields a contradiction. Hence, dom f ∗∗⊂X.

(c)⇒ (a). Let us fix r ∈ R and n ∈ N such that Levr,n( f ) is a non-empty set, otherwisethe claim is trivial. Take an arbitrary net {xα}α∈A ⊂ Levr,n( f ). Since ‖xα‖ = ‖xα‖ andf (xα) = f ∗∗(xα) by (2.57), we have xα ∈ S := {x∗∗ ∈ X∗∗ : f ∗∗(x∗∗) ≤ r} ∩ nBX∗∗ . Thelater set being obviously w(X∗∗, X∗) compact one may extract from {xα}α∈A a subnet{xs(γ)}γ∈Γ that converges to some x∗∗ ∈ S in the w(X∗∗, X∗) topology. The definitionof S yields that x∗∗ ∈ dom f ∗∗. By the assumption of (c) we have that x∗∗ = xfor some x ∈ X, and x∗∗ ∈ nBX∗∗ implies that x ∈ nBX. Obviously, {xs(γ)} tends tox in the w(X, X∗) topology and x ∈ Levr,n( f ) since f (x) = f ∗∗(x) by (2.57). Fromthe arbitrary net {xα}α∈A ⊂ Levr,n( f ) we have shown how to obtain a subnet thatis w(X, X∗) convergent to an element of Levr,n( f ). This means that the latter set isw(X, X∗) compact. The proof is then complete. �

To finish, let us recall that a concave function g on X is an everywhere definedfunction with values in the extended real interval [−∞,+∞] such that the function(−g) is convex.

2.4.1 Saddle functions. Properties

We will work in a space Z, which is a product of two real Banach spaces X andY , i.e., Z = X × Y . Setting any reasonable norm on X × Y we have that Z is a Banachspace (take for instance ‖(x, y)‖ := max{‖x‖, ‖y‖}) and its dual can be identified withX∗ × Y∗ using the pairing 〈(x, y), (x∗, y∗)〉 = 〈x, x∗〉 + 〈y, y∗〉.

A saddle function on Z is an everywhere defined function K with values in[−∞,+∞] such that K(·, y) is convex on X for each y ∈ Y and K(x, ·) is concaveon Y for each x ∈ X. We denote by cl 1K the saddle function obtained by closingK(x, y) as a convex function of x for each fixed y, i.e., cl 1K(x, y) := cl K(·, y)(x).Similarly, we denote by cl 2K the saddle function obtained by closing (−K)(x, y) as


a convex function of y for each fixed x and after that by taking its negative, i.e.,cl 2K(x, y) := −cl (−K)(x, ·)(y). Clearly, cl 1K ≤ K ≤ cl 2K.

Two saddle functions K and L on Z are said to be equivalent (written K ∼ L) ifcl 1K = cl 1L and cl 2K = cl 2L (see Gossez [79] and Rockafellar [148, 149]). WhenK ∼ L, we say that K and L belong to the same equivalence class and that K and Lare representatives of the class. There exists another definition for the relation K ∼ Lwhich is, of course, equivalent to the former. For the second one we need to introducetwo more notions.

The function on X × Y∗ obtained by taking the conjugate of (−K)(x, y) in thesecond argument (its convex argument) when the first argument is fixed, i.e., F(x, ·) :=[−K(x, ·)]∗, or

(2.61) F(x, y∗) := supy∈Y{K(x, y) + 〈y, y∗〉},

will be called the convex parent of K (see Rockafellar [145]). It is a convex functionover X × Y∗. Dually, the concave parent of K is defined by G(·, y) := −[K(·, y)]∗, or

(2.62) G(x∗, y) := infx∈X{K(x, y) − 〈x, x∗〉}.

All parent functions that appear in the section are considered as functions of thejoint variable belonging to the product Banach space, and their effective domains andsubdifferentials are taken in this setting.

It is shown by Rockafellar in [148] that two saddle functions are equivalent exactlywhen they have the same parent functions. Here we give a proof for completenessand for the convenience of the reader.

Lemma 2.4.3. Let K, L : X × Y → [−∞,+∞] be two saddle functions. Then K ∼ L,if and only if, they have the same parent functions.

Proof. First, let us suppose that K ∼ L, i.e., cl 1K = cl 1L and cl 2K = cl 2L accordingto the original definition. Denote by F the convex parent of K and by F′ the convexparent of L. Then

(2.63) F(x, y∗) := supy∈Y{K(x, y) + 〈y, y∗〉} = sup

y∈Y{cl 2K(x, y) + 〈y, y∗〉},

where for the latter equality we use the fact that a convex function and its closurehave the same conjugate function (see Moreau [129]). Hence,

F(x, y∗) = supy∈Y{cl 2K(x, y) + 〈y, y∗〉} = sup

y∈Y{cl 2L(x, y) + 〈y, y∗〉} = F′(x, y∗).


Analogously, for the concave parent G of K and the concave parent G′ of L oneobtains that G(x∗, y) = G′(x∗, y).

Now, let us suppose that F = F′ and G = G′. From Fenchel duality (2.57) wehave

−cl 2K(x, y) = supy∗∈Y∗{〈y, y∗〉 − F(x, y∗)} = sup

y∗∈Y∗{〈y, y∗〉 − F′(x, y∗)} = −cl 2L(x, y),

hence cl 2K = cl 2L. Analogously, cl 1K = cl 1L and the proof is complete. �

If cl 1K and cl 2K are both equivalent to K (written K ∼ cl 1K ∼ cl 2K), then K issaid to be a closed saddle function (see Gossez [79] and Rockafellar [148, 149]). Anecessary and sufficient condition for K to be closed is

(2.64) cl 1cl 2K = cl 1K and cl 2cl 1K = cl 2K.

It is easy to see that when K is a closed saddle function and L is a saddle functionequivalent to K, i.e., L ∼ K, then L is closed too.

The effective domain of a saddle function K (see Rockafellar [145, 146, 148]) isdefined as the set Dom ′K = C′K × D′K , where

C′K = {x ∈ X : K(x, y) < +∞, ∀y ∈ Y},

D′K = {y ∈ Y : K(x, y) > −∞, ∀x ∈ X}.

A basic disadvantage of this definition comes from the fact that Dom ′K depends onthe representative of the equivalence class of K, as it can be seen from an exampledue to Rockafellar (see Gossez [79]). Hence, one introduces the following moreappropriate notion for the domain of a saddle function K that depends only on theequivalence class to which K belongs (and not on the representatives of the class).The domain of a saddle function K (see Gossez [79] and Rockafellar [149]) is definedby Dom K = CK × DK , where

CK = {x ∈ X : cl 2K(x, y) < +∞, ∀y ∈ Y},

DK = {y ∈ Y : cl 1K(x, y) > −∞, ∀x ∈ X}.

Clearly, Dom K ⊂ Dom ′K. If K is closed, then Dom K is dense in Dom ′K, andDom K = Dom ′cl 1K ∩Dom ′cl 2K (see Gossez [79, Proposition 1]). The saddle func-tion K is said to be proper if Dom ′K is a non-empty set. From the preceding, if K isclosed, it is proper exactly when Dom K is a non-empty set.


Lemma 2.4.4. Let K : X × Y → [−∞,+∞] be a saddle function. Then

Dom K = {(x, y) ∈ X × Y : −∞ < cl 1K(x, y) ≤ K(x, y) ≤ cl 2K(x, y) < +∞}.

Proof. Let us denote

V := {(x, y) ∈ X × Y : −∞ < cl 1K(x, y) ≤ K(x, y) ≤ cl 2K(x, y) < +∞}.

The inclusion V ⊂ Dom K being obvious for V = ∅, we may suppose that V , ∅.Take any (x0, y0) ∈ V . Suppose that y0 < DK . Then there exists x ∈ X such thatcl 1K(x, y0) = −∞. Then, by the definition of the closure of a convex function, itfollows that cl 1K(x, y0) = −∞ for all x ∈ X, which yields a contradiction, sincecl 1K(x0, y0) is finite. Hence, y0 ∈ DK . Analogously, one shows that x0 ∈ CK and then(x0, y0) ∈ dom K, so the inclusion V ⊂ Dom K is established.

The opposite inclusion, i.e., Dom K ⊂ V , being obvious for Dom K = ∅, wesuppose it is non-empty and take any (x0, y0) ∈ Dom K. Then

x0 ∈ CK =⇒ cl 2K(x0, y) < +∞, ∀y ∈ Y =⇒ cl 2K(x0, y0) < +∞,

y0 ∈ DK =⇒ cl 1K(x, y0) > −∞, ∀x ∈ X =⇒ cl 1K(x0, y0) > −∞,

hence (x0, y0) ∈ V . The proof is then complete. �

The following statement concerns a useful property of a proper closed saddlefunction.

Lemma 2.4.5. Let K : X × Y → [−∞,+∞] be a proper closed saddle function. Then(a) for any y ∈ DK , the function cl 1K(·, y) is a proper lower semicontinuous convex

function with dom cl 1K(·, y) ⊃ CK and, for any y < DK , the function cl 1K(·, y) = −∞;(b) for any x ∈ CK , the function (−cl 2K)(x, ·) is a proper lower semicontinuous

convex function with dom(−cl 2K)(x, ·) ⊃ DK and, for any x < CK , the function−cl 2K(x, ·) = −∞.

Proof. We will prove (a), the proof of (b) being similar. Since K is proper and closed,Dom K = CK × DK is non-empty.

Take y ∈ DK . By definition, cl 1K(x, y) > −∞ for all x ∈ X and for any x ∈ CK ,cl 1K(x, y) is finite by Lemma 2.4.4. Hence, cl 1K(·, y) is a proper lower semicontinu-ous convex function and CK ⊂ dom cl 1K(·, y).

Take y < DK . Then there exists x ∈ X such that cl 1K(x, y) = −∞ and the definitionof the closure of a convex function implies that cl 1K(·, y) = −∞. �


The following result is a simple extension of Rockafellar [145, Lemma 1].

Lemma 2.4.6. Let K : X × Y → [−∞,+∞] be a proper closed saddle function and letF and G be its convex and concave parent, respectively. Then F : X×Y∗ → R∪{+∞}and (−G) : X∗ × Y → R ∪ {+∞} are proper lower semicontinuous convex functions,such that dom F ⊂ CK ×Y∗, dom(−G) ⊂ X∗×DK . Moreover, F is the restriction of theconjugate function (−G)∗ : X∗∗ × Y∗ → R∪ {+∞} to the norm closed subspace X × Y∗,i.e.,

F(x, y∗) = (−G)∗(x, y∗),

and (−G) is the restriction of the conjugate function F∗ : X∗ × Y∗∗ → R∪ {+∞} to thenorm closed subspace X∗ × Y , i.e.,

−G(x∗, y) = F∗(x∗, y).

Proof. Take a proper closed saddle function K. Then K, cl 1K and cl 2K have thesame convex parent F according to Lemma 2.4.3. Hence, F is closed and convexbecause it is a supremum of closed convex functions cl 1K(·, y) + 〈·, y〉 on X × Y∗. Ina similar way one has that (−G) is closed and convex.

The properness and closedness of K imply that the set Dom K = CK × DK is non-empty. The inclusions of the domains hold from Lemma 2.4.5 and from the closednessof K. For x ∈ CK , the function (−cl 2K)(x, ·) is a proper lower semicontinuous convexfunction according to Lemma 2.4.5 and F(x, ·) is its conjugate. Hence, F(x, ·) alsois a proper function (see Aubin and Ekeland [8, p. 201, Theorem 2]). This and theclosedness of F imply that F is a proper function. Analogously one obtains that (−G)is a proper function.

Further, since F(x, ·) is the conjugate of the closed convex function (−cl 2K)(x, ·)(see (2.63) and Lemma 2.4.5), from Fenchel duality (2.57) we have [F(x, ·)]∗ =

−cl 2K(x, ·) on Y , i.e.,

−cl 2K(x, y) = supy∗∈Y∗{〈y, y∗〉 − F(x, y∗)}.

For similar reasons,

(2.65) cl 1K(x, y) = supx∗∈X∗{G(x∗, y) + 〈x, x∗〉}.

As we said above, F is also the convex parent of cl 1K, i.e.,

F(x, y∗) = supy∈Y{cl 1K(x, y) + 〈y, y∗〉}.


Combining the latter with (2.65) one obtains

(2.66) F(x, y∗) = sup(x∗,y)∈X∗×Y

{G(x∗, y) + 〈y, y∗〉 + 〈x, x∗〉},

hence, F(x, y∗) is the restriction of the conjugate function (−G)∗ : X∗∗×Y∗ → R∪{+∞}to the norm closed subspace X × Y∗. Similarly one obtains that

(2.67) −G(x∗, y) = sup(x,y∗)∈X×Y∗

{−F(x, y∗) + 〈y, y∗〉 + 〈x, x∗〉},

so (−G)(x∗, y) is the restriction of the conjugate function F∗ : X∗ × Y∗∗ → R ∪ {+∞}to the norm closed subspace X∗ × Y and the proof is then complete. �

2.4.2 Subdifferential of a saddle function. Partially ball weaklyinf-compact saddle functions. Definition and properties

The notion of the subdifferential of a saddle function K : X × Y → [−∞,+∞] isintroduced by Rockafellar as the multivalued mapping ∂K : X × Y → 2X∗×Y∗ definedby

∂K(x, y) := {(x∗, y∗) ∈ Z∗ : x∗ is a subgradient of the convex function K(·, y) at x and−y∗ is a subgradient of the convex function −K(x, ·) at y}.

The (possibly empty) set ∂K(x, y) is called the subdifferential of K at (x, y) (seeRockafellar [148, 142]). The domain of ∂K is defined by Dom ∂K := {(x, y) ∈ X × Y :∂K(x, y) , ∅}. It is clear from the definitions and from (2.60) that when K is proper,

(2.68) Dom ∂K ⊂ Dom K.

The original finite dimensional version of the following lemma is due to Rock-afellar and can be found in Rockafellar [145, Lemma 4]. Our proof follows the samesteps.

Lemma 2.4.7. For a proper closed saddle function K : X × Y → [−∞,+∞] thefollowing are equivalent:(a) (x∗, y) ∈ ∂F(x, y∗);(b) (x, y∗) ∈ ∂(−G)(x∗, y);(c) (x∗,−y∗) ∈ ∂K(x, y).

Any of these conditions implies that the values F(x, y∗), G(x∗, y) and K(x, y) arefinite.


Proof. First, we will show that (a) and (b) are equivalent. Since by Lemma 2.4.6 Fis a proper closed convex function, we have by (2.59) that

(x∗, y) ∈ ∂F(x, y∗) ⇐⇒ (x, y∗) ∈ ∂F∗(x∗, y) ⇐⇒ F∗(x∗, y)+ F(x, y∗) = 〈x, x∗〉+ 〈y, y∗〉.

By Lemma 2.4.6 again, F∗(x∗, y) = (−G)(x∗, y), hence

(x∗, y) ∈ ∂F(x, y∗) ⇐⇒ (−G)(x∗, y) + F(x, y∗) = 〈x, x∗〉 + 〈y, y∗〉,

which is(a) ⇐⇒ (−G)(x∗, y) + F(x, y∗) = 〈x, x∗〉 + 〈y, y∗〉.

Since by Lemma 2.4.6 (−G) is a proper closed convex function we have by (2.59)that

(x, y∗)∈∂(−G)(x∗, y)⇐⇒ (x∗, y)∈∂(−G)∗(x, y∗)⇐⇒ (−G)∗(x, y∗)−G(x∗, y)=〈x, x∗〉+〈y, y∗〉.

By Lemma 2.4.6, (−G)∗(x, y∗) = F(x, y∗), hence

(x, y∗)∈∂(−G)(x∗, y) ⇐⇒ F(x, y∗)−G(x∗, y)=〈x, x∗〉+〈y, y∗〉,

which is(b) ⇐⇒ F(x, y∗) −G(x∗, y) = 〈x, x∗〉+〈y, y∗〉.

Finally, (a) and (b) are equivalent, since

(2.69) (a) ⇐⇒ F(x, y∗) −G(x∗, y) = 〈x, x∗〉 + 〈y, y∗〉 ⇐⇒ (b)

and either of those implies the finiteness of F(x, y∗) and G(x∗, y).Next, we will show that (c) implies (a) and (b).When (c) holds it implies that K(x, y) is finite and by the definition of ∂K we

have that x∗ ∈ ∂1K(x, y), y∗ ∈ ∂2(−K)(x, y), i.e.,

K(z, y) ≥ K(x, y) + 〈z − x, x∗〉, ∀z ∈ X−K(x,w) ≥ −K(x, y) + 〈w − y, y∗〉, ∀w ∈ Y

⇐⇒K(x, y) − 〈x, x∗〉 ≤ K(z, y) − 〈z, x∗〉, ∀z ∈ XK(x, y) + 〈y, y∗〉 ≥ K(x,w) + 〈w, y∗〉, ∀w ∈ Y

⇐⇒

(2.70)K(x, y) − 〈x, x∗〉 ≤ G(x∗, y),K(x, y) + 〈y, y∗〉 ≥ F(x, y∗).


Adding the last two inequalities we obtain F(x, y∗) − G(x∗, y) ≤ 〈x, x∗〉 + 〈y, y∗〉. Theopposite inequality being obvious, we have F(x, y∗) − G(x∗, y) = 〈x, x∗〉 + 〈y, y∗〉. Itfollows from (2.69) that (a) and (b) hold.

Finally, let us suppose that (a) and (b) hold. The function F(x, ·) is a proper lowersemicontinuous convex function and y ∈ ∂2F(x, y∗). Since F(x, ·) is the conjugate of(−cl 2K)(x, ·), it follows from (2.59) that

y ∈ ∂2F(x, y∗) ⇐⇒ y∗ ∈ ∂2(−cl 2K)(x, y) ⇐⇒ F(x, y∗) − 〈y, y∗〉 = cl 2K(x, y).

The function (−G)(·, y) is a proper lower semicontinuous convex function andx ∈ ∂1(−G)(x∗, y). Since (−G)(·, y) is the conjugate of cl 1K(·, y), we have from (2.59)

x ∈ ∂1(−G)(x∗, y) ⇐⇒ x∗ ∈ ∂1cl 1K(x, y) ⇐⇒ G(x∗, y) + 〈x, x∗〉 = cl 1K(x, y).

From (2.69), we have that the left hand sides of the last two equalities are equal,hence we obtain that cl 1K(x, y) = cl 2K(x, y), and that, in particular, they are bothequal to K(x, y). Hence,

F(x, y∗) = K(x, y) + 〈y, y∗〉,

G(x∗, y) = K(x, y) − 〈x, x∗〉,

which entails (2.70). But it was already shown that (2.70) is equivalent to (c). Theproof is then complete. �

Combining Lemma 2.4.3 and Lemma 2.4.7, one easily obtains that if K is aproper closed saddle function and L is a saddle function equivalent to K, i.e., L ∼ K,then ∂K = ∂L. Hence, the subdifferential of a proper closed saddle function Kdepends only on the equivalence class to which K belongs and does not depend onits representatives.

It is established by Rockafellar that the domain of the subdifferential of a properclosed saddle function K : X × Y → [−∞,+∞] is nonempty when one of the spacesX and Y is reflexive (see Rockafellar [146]). We will extend this property (seeTheorem 2.4.11) for a class of closed saddle functions defined on product of Banachspaces introduced by the following

Definition 2.4.8. Let X, Y be Banach spaces and K : X × Y → [−∞,+∞] be a saddlefunction.

We say that K is X-bwc if for some y0 ∈ DK the function cl 1K(·, y0) is bwc.Respectively, we say that K is Y-bwc if for some x0 ∈ CK the function −cl 2K(x0, ·) isbwc.


The function K is said to be partially ball weakly inf-compact (pbwc for short) ifit is X-bwc or Y-bwc.

When for any (x0, y0) ∈ Dom K it holds that cl 1K(·, y0) or −cl 2K(x0, ·) is bwc,then K is said to be totally partially ball weakly inf-compact (tpbwc for short).

Let us note that when one of the spaces X and Y , say X, is reflexive then anysaddle function K on X × Y is tpbwc because cl 1K(·, y) is X-bwc for all y ∈ Y whichis ensured by the weak compactness of the closed unit ball BX.

From the very definition it is clear that K is pbwc whenever it is tpbwc, but whenboth spaces are not reflexive there exist pbwc saddle functions which are not tpbwcas we can see from the following

Example 2.4.9 (suggested by M. Ivanov). Let X and Y be two Banach spaceswhich are non reflexive. Fix any function g : X → R ∪ {+∞} that is convex, lowersemicontinuous and bwc function with g(0) = 0 and g(x) > 0 for all x ∈ X \ {0}, andsuch that for C := dom g the set BX ∩ cl C is not w(X, X∗) compact and is non empty.

Define a function K from X × Y into [−∞,+∞] by

K(x, y) =

(1 − ‖y‖)g(x), if x ∈ C, y ∈ BY

−∞, if x ∈ C, y < BY

+∞, otherwise.

Since cl 2K(x, ·) = K(x, ·) for all x ∈ X and since

cl 1K(·, y) =

K(·, y), if ‖y‖ < 1ψcl C(·), if ‖y‖ = 1−∞, if ‖y‖ > 1

(here, for a subset S , ψS denotes the indicator function, i.e, ψS (x) = 0 if x ∈ S andψS (x) = +∞ otherwise) it is not difficult to check that K is a proper closed saddlefunction with Dom K = C × BY .

As for y0 = 0 ∈ BY one has cl 1K(·, y0) = K(·, y0) = g(·) which is bwc, we get thatthe function K is pbwc.

We claim that K is not tpbwc. Fix any (x0, y0) ∈ Dom K = C × BY with ‖y0‖ = 1.On the one hand, cl 1K(·, y0) = ψcl C(·) which is not bwc since BX ∩ cl C is not

w(X, X∗) compact.On the other hand,

cl 2K(x0, y) =

{(1 − ‖y‖)g(x0), if y ∈ BY

−∞, if y < BY ,


hence for r = 0

BY ∩ {y ∈ Y : −cl 2K(x0, y) ≤ r} = BY ∩ BY = BY

which is not w(Y,Y∗) compact. So the claim is established.

An example of such a function g : X → R∪ {+∞} with the properties listed aboveis given in X := l1(N) by

g(x) =

∞∑n=1

2n|xn| for all x ∈ l1(N), x = (xn)∞n=1.

Indeed, one has {g ≤ r} = ∅ for r < 0, {g ≤ 0} = {0} for r = 0, and {g ≤ r} ⊂ [−a, a]with a := (r/2n)∞n=1 and [−a, a] := {x ∈ l1(N) : −(r/2n) ≤ xn ≤ r/2n,∀n} is easilyseen to be totally bounded and closed and hence ‖.‖ compact. So, the function g isa convex, lower semicontinuous bwc function with g(0) = 0 and g(x) > 0 for allx ∈ X \ {0}.

Concerning the above properties required for g, it remains to show that L :=BX ∩ cl C is not w(X, X∗) compact. Let ek with k ≥ 1 be the standard basis ofl1(N) = X and c = (cn)∞n=1 ∈ l∞(N) = X∗ with 0 < cn < cn+1 and cn → 1, so ‖c‖∞ = 1.Since ek ∈ L and 〈c, ek〉 = ck → 1, one has sup

x∈L〈c, x〉 = 1. On the other hand, for each

x ∈ L \ {0}

〈c, x〉 ≤∞∑

n=1

cn|xn| <

∞∑n=1

|xn| ≤ 1,

so 〈c, ·〉 does not attain its maximum over L, which means that L is not w(X, X∗)compact according to James theorem.

Below we will find out that the pbwc saddle functions defined on a productBanach space possess many of the well-known properties of the saddle functionsdefined on a product of Banach spaces one of which is reflexive.

Lemma 2.4.10. Let X, Y be Banach spaces, let K : X × Y → [−∞,+∞] be a properclosed X-bwc saddle function and let G be its concave parent. Then the followingproperties hold:

(a) dom(−G)∗ ⊂ X × Y∗;(b) the range of ∂(−G) is a non-empty subset of X × Y∗;(c) (x∗, y∗∗) ∈ ∂F(x, y∗) ⇐⇒ (x∗, y∗∗) ∈ ∂(−G)∗(x, y∗).


Proof. (a) Take any (x∗∗, y∗) ∈ dom(−G)∗ and set r := (−G)∗(x∗∗, y∗). Then r ∈ R andthe equality (−G)∗(·, y) = [cl 1K(·, y)]∗ yields

r := (−G)∗(x∗∗, y∗) = sup(x∗,y)∈X∗×Y

{〈y, y∗〉 + 〈x∗, x∗∗〉 + G(x∗, y)}

= sup(x∗,y)∈X∗×Y

{〈y, y∗〉 + 〈x∗, x∗∗〉 − [cl 1K(·, y)]∗}

= sup(x∗,y)∈X∗×DK

{〈y, y∗〉 + 〈x∗, x∗∗〉 − [cl 1K(·, y)]∗(x∗)}

= supy∈DK

{〈y, y∗〉 + supx∗∈X∗{〈x∗, x∗∗〉 − [cl 1K(·, y)]∗(x∗)}}

= supy∈DK

{〈y, y∗〉 + [cl 1K(·, y)]∗∗(x∗∗)},

where the third equality is due to the fact that, for any y < DK , one has [cl 1K(·, y]∗ =

+∞, since cl 1K(·, y) = −∞ according to Lemma 2.4.5. Hence, x∗∗ is in the effectivedomain of the biconjugate of cl 1K(·, y) for any y ∈ DK . Since for some y0 ∈ DK thefunction cl 1K(·, y0) is bwc, Theorem 2.4.2 (c) ensures that x∗∗ ∈ X.(b) From Lemma 2.4.6, (−G) is a proper closed convex function, hence the range of∂(−G) is non-empty.

Take any (x∗∗, y∗) ∈ ∂(−G)(x∗, y). By (2.59), (x∗, y) ∈ ∂(−G)∗(x∗∗, y∗). From (a)we have dom ∂(−G)∗ ⊂ dom(−G)∗ ⊂ X × Y∗. So, there exists some x ∈ X such thatx∗∗ = x.(c) follows from Lemma 2.4.6. �

Theorem 2.4.11. Let X, Y be Banach spaces and let K : X × Y → [−∞,+∞] be aproper closed pbwc saddle function. Then Dom ∂K , ∅.

Proof. Without loss of generality we suppose that K is X-bwc. For the proper closedconvex function (−G) there exists (x∗∗, y∗) ∈ ∂(−G)(x∗, y). From Lemma 2.4.10 (b)we have that x∗∗ = x for some x ∈ X, i.e., we have that (x, y∗) ∈ ∂(−G)(x∗, y).Lemma 2.4.7 ensures that this is equivalent to (x∗,−y∗) ∈ ∂K(x, y). The latter says inparticular that Dom ∂K is a non-empty set. The proof is then complete. �

From Theorem 2.4.11 we can derive the density of the domain of the subdifferen-tial in the domain of a proper closed tpbwc saddle function. This result is conjecturedfor any proper closed saddle function by Rockafellar (see Rockafellar [146, p. 249]).It is established by him when X and Y are finite dimensional spaces (see Rockafel-lar [142, 148]) and extended by Gossez (see Gossez [79, Theorem 1]) to the casewhen one of the spaces X and Y is reflexive. The reflexivity assumption enters thelatter proof essentially to ensure that in this case Dom ∂K is a non-empty set. We willfollow the proof of Gossez. To this end let us recall Gossez [79, Corollary 1] which


states: Let K be a closed saddle function on the product Banach space Z = X × Y andlet C ⊂ X and D ⊂ Y be closed convex sets such that the interior of (C × D) meetsDom K. Define

(2.71) K0(x, y) :=

K(x, y), if x ∈ C, y ∈ D

+∞, if x < C, y ∈ D−∞, if y < D.

Then K0 is a closed saddle function with

(2.72) Dom K0 = Dom K ∩ [C × D]

and for each (x, y) ∈ X × Y

(2.73) ∂K0(x, y) = ∂K(x, y) + [NC(x) × (−ND(y))],

where NC(x) (resp. ND(y)) denotes the normal cone to C (resp. D) at x (resp. y).

Theorem 2.4.12. Let X, Y be Banach spaces and let K : X × Y → [−∞,+∞] be aproper closed tpbwc saddle function. Then Dom ∂K is dense in Dom K.

Proof. We will use Theorem 2.4.11 and will follow Gossez’s proof of [79, Theo-rem 1].

Take any (x0, y0) ∈ dom K. Without loss of generality we suppose that cl 1K(·, y0)is bwc.

Fix ε > 0 and consider the function K0 as above with C := x0 + εBX and D :=y0 + εBY . Observe that (2.72) entails that DK0 ⊂ DK and that, for any y ∈ DK0 , wehave by (2.71) that

K(·, y) ≤ K0(·, y), and hence cl 1K(·, y) ≤ cl 1K0(·, y).

So {cl 1K0(·, y) ≤ r} ⊂ {cl 1K(·, y) ≤ r} for all r ∈ R and y ∈ DK0 . This says inparticular that cl 1K0(·, y0) is bwc, hence K0 is pbwc. Then Theorem 2.4.11 ensuresthat Dom ∂K0 , ∅ and hence (2.73) gives some (x, y) ∈ C ×D such that ∂K(x, y) , ∅.The proof is then complete. �

Another interesting property of a closed proper pbwc saddle function is that thegraphs of the subdifferentials of its parent functions are completely determined by thegraph of its subdifferential.

Lemma 2.4.13. Let X, Y be Banach spaces and let K : X × Y → [−∞,+∞] bea proper closed pbwc saddle function. Then gph ∂F and gph ∂(−G) are completelydetermined by the elements of gph ∂K.


Proof. Suppose that K is X-bwc.Lemma 2.4.7 shows that (x, y∗) ∈ ∂(−G)(x∗, y) exactly when (x∗,−y∗) ∈ ∂K(x, y),

so by Lemma 2.4.10 (b) gph ∂(−G) is completely determined by gph ∂K.From Lemma 2.4.6 we know that F is a proper closed convex function. The

density of dom ∂F in the non-empty set dom F implies that dom ∂F is non-empty.Let (x∗, y∗∗) ∈ ∂F(x, y∗). From Lemma 2.4.10 (c) we have that (x∗, y∗∗) ∈ ∂F(x, y∗)exactly when (x∗, y∗∗) ∈ ∂(−G)∗(x, y∗). Set W to be the Banach space on which theconcave parent G of K is defined, i.e., W := X∗×Y . For the conjugate function (−G)∗

Lemma 2.4.10 (a) gives that dom ∂(−G)∗ ⊂ X×Y∗. This combined with Theorem 2.4.1ensures that:

(2.74)

(x∗, y∗∗) ∈ ∂(−G)∗(x, y∗) ⇐⇒there exists a net {(xα, yα∗)}α∈A ∈ W∗ converging to (x, y∗)in the norm topology of W∗ anda bounded net {(x∗α, yα)}α∈A ∈ W converging to (x∗, y∗∗)in the w(W∗∗,W∗) topology such that(xα, y∗α) ∈ ∂(−G)(x∗α, yα), ∀α ∈ A.

Lemma 2.4.7 implies that (xα, y∗α)∈∂(−G)(x∗α, yα) exactly when (x∗α,−y∗α) ∈ ∂K(xα, yα).Hence, (2.74) may be rewritten as:

(2.75)

(x∗, y∗∗) ∈ ∂F(x, y∗) ⇐⇒there exists a net {(xα, yα∗)}α∈A ∈ W∗ converging to (x, y∗)in the norm topology of W∗ anda bounded net {(x∗α, yα)}α∈A ∈ W converging to (x∗, y∗∗)in the w(W∗∗,W∗) topology such that(x∗α,−y∗α) ∈ ∂K(xα, yα), ∀α ∈ A.

The latter says that gph F is completely determined by the elements of gph ∂K. �

With any saddle function K : X × Y → [−∞,+∞] one may associate a monotone(see below) multivalued operator TK : X × Y → 2X∗×Y∗ given by

TK(x, y) := {(x∗, y∗) ∈ X∗ × Y∗ : (x∗,−y∗) ∈ ∂K(x, y)}.

When K is proper and closed, we saw in Lemma 2.4.7 that ∂K depends only on theequivalence class. The above definition ensures that in this case the same holds forthe operator TK . This operator was introduced in relation with minimax problems byRockafellar [146] who proved that in the case when one of the Banach spaces in-volved is assumed to be reflexive, TK is maximal monotone whenever K is proper andclosed (see Rockafellar [146, Theorem 3]). We will show that this is still true when


K is a proper closed partially ball weakly inf-compact saddle function defined on aproduct Banach space. Closed relationship between proper closed saddle functions onX × X and maximal monotone operators on X is established by Krauss in [105].

Let us recall that a set-valued mapping S from a Banach space X to its dualX∗ is said to be monotone if x∗ ∈ S (x), y∗ ∈ S (y) imply 〈x − y, x∗ − y∗〉 ≥ 0. Theoperator S is said to be maximal monotone if S is monotone and S has no propermonotone extension, i.e., if (x, x∗) ∈ X × X∗ is such that the monotone relation〈x − y, x∗ − y∗〉 ≥ 0 holds for all (y, y∗) ∈ gph S , then (x, x∗) ∈ gph S (cf., e.g.,Phelps [133], Rockafellar [147] and Simons [156]).

Theorem 2.4.14. Let X, Y be Banach spaces and let K : X × Y → [−∞,+∞] be aclosed proper pbwc saddle function. Then the operator TK is maximal monotone.

Proof. Let K be X-bwc. By Lemma 2.4.10, (b) we know that the range of ∂(−G) liesin X×Y∗. By Lemma 2.4.7 and by the definition of TK it is clear that (x∗, y∗) ∈ TK(x, y)exactly when (x, y∗) ∈ ∂(−G)(x∗, y). The latter, being the subdifferential of a properclosed convex function, is maximal monotone (see Rockafellar [147]). This impliesthe maximal monotonicity of TK . �

Let us note that in the paper of Pak [131] one can find the statement of the aboveresult for arbitrary proper closed saddle function defined on a product of arbitraryBanach spaces, but the presented there proof implicitly presumes reflexivity.

2.4.3 Integrability of the subdifferential of a proper closed par-tially ball weakly inf-compact saddle function

Here we are interested in the integrability of the subdifferential of a saddle func-tion on the product of Banach spaces. The result is established for Lipschitz saddlefunction by Correa and Thibault in [54]. Some generalizations for directionally Lips-chitz saddle functions are already presented in Section 2.3.

We consider two proper closed saddle functions K, L : X×Y → [−∞,+∞] definedon a product Banach space and we are interested whether the pbwc condition on oneof the functions K and L and the inclusion ∂L ⊂ ∂K entail that K and L are equivalentup to a finite additive constant.

First we will consider the case when the inside for the subdifferential inclusionfunction L is supposed to be pbwc.

Theorem 2.4.15. Let X, Y be Banach spaces and let L : X × Y → [−∞,+∞] be aproper closed pbwc saddle function.


Then for any proper closed saddle function K : X × Y → [−∞,+∞] the condition

∂L(x, y) ⊂ ∂K(x, y), ∀(x, y) ∈ Dom ∂L

implies that K and L are equivalent up to a finite additive constant c, i.e., K ∼ (L+c).

Proof. Let K and L be saddle functions satisfying the assumptions of the theorem.Let us denote by F′ and F the convex parents of L and K, respectively and by G′

and G the concave parents of L and K, respectively.Since L and K are proper and closed we know by Lemma 2.4.6 that the functions

F′, F, (−G′) and (−G) are proper and lower semicontinuous convex functions.Since L is supposed to be pbwc, without loss of generality we suppose that L is

X-bwc.First, we will show that

(2.76) ∂(−G′) ⊂ ∂(−G).

Since L is X-bwc, by Lemma 2.4.10 (b) we have Rge ∂(−G′) ⊂ X × Y∗. Takeany (x, y∗) ∈ ∂(−G′)(x∗, y). Lemma 2.4.7 for L ensures that (x∗,−y∗) ∈ ∂L(x, y), so(x∗,−y∗) ∈ ∂K(x, y). Lemma 2.4.7 for K yields that (x, y∗) ∈ ∂(−G)(x∗, y). Hence, wehave proved (2.76).

Second, we will show that

(2.77) ∂(F′) ⊂ ∂(F).

Take any (x∗, y∗∗) ∈ ∂F′(x, y∗). Since L is X-bwc, Lemma 2.4.10 (c) gives that thelatter implies (x∗, y∗∗) ∈ ∂(−G′)∗(x, y∗). Since (−G′)∗ is the conjugate function of thelower semicontinuous convex function (−G′), then Theorem 2.4.1 applied to it givesthat (recalling the notation W := X∗ × Y)

(2.78)

(x∗, y∗∗) ∈ ∂(−G′)∗(x, y∗)⇒there exists a net {(xα∗∗, yα∗)}α∈A ∈ W∗ converging to (x, y∗)in the norm topology of W∗ anda bounded net {(x∗α, yα)}α∈A ∈ W converging to (x∗, y∗∗)in the w(W∗∗,W∗) topology such that(xα∗∗, y∗α) ∈ ∂(−G′)(x∗α, yα), ∀α ∈ A.

The X-bwc property of L yields according to Lemma 2.4.10 (b) that x∗∗α = xα forsome xα ∈ X. From Lemma 2.4.7 (xα, y∗α) ∈ ∂(−G′)(x∗α, yα) implies that (xα∗,−y∗α) ∈∂L(xα, yα). From ∂L ⊂ ∂K we have (xα∗,−y∗α) ∈ ∂K(xα, yα) which by Lemma 2.4.7yields that (xα, yα∗) ∈ ∂(−G)(x∗α, yα).


Since (xα, y∗α) ∈ ∂(−G)(x∗α, yα) for all α ∈ A and since the net {(xα, yα∗)}α∈A ∈ W∗

converges to (x, y∗) in the norm topology of W∗ and the bounded net {(x∗α, yα)}α∈A ∈

W converges to (x∗, y∗∗) in the w(W∗∗,W∗) topology, Theorem 2.4.1 implies that(x∗, y∗∗) ∈ ∂(−G)∗(x, y∗). As F(u, v∗) = (−G)∗(u, v∗) for all (u, v∗) ∈ X × Y∗ byLemma 2.4.6, we deduce that (x∗, y∗∗) ∈ ∂F(x, y∗). Hence, (2.77) is established.

The functions F and F′ are both proper lower semicontinuous convex functions onthe Banach space X ×Y∗ satisfying ∂F′ ⊂ ∂F. Hence, we can apply Rockafellar [147,Theorem B] (see also Thibault and Zagrodny [160, Corollary 2.2]) to deduce thatthere exists a finite constant c such that

(2.79) F(x, y∗) = F′(x, y∗) + c, ∀(x, y∗) ∈ X × Y∗.

The functions (−G′) and (−G) are both proper lower semicontinuous convex functionssatisfying ∂(−G′) ⊂ ∂(−G). Analogous reasoning gives a finite constant d such that

(2.80) G(x∗, y) = G′(x∗, y) + d, ∀(x∗, y) ∈ X∗ × Y.

Now using (2.66) from Lemma 2.4.6, (2.80) and (2.79) we obtain that

F(x, y∗) = sup(x∗,y)∈X∗×Y

{G(x∗, y) + 〈y, y∗〉 + 〈x, x∗〉}

= sup(x∗,y)∈X∗×Y

{G′(x∗, y) + 〈y, y∗〉 + 〈x, x∗〉} + d

= F′(x, y∗) + d = F(x, y∗) − c + d,

and c = d because F′ is finite at some point. From (2.79) and (2.80) it is clear that thefunctions K and L + c have the same parent functions. By Lemma 2.4.3 one obtainsthat K ∼ (L + c) and K and L are equivalent up to the additive constant c. The proofis then complete. �

Second we consider the case when the outside for the subdifferential inclusionfunction K is supposed to be pbwc.

Theorem 2.4.16. Let X, Y be Banach spaces and let K : X × Y → [−∞,+∞] bea proper closed pbwc saddle function. Then for any proper closed saddle functionL : X × Y → [−∞,+∞] the conditions

(2.81) ∂1cl 1L(·, y) ⊂ ∂1K(·, y), and ∂2(−cl 2L)(x, ·) ⊂ ∂2(−K)(x, ·),∀(x, y) ∈ X × Y

imply that K and L are equivalent up to a finite additive constant c, i.e., K ∼ (L + c).


Proof. Let (x, y) ∈ Dom L. By Lemma 2.4.5 we know that the function cl 1L(·, y) isa proper lower semicontinuous convex function and that x ∈ dom cl 1L(·, y). By thedensity of the domain of the subdifferential of a closed convex function in its effectivedomain (see (2.58)) it follows that there exists x ∈ dom ∂1cl 1L(·, y) ⊂ dom ∂1K(·, y),the latter by (2.81). From (2.60) we have that x ∈ dom cl 1K(·, y). Hence, cl 1K(·, y)is a proper closed convex function too and further the first inclusion of (2.81) yields∂1cl 1L(·, y) ⊂ ∂1cl 1K(·, y). We can then apply Rockafellar [147, Theorem B] (seealso Thibault and Zagrodny [160, Corollary 2.2]) to obtain that there exists a finiteconstant c(y) such that

(2.82)cl 1L(x, y) = cl 1K(x, y) + c(y), ∀x ∈ X anddom cl 1L(·, y) ≡ dom cl 1K(·, y).

Analogously, one obtains that there exists a finite constant d(x) such that

(2.83)cl 2L(x, y) = cl 2K(x, y) + d(x), ∀y ∈ Y anddom(−cl 2L)(x, ·) ≡ dom(−cl 2K)(x, ·).

Recall that (2.82) and (2.83) hold for all (x, y) ∈ Dom L. From (2.82), (2.83) andLemma 2.4.4 it follows that (x, y) ∈ Dom K, so Dom L ⊂ Dom K.

To establish the opposite inclusion of domains, let us take any (x, y) ∈ Dom K.Suppose that y < DL. Then cl 1L(·, y) = −∞ and ∂1cl 1L(x, y) ≡ X∗ for any x ∈ X. Thefirst inclusion in (2.81) ensures that K(·, y) is everywhere subdifferentiable, hencefrom (2.60), K(·, y) = cl 1K(·, y), and ∂1cl 1K(·, y) = ∂1K(·, y) = X∗, which yieldsa contradiction, because cl 1K(·, y) is a proper closed convex function. By similarreasons we obtain that x ∈ CL, hence Dom K ⊂ Dom L and, finally, Dom L = Dom K.

As K is pbwc, we may suppose that K is X-bwc. Take y0 ∈ DK = DL suchthat cl 1K(·, y0) is bwc. Fix r ∈ R and n ∈ N and consider the sublevel set Pr,n :={cl 1L(·, y0) ≤ r} ∩ nBX. Since the function cl 1L(·, y0) is convex and lower semicontin-uous the latter set is weakly closed. By (2.82), Pr,n ⊂ {cl 1K(·, y0) ≤ r − c(y0)} ∩ nBX

which is weakly compact. Hence, Pr,n is weakly compact, which implies that theclosed saddle function L is X-bwc too. Then by Theorem 2.4.11 Dom ∂L , ∅. Takeany (x∗, y∗) ∈ ∂L(x, y). In particular, x∗ ∈ ∂1L(·, y)(x). From (2.60) and the propernessof L(·, y) we have that L(x, y) = cl 1L(x, y) and that ∂1cl 1L(·, y)(x) = ∂1L(·, y)(x). Thelatter says that x∗ ∈ ∂1cl 1L(·, y)(x). The assumption (2.81) gives that x∗ ∈ ∂1K(·, y)(x).Analogously one obtains that −y∗ ∈ ∂2(−K(x, ·))(y). Hence, (x∗, y∗) ∈ ∂K(x, y). Thearbitrariness of (x∗, y∗) ensures that ∂L(x, y) ⊂ ∂K(x, y) for all (x, y) ∈ Dom ∂L. Theassumptions of Theorem 2.4.15 being satisfied, we conclude that the result is estab-lished. �

Chapter 3

Variational analysis of multivaluedmaps

In this chapter we study certain multivalued maps, as well as, multivalued mapsdepending on parameter. Such maps are considered in optimization and are intensivelystudied last years.

In Section 3.1 we study Lipschitz continuity with respect to the parameter of theset of solutions of a parameterized minimax problem on a product Banach space.We present a sufficient condition, ensuring that the map which to any value of theparameter assigns the set of solutions of the problem (possibly multi-valued, andunbounded) possesses Aubin property. Results are published by Quincampoix andZlateva in [140].

In Section 3.2 we present a derivative criterion for metric regularity of set-valuedmappings that is based on works of J.-P. Aubin and co-authors. A related implicitmapping theorem is also obtained. As applications, we first show that Aubin criterionleads directly to the known fact that the mapping describing an equality/inequalitysystem is metrically regular if and only if the Mangasarian-Fromovitz condition holds.We also derive a new necessary and sufficient condition for strong regularity ofvariational inequalities over polyhedral sets. A new proof of the radius theorem formetric regularity based on Aubin criterion is given as well. Results are published byDontchev, Quincampoix and Zlateva in [63].

In Section 3.3 is proved Long orbit or empty value principle for a multivaluedmap ant it is applied to provide unified approach to several fixed point and surjectivityresults. All of the latter are derived from a novel general result. Results are publishedby Ivanov and Zlateva in [96].

157

158 Chapter 3. Variational analysis of multivalued maps

3.1 Parameterized minimax problem: on Lipschitz-like dependence of the solution with respect toparameter

In this section we establish quite general sufficient conditions for Aubin continuityof the saddle point map S : λ⇒ S(λ) arising from the parameterized minimax prob-lem M(λ). Examples illustrating these conditions are presented. Several corollariesrelated to the case of convex-concave smooth data are also sketched.

Consider the parameterized minimax problem

M(λ) infx∈K

supy∈L

f (x, y, λ),

where λ ∈ Λ is a parameter.Here K and L are nonempty closed subsets of the Banach spaces X and Y ,

respectively; { f (·, ·, λ) : X × Y → R, λ ∈ Λ} is a family of real-valued functionsparameterized by λ ∈ Λ, where Λ is a subset of the Banach space Z.

Saddle point of f (·, ·, λ) on K × L is any point (x, y) ∈ K × L that satisfies

f (x, y, λ) ≤ f (x, y, λ) ≤ f (x, y, λ) ∀x ∈ K, ∀y ∈ L.

A saddle point (x, y) of f (·, ·, λ) on K × L can be considered as a solution of the min-imax problem M(λ) by reason of (x, y) ∈ K × L and f (x, y) = infx∈K supy∈L f (x, y, λ).Let us denote the (possibly empty) set of all saddle points of the function f (·, ·, λ) onK × L by

(3.1) S(λ) := {(x, y)∈K×L : f (x, y, λ) ≤ f (x, y, λ) ≤ f (x, y, λ), ∀x ∈ K, ∀y ∈ L}.

That S(λ) is nonempty can be ensured in several cases. For example, if K andL are convex sets, f (x, y, λ) is convex and lower semicontinuous in x, concave andupper semicontinuous in y, and there are x0 ∈ K and y0 ∈ L such that f (·, y0, λ) isinf-compact and f (x0, ·, λ) is sup-compact, then S(λ) , ∅ by a minimax result due toHartung [81, Theorem 1] (see also Aubin and Ekeland [8, Theorem 6.2.8]). When,moreover, f (x, y, λ) is strictly convex in x and strictly concave in y, then S(λ) is asingleton.

In the present section we presume the existence of saddle points for M(λ) andfocus our attention on studying Lipschitz-like dependence of the solution set S(λ) onthe parameter λ. That is, we find sufficient conditions for Lipschitz-like continuity ofthe set-valued map

S : λ⇒ S(λ)

from Λ to nonempty subsets of K × L.

3.1. Parameterized minimax problem: on Lipschitz-like dependence of the . . . 159

Of course, when the map S is single-valued, the Lipschitz continuity is understoodin the classical sense. However, the map S could be multivalued. Moreover, its valuesS(λ) could be unbounded sets. A notion of Lipschitz-like continuity very appropriatefor such a case is due to Aubin [4, 5]:

The multivalued map S : Λ ⇒ X has Aubin property, or it is Aubin continuous,near (λ, x) ∈ gph S , if there are positive constant κ and neighborhoods U of x, and Vof λ, such that

(3.2) e(S (λ) ∩ U, S (µ)) ≤ κ‖λ − µ‖, ∀λ, µ ∈ Λ ∩ V,

where e(A, B) := supx∈A d(x, B) is the excess from set A to set B with e(∅, B) = +∞. Sis said to be Aubin continuous if S is Aubin continuous near any point (λ, x) ∈ gph S .

For various applications of Aubin continuity in the field of nonlinear analysis andoptimization the reader is referred, e.g., to Aubin [4, 5], Aubin and Frankowska [10]and Rockafellar and Wets [152]. The Aubin property of a map S near (λ, x) is knownto be equivalent to the metric regularity of S −1 near (x, λ) and was originally intro-duced in Aubin [5] under the name of pseudo-Lipschitz continuity. For bibliographicaldetails see Rockafellar and Wets book [152].

Whenever S is locally bounded, Aubin continuity coincides with the classicalnotion for Lipschitz continuity of set-valued maps (see Aubin and Frankowska [10]and Rockafellar and Wets [152])

e(S (λ), S (µ)) ≤ κ‖λ − µ‖, ∀λ, µ,

but Aubin property works without any boundedness imposed on the values of S .Aubin property is in fact Lipschitzean property localized in the range space, as wellas in the domain space.

In Subsection 3.1.1, after a short subsection devoted to preliminaries, we formulateand prove a sufficient condition for Aubin continuity of the solution map S : Λ⇒ Xof a parameterized minimization problem

P(λ) infx∈K

f (x, λ).

Many authors study Lipschitz-like dependence on λ of the solutions of the associatedgeneralized Euler equation

0 ∈ ∇x f (x, λ) + NK(x);

see Bonnans and Shapiro [22], Dontchev and Rockafellar [65], Shapiro [155] andthe references therein for recent developments. Here we do not follow that approach,because the map S t : λ ⇒ S t(λ), which to any λ assigns the set S t(λ) of solutionsof the generalized Euler equation, does not inherit Aubin continuity property fromS (see Example 3.1.6 for a parameterized problem such that the corresponding S isAubin continuous while S t is not).


In Subsection 3.1.2 we present our main result (Theorem 3.1.10), which is asufficient condition for Aubin continuity of the saddle point map S : Λ⇒ X × Y of aparameterized minimax problem M(λ).

It is clear that the results of Subsection 3.1.1 are contained in the more generalframework of Subsection 3.1.2. Nevertheless, we think that presenting the proof ofthe former simple case will help the understanding of the more technical proof of thelatter general case.

Subsection 3.1.3 relates the obtained results to some questions in the field oftwo-player zero sum differential games.

3.1.1 Parameterized minimization problem

We work in a Banach space (X, ‖ · ‖).For C ⊂ X the distance function to C is d(x,C) := infc∈C ‖x − c‖ if C , ∅, and

d(x,C) := +∞ if C = ∅.Function f : X → R is Gateaux differentiable at x ∈ X if there exists ∇ f (x) ∈ X∗,

called the Gateaux derivative of f at x, such that for any h ∈ X,

limt→0

f (x + th) − f (x)t

= 〈∇ f (x), h〉.

Also, f is said to be strictly differentiable at x whenever

limx→xt→0

f (x + th) − f (x)t

= 〈∇ f (x), h〉.

Given an open set U ⊂ X we denote by C1,α(U) the class of all Gateaux differ-entiable functions f : U → R such that ∇ f : U → X∗ is α-Holder on U, that is, forsome constant L > 0,

‖∇ f (x) − ∇ f (y)‖ ≤ L‖x − y‖α, ∀x, y ∈ U.

Let Z be a Banach space, whose norm is also denoted by ‖ · ‖. Let S be a mapfrom Λ ⊂ Z to X. If not stated otherwise, map means set-valued map. In order tooutline the multivaluedness we write S : Z ⇒ X. The inverse S −1 : X ⇒ Z of S isdefined by λ ∈ S −1(x) ⇐⇒ x ∈ S (λ). The graph, domain, and range sets of S aregiven by

gph S := {(λ, x) | x ∈ S (λ)}, dom S := {λ | S (λ) , ∅}, rge S := dom S −1,

respectively.Any product space X × Z of Banach spaces X and Z is considered with the

supremum norm ‖(x, z)‖ := max{‖x‖, ‖z‖}.


Assumptions

Let { f (·, λ) : X → R, λ ∈ Λ} be a family of functions parameterized by λ ∈ Λ ⊂ Z.We look for sufficient conditions to ensure Aubin continuity of the solutions of theparameterized family of constrained minimization problems:

P(λ) infx∈K

f (x, λ),

where K is a given nonempty closed set in X.For λ ∈ Λ, the (possibly empty) set of solutions of the minimization problem P(λ)

is denoted by

S (λ) :={x ∈ K : f (x, λ) = inf

x∈Kf (x, λ)

},

and its optimal value bym(λ) := inf

x∈Kf (x, λ).

It is well known that even for smooth parameterized problem P(λ) the solutionS : Λ ⇒ X may fail Lipschitz continuity. For example, for f (x, λ) = 1

4 x4 − λx, wherex, λ ∈ R, and K = [−1, 1], we see that for λ ∈ (−1, 1) the solution is S (λ) = {

3√λ},

and it is not Lipschitz continuous at λ = 0 (see Bonnans and Shapiro [22, Example4.31]).

Hence, to establish Lipschitz behavior of S one needs something more than thestandard requirements. We now turn to relevant analysis of P(λ).

Definition 3.1.1. Let X and Z be Banach spaces. Let U ⊂ X, V ⊂ Z be non-empty.We denote by Lα,β(U; V), α, β ∈ [0, 1], the class of all functions g : U × U × V → Rsuch that there exists a constant kg > 0 such that for all x, x′ ∈ U and all λ, λ′ ∈ V ,

|g(x, x′, λ) − g(x, x′, λ′)| ≤ kg‖x − x′‖α‖λ − λ′‖β.

For example, g ∈ L1,1(U; V) means that g(x, x′, ·) is Lipschitz on V and its bestLipschitz constant L(x, x′) satisfies L(x, x′) ≤ k‖x − x′‖ for some positive constant kand all x, x′ ∈ U.

With the parameterized family of functions { f (·, λ), λ ∈ Λ} one may associate twodifference functions: the function f1 : X × X × Λ→ R defined by

f1(x, x′, λ) := f (x, λ) − f (x′, λ),

and the function f2 : Λ × Λ × X → R defined by

f2(λ, λ′, x) := f (x, λ) − f (x, λ′).

The above notions are linked through the following:


Proposition 3.1.2. For any U ⊂ X, V ⊂ Z the function f1 ∈ Lα,β(U; V) if and only if

f2 ∈ Lβ,α(V; U).

Proof. Let f1 ∈ Lα,β(U; V). Take any x, x′ ∈ U, and any λ, λ′ ∈ V . Since

f2(λ, λ′, x) − f2(λ, λ′, x′) = [ f (x, λ) − f (x, λ′)] − [ f (x′, λ) − f (x′, λ′)]= [ f (x, λ) − f (x′, λ)] − [ f (x, λ′) − f (x′, λ′)]= f1(x, x′, λ) − f1(x, x′, λ′) ≤ k f1‖x − x′‖α‖λ − λ′‖β,

one can take k f2 := k f1 to conclude that f2 ∈ Lβ,α(V; U). The proof of the other

direction is similar. �

We are ready to present the sufficient condition for Aubin continuity of the solu-tion map.

Given a (λ, x) ∈ gph S , consider the following local assumption A at (λ, x):

(A)

there exist neighborhoods U of x and V of λ, such that

1. S (λ) ∩ U , ∅ for all λ ∈ Λ ∩ V;

and there exist constants c > 0 and α ∈ [0, 1] such that

2. f (x, λ′) ≥ m(λ′) + cd1+α(x, S (λ′)), ∀λ, λ′ ∈ Λ ∩ V, ∀x ∈ S (λ) ∩ U;

3. f1 ∈ Lα,1(K; Λ ∩ V).

By Proposition 3.1.2 it is clear that assumption A3 could be replaced withf2 ∈ L

1,α(Λ ∩ V; K).It is clear that A1 implies that m(λ) is finite for all λ ∈ Λ ∩ V .In the case α = 1, assumption A2 can be considered as a relaxed (in x) uniform

(in λ′) version of the so-called second-order growth condition. One says that thesecond-order growth condition holds for the problem

infx∈K

f (x)

in a neighborhood N of the solution set S 0, if there exists a constant c > 0 such that

(3.3) f (x) ≥ infK

f + cd2(x, S 0), ∀x ∈ K ∩ N.

This condition is involved in a number of works (see Bonnans and Ioffe [20],Bonnans and Shapiro [21, 22], Klatte and Henrion [101], Shapiro [154]) in orderto ensure Lipschitz stability of the solution map S of the constrained minimizationproblem. Let us recall that S is said to be Lipschitz stable or, equivalently, upperLipschitz at a point λ ∈ Λ, if there exist a constant κ > 0 and a neighborhood V of λsuch that it holds that

e(S (λ), S (λ)) ≤ κ∥∥∥λ − λ∥∥∥ , ∀λ ∈ Λ ∩ V.


Let us note that Lipschitz stability is a point property: it holds for S at a fixed point λ,while Aubin continuity we wish to obtain, is a local property, and it holds uniformlyat all points µ in some neighborhood V of the referenced point λ. Obviously, Aubincontinuity of S near (λ, x) implies Lipschitz stability of S ∩U at λ while the oppositeimplication is not always true.

A stronger version of uniform second-order growth condition than A2 with α = 1is given in Bonnans and Shapiro [22, Definition 5.16]. It implies single-valuednessand local Lipschitz continuity of S (cf. Bonnans and Shapiro [22, Theorem 5.17 andRemark 5.19]). In contrast, assumption A2 does not imply neither single-valuednessnor local boundedness of the solution map S (see Example 3.1.6).

We would now give a few examples of parameterized families of functions{ f (·, λ), λ ∈ Λ} for which A holds, in this way showing the consistency of our mainassumption.

Obviously, the compactness of K and lower semicontinuity of f (·, λ) on K aresufficient to ensure A1 (note that weak compactness and weak lower semicontinuitywould do just as well).

A2 with α = 1 is satisfied at any (λ, x) ∈ gph S provided that, for example, U = X,V = Z, and K ⊂ X is a nonempty closed convex set, the functions f (·, λ) are lowersemicontinuous, and uniformly on λ ∈ Λ strongly convex on K, that is, for someconstant c > 0 the inequality

f (tx′ + (1 − t)x′′, λ) ≤ t f (x′, λ) + (1 − t) f (x′′, λ) − ct(1 − t)‖x′ − x′′‖2

holds for every t ∈ [0, 1], every x′, x′′ ∈ K and every λ ∈ Λ.Lemma 3.1.3 below provides examples of parameterized families of functions

satisfying A3. However, we need a few more definitions before stating this lemma.Recall that the Clarke generalized derivative of Lipschitz function f : X → R at

x ∈ X in direction h ∈ X is

f ◦(x; h) := lim supx→xt↓0

f (x + th) − f (x)t

,

and the Clarke subdifferential at x is the nonempty w∗ compact set

∂ f (x) := {x∗ ∈ X∗ : 〈x∗, h〉 ≤ f ◦(x; h), ∀h ∈ X};

see Clarke [44]. It is well known that for any h ∈ X there exists some x∗ ∈ ∂ f (x)such that

〈x∗, h〉 = f ◦(x; h).

Lipschitz function f : U → R is said to be regular on an open set U ⊂ X if forany h ∈ X and any x ∈ U its directional derivative

f ′(x; h) := limt↓0

f (x + th) − f (x)t


exists and is equal to f ◦(x; h). Convex continuous functions and strictly differentiablefunctions are examples of regular functions.

Let f (x, λ) be Lipschitz on each variable bivariate function. Denote by f ◦x (x, λ; h)and by f ′x(x, λ; h) the generalized derivative and the directional derivative of f (·, λ)at x in direction h, respectively. Also, denote by ∂x f (x, λ) the partial Clarke subdif-ferential of f (·, λ) at x, and by ∂λ f (x, λ) the partial Clarke subdifferential of f (x, ·)at λ.

Lemma 3.1.3. Let (λ, x) ∈ gph S and let U ⊂ X and V ⊂ Z be convex neighborhoodsof K and λ, respectively. Consider the conditions:

(F1)

for λ ∈ Λ ∩ V, f (·, λ) is Lipschitz and regular on U and

∂x f (x, ·) : Λ ∩ V→X∗ is a k-Lipschitz map on Λ ∩ V

with k that does not depend on x ∈ U,

(F2)

for x ∈ K, f (x, ·) is Lipschitz and regular on V and

∂λ f (·, λ) : K→Z∗ is a k-Lipschitz map on K

with k that does not depend on λ ∈ V.

If f satisfies F1 or F2, then A3 holds with α = 1.

Proof. Let f satisfy F1. Fix x, y ∈ K and λ, µ ∈ Λ ∩ V . Consider the functionr(t) := f (y + t(x − y), λ) which is well-defined on an open interval I containing [0, 1].Since the function f (·, λ) is assumed to be Lipschitz on U, we have that r is Lipschitzon I. By Rademacher’s theorem, for almost all t ∈ [0, 1] there exists

r′(t) = lims→0

r(t + s) − r(t)s

= lims↓0

f (y + t(x − y) + s(x − y), λ) − f (y + t(x − y), λ)s

= f ′x(y + t(x − y), λ; x − y) = f ◦x (y + t(x − y), λ; x − y).

The last equality holds because f (·, λ) is regular on U.Hence,

(3.4) f (x, λ)− f (y, λ) = r(1)−r(0)=∫ 1

0r′(t) dt=

∫ 1

0f ◦x (y+t(x−y), λ; x−y) dt.

Similarly,

(3.5) f (x, µ) − f (y, µ) =

∫ 1

0f ◦x (y + t(x − y), µ; x − y) dt.

There exists x∗λ(t) ∈ ∂x f (y+t(x−y), λ) such that f ◦x (y+t(x−y), λ; x−y) = 〈x∗λ(t), x−y〉,so (3.4) becomes

(3.6) f (x, λ) − f (y, λ) =

∫ 1

0〈x∗λ(t), x − y〉 dt.


Since xλ(t) ∈ ∂x f (y + t(x − y), λ) and the multivalued map ∂x f (x, ·) : Λ ∩ V→X∗ isk–Lipschitz continuous with w∗ compact images, there is x∗µ(t) ∈ ∂x f (y + t(x − y), µ)such that ‖x∗λ(t)− x∗µ(t)‖ ≤ k‖λ− µ‖. Note that k does not depend on either t ∈ [0, 1] orx, y ∈ U.

Let us use these for estimating f1(x, y, λ) − f1(x, y, µ).From (3.6) we get

f (x, λ) − f (y, λ) =

∫ 1

0〈x∗λ(t) − x∗µ(t), x − y〉 dt +

∫ 1

0〈x∗µ(t), x − y〉 dt

≤

∫ 1

0‖x∗λ(t) − x∗µ(t)‖ ‖x − y‖ dt +

∫ 1

0〈x∗µ(t), x − y〉 dt

≤ k‖λ − µ‖ ‖x − y‖ +

∫ 1

0〈x∗µ(t), x − y〉 dt.

Since x∗µ(t) ∈ ∂x f (y + t(x − y), µ), it holds that 〈x∗µ(t), x − y〉 ≤ f ◦x (y + t(x − y), µ; x − y),and by (3.5) we have∫ 1

0〈x∗µ(t), x − y〉 dt ≤

∫ 1

0f ◦x (y + t(x − y), µ; x − y) dt = f (x, µ) − f (y, µ).

Hence,f (x, λ) − f (y, λ) ≤ f (x, µ) − f (y, µ) + k‖λ − µ‖‖x − y‖;

that is, f1(x, y, λ) ≤ f1(x, y, µ) + k‖λ − µ‖‖x − y‖, or

f1(x, y, λ) − f1(x, y, µ) ≤ k‖λ − µ‖‖x − y‖,

which means that f1 ∈ L1,1(K; Λ∩V).

If f satisfies F2, then by the same reasoning one obtains that f2 ∈ L1,1(Λ∩V; K)

and by Proposition 3.1.2, A3 holds. �

It is interesting to note here that the regularity (in particular, the differentiability)can be asked for the argument x as in F1, or for the parameter λ as in F2.

It is clear that both F1 and F2 hold whenever f ∈ C1,1(U × V).

Lipschitz-like continuity of the solution map

Here we prove that given (λ, x) ∈ gph S , assumption A is sufficient to ensureAubin continuity of the solution map S near (λ, x).

Proposition 3.1.4. Assume that X and Z are Banach spaces and consider a family ofconstraint minimization problems P(λ) parameterized by λ ∈ Λ, a nonempty subsetof Z.


If for some (λ, x) ∈ gph S assumption A holds, then

(3.7) e(S (λ) ∩ U, S (µ)) ≤k f1

c‖λ − µ‖, ∀λ, µ ∈ Λ ∩ V,

and the solution map S is Aubin continuous near (λ, x) ∈ gph S .

Proof. Take any λ ∈ Λ ∩ V and any xλ ∈ S (λ) ∩ U (which is a nonempty set thanksto A1). By A2, for arbitrary µ ∈ Λ ∩ V

(3.8) f (xλ, µ) ≥ m(µ) + cd1+α(xλ, S (µ)).

Since by A1 the set S (µ) ∩U is nonempty, for any ε > 0 there exists some xεµ ∈ S (µ)such that

(3.9) ‖xλ − xεµ‖ ≤ d(xλ, S (µ)) + ε.

As xεµ ∈ S (µ) we have m(µ) = f (xεµ, µ) and inequality (3.8) reads

(3.10) f (xλ, µ) ≥ f (xεµ, µ) + cd1+α(xλ, S (µ)).

Since xλ ∈ S (λ), we have that

(3.11) f (xεµ, λ) ≥ f (xλ, λ).

By adding (3.10) and (3.11) and rearranging, we obtain

[ f (xλ, µ) − f (xεµ, µ)] − [ f (xλ, λ) − f (xεµ, λ)] ≥ cd1+α(xλ, S (µ)).

That is,

(3.12) f1

(xλ, xεµ, µ

)− f1

(xλ, xεµ, λ

)≥ cd1+α(xλ, S (µ)).

Using A3, that is, f1 ∈ Lα,1(K; Λ ∩ V), we estimate the left-hand side of (3.12):

f1

(xλ, xεµ, µ

)− f1

(xλ, xεµ, λ

)≤ k f1‖xλ − xεµ‖

α‖λ − µ‖.

Hence, we have that k f1‖λ − µ‖ ‖xλ − xεµ‖α ≥ cd1+α(xλ, S (µ)). From this and (3.9) it

follows thatk f1‖λ − µ‖[d(xλ, S (µ)) + ε]α ≥ cd1+α(xλ, S (µ)).

Letting ε ↓ 0 and then dividing by dα(xλ, S (µ)) > 0 (if = 0 the inequality below istrivial), we obtain k f1‖λ − µ‖ ≥ cd(xλ, S (µ)), or

d(xλ, S (µ)) ≤k f1

c‖λ − µ‖.

As xλ was an arbitrary point in S (λ) ∩ U, the latter yields

e(S (λ) ∩ U, S (µ)) ≤k f1

c‖λ − µ‖,

completing the proof. �


Examples and corollaries

The following is a basic example of non-smooth parameterized minimizationproblem with Lipschitz continuous solution map with unbounded values. We showthat it is within the scope of Proposition 3.1.4.

Example 3.1.5. Let K = R2 and

f (x1, x2, λ) := |x1 − x2 − λ|,

x1, x2, λ ∈ R. Consider the parameterized family of unconstrained minimization prob-lems over the plane

P(λ) infx1,x2

f (x1, x2, λ).

Then the solution map S : λ⇒ S (λ) is Lipschitz continuous.

Proof. Obviously, for any λ ∈ R the solution set consists of a single line, i.e.,S (λ) = {(x1, x2) : x1 − x2 = λ}. Moreover, for λ and µ the solution sets S (λ) and S (µ)are parallel lines. The distance between S (λ) and S (µ) is the distance from any point(x1, x2) ∈ S (λ) to the line x1 − x2 = µ which is equal to |x1−x2−µ|

√2

=|λ−µ|√

2, so the map S

is Lipschitz continuous with Lipschitz constant 1√

2.

Note that the sufficient condition A holds. IndeedA1 holds with U ≡ R2;A2 holds with α = 0, c =

√2, and U = R2, V = R;

A3 holds because f1 ∈ L0,1(R2,R) with k f1 = 2.

The Lipschitz constant provided by Proposition 3.1.4 isk f1c =

√2. �

The next example shows that studying the generalized Euler equation may some-times be inadequate for obtaining Aubin continuity of the solution map. This isbecause the set of the stationary points may be larger than the set of minima.

Example 3.1.6. Let K = R2 and

f (x1, x2, λ) := (x1 + λx2 − 1)2(x2 + λx1 + 1)2,

x1, x2, λ ∈ R. Consider the parameterized family of unconstrained minimization prob-lems over the plane

P(λ) infx1,x2

f (x1, x2, λ).

Then at the point λ = 1 the set of solutions S (λ) is smaller than the set ofstationary points S t(λ) := {x ∈ R2 : 0 ∈ ∇x f (x, λ)}. Moreover, the map S is Aubincontinuous near any point in his graph while S t is not Aubin continuous near thepoint (λ, x) ∈ gph S t where λ = 1 and x = (0, 0).


Proof. Straightforward computations show that for any λ ∈ R the solution set

S (λ) = {(x1, x2) : x2 + λx1 = −1, or x1 + λx2 = 1}

is the union of two lines—the line p1(λ) with equation x2 + λx1 = −1 and the linep2(λ) with equation x1 + λx2 = 1. Because of

∇x f (x, λ) = [2(x1 + λx2 − 1)(x2 + λx1 + 1)(2λx1 + (1 + λ2)x2 + 1 − λ),2(x1 + λx2 − 1)(x2 + λx1 + 1)((1 + λ)2x1 + 2λx2 + λ − 1)],

the set of the stationary points at λ = 1 consists of three parallel lines

S t(1) = {(x1, x2) : x1 + x2 = 1, or x1 + x2 = −1, or x1 + x2 = 0},

while for λ , 1, S t(λ) ≡ S (λ).It is not difficult to see that S is Aubin continuous near an arbitrary point (λ, x) ∈

gph S (we note, by the way, that S is not Lipschitz continuous). Indeed, fix λ ∈ Rand take x = (x1, x2) ∈ S (λ) = p1(λ) ∪ p2(λ). Obviously, x , 0.

Take λ such that |λ − λ| < 1/2. If x ∈ p1(λ), then

d(x, S (λ)) ≤ d(x, p1(λ)) =

∣∣∣∣(λ − λ) x1

∣∣∣∣√

1 + λ2≤

∣∣∣λ − λ∣∣∣ |x1| ,

and if x ∈ p2(λ), then

d(x, S (λ)) ≤ d(x, p2(λ)) =

∣∣∣∣(λ − λ) x2

∣∣∣∣√

1 + λ2≤

∣∣∣λ − λ∣∣∣ |x2| ,

which yields

d(x, S (λ)) ≤∣∣∣λ − λ∣∣∣ max{|x1|, |x2|} ≤

∣∣∣λ − λ∣∣∣ ‖x‖ < ‖x‖/2.This implies that for all λ such that |λ− λ| < 1/2 the intersection of S (λ) with the

neighborhood U := x + ‖x‖B◦ is nonempty.Take x = (x1, x2) ∈ S (λ) ∩ U and µ such that |µ − λ| < 1/2. Similarly we get

d(x, S (µ)) ≤ |λ − µ| ‖x‖ ≤ |λ − µ| [‖x − x‖ + ‖x‖] ≤ 2‖x‖ |λ − µ|.

Hence,

e(S (λ) ∩ U, S (µ)) ≤ 2‖x‖ |λ − µ|, ∀λ, µ ∈ λ +12

B◦,

which means that S is Aubin continuous near (λ, x) ∈ gph S .


In contrast, S t is not Aubin continuous near the point, (λ, x) ∈ gph S t where λ = 1and x = (0, 0). Indeed, if S t is Aubin continuous near that point, then d(x, S t(λ))tends to zero as λ tends to 1. But the distance

d(x, S t(λ)) = min{d(x, p1(λ)), d(x, p2(λ))} =1

√1 + λ2

tends to 1√

2as λ tends to 1, which means that S t is not Aubin continuous near

(λ, x) ∈ gph S t. �

As an immediate consequence of Proposition 3.1.4 we get the following.

Corollary 3.1.7. Let for the parameterized family of minimization problems P(λ) thefollowing assumption hold

(A′)

for all λ ∈ Λ, all x ∈ K, and some c > 0

1. S (λ) , ∅;

2. f (x, λ) ≥ m(λ) + cd2(x, S (λ));

3. f ∈ C1,1(X × Z).

Then the solution map S : Λ⇒ X is Lipschitz continuous on Λ.

In a Banach space X with separable dual X∗ the notion of a second-order subdiffe-rential for a function f ∈ C1,1(X) is introduced in Georgiev and Zlateva [78] (see alsothe previous work Hiriart-Urruty, Strodiot and Nguyen [85] for the finite dimensionalcase). For any x ∈ X the second-order subdifferential ∂2 f (x) of f at x is a nonempty,convex, and w∗ compact set in L(X × X) (the Banach space of all bilinear continuousfunctionals M : X × X → R with the norm ‖M‖ := sup‖h1‖=‖h2‖=1 |M[h1, h2]|), which issingleton exactly when f is twice strictly Gateaux differentiable at x.

Setting a simple condition on the second subdifferential is sufficient to get afamily of functions satisfying assumption A′ in the above corollary.

Indeed, let X be a Banach space with separable dual. Let in the parameterizedfamily of minimization problems P(λ), f ∈ C1,1(X × Z), and let the constraint set Kbe closed and convex. If there exist c > 0 with

(3.13) 〈M(y−x), y−x〉 ≥ c‖y − x‖2 for all λ ∈ Λ, x, y ∈ K, M ∈ ∂2 f (·, λ)(x),

then the solution map S : Λ → X will be single-valued and Lipschitz continuouson Λ.

It is easily seen that (3.13) implies uniform on λ ∈ Λ strong convexity of f (·, λ)on K. By this and continuity of f (·, λ), for every λ the infimum of f (·, λ) is attainedat unique xλ ∈ K and A′1 holds.


For any x ∈ K and λ ∈ Λ there exists some zλ ∈ K and Mzλ ∈ ∂2 f (·, λ)(zλ) with

f (x, λ) = f (xλ, λ) + 〈∇x f (xλ, λ), x − xλ〉 +12〈Mzλ(x − xλ), x − xλ〉

(see Georgiev and Zlateva [78]). Since xλ is a minimum point for f (·, λ) on K and Kis convex, then for all x ∈ K, 〈∇x f (xλ, λ), x − xλ〉 ≥ 0 and from above equality and(3.13)

f (x, λ) ≥ m(λ) +12

c‖x − xλ‖2,

so A′2 holds.

We will use Corollary 3.1.7 to obtain existence and Lipschitz continuity of theoptimal solution for a linearly perturbed optimization problem, assuming a slightlyweaker version (see (3.14) below) of the uniform second-order growth condition(Definition 5.19 in Bonnans and Shapiro [22]), and C1,1 data. In this way we extendBonnans and Shapiro [22, Theorem 5.17] (see also Bonnans and Shapiro [22, Remark5.19]), where C2 data are assumed.

Recall that the Banach space X has Radon–Nikodym property (RNP) if for everybounded set C and every ε > 0, there exists an x ∈ C that does not belong to theclosed convex hull of C \ {x +εB◦}. All Banach spaces which have separable dual andall reflexive Banach spaces have RNP. In Diestel and Uhl [59, p. 157] there is a longlist of equivalent definitions of RNP. A good introductory survey on RNP is Diesteland Uhl [60].

An efficient tool in dealing with minimization problems on Banach space X withRNP is Stegall’s variational principle [157] (see also Phelps [133, Theorem 5.15]):Let C ⊂ X be a non-empty closed and bounded convex set and let f : C → R∪ {+∞}be a lower semicontinuous function, bounded below on C, then for every ε > 0, thereexists x∗ ∈ X∗ with ‖x∗‖ ≤ ε such that f + x∗ attains its strong minimum on C. Let usremind that x0 ∈ C is said to be a strong minimum for function g : C → R ∪ {+∞} onthe set C if g(x0) = infC g and ‖xn − x0‖ → 0 whenever g(xn)→ g(x0).

Corollary 3.1.8. Let the Banach space X have Radon–Nikodym property. Considera parameterized family of minimization problems P(λ), where the parameter space isX∗ and f : X × X∗ → R is defined by f (x, λ) := f (x) + 〈λ, x〉.

Assume that the constraint set K is closed and convex, f ∈ C1,1(X), and S (0) isnonempty.

Suppose that there exist neighborhood V of the origin 0 of X∗ and a constantc > 0 such that for all λ ∈ V and all xλ ∈ S (λ) it holds that

(3.14) f (x, λ) ≥ f (xλ, λ) + c‖x − xλ‖2, ∀x ∈ K.

Then there exists a neighborhood W of the origin 0 of X∗ such that S (λ) is single-valued and Lipschitz continuous on W.


Proof. From (3.14) it is clear that S (λ) contains at most one point for λ ∈ V .We will show that S (λ) is nonempty for λ belonging to some neighborhood of 0.

Fix γ > 0 such that W := 2γB◦ ⊂ V .Given λ ∈ γB◦, let εk be a sequence of positive numbers less than γ, tending to

zero.Thanks to (3.14) with λ = 0, f (·, λ) is bounded below on K.If K is bounded we could apply directly Stegall’s variational principle for the

function f (·, λ) : K → R and εk to find x∗k ∈ X∗ with ‖x∗k‖ ≤ εk and a strong minimumxk of f (·, λ) + x∗k on K.

If K is not bounded, a variant of Stegall’s variational principle still holds thanksto (3.14). Indeed, (3.14) for λ = 0 reads

f (x) ≥ f (x0) + c‖x − x0‖2, ∀x ∈ K,

which yields that for all x ∈ K,

f (x, λ) ≥ f (x0) + 〈λ, x〉 + c‖x − x0‖2 = f (x0, λ) + 〈λ, x − x0〉 + c‖x − x0‖

2

≥ f (x0, λ) + ‖x − x0‖[c‖x − x0‖ − ‖λ‖](3.15)

≥ f (x0, λ) + ‖x − x0‖[c‖x − x0‖ − γ].

Set r := 3γc . Now, we apply Stegall’s variational principle for the function f (·, λ) on

the closed bounded set K ∩ {x0 + rB} and εk. Thus, there exists x∗k ∈ X∗, ‖x∗k‖ < εk,and a point xk ∈ K ∩ {x0 + rB} such that f (·, λ) + x∗k attains a strong minimum onK ∩ {x0 + rB} at xk. Moreover, xk is a strong minimum of f (·, λ) + x∗k on K. Indeed, ifwe assume that x ∈ K is such that

f (x, λ) + 〈x∗k, x〉 ≤ f (xk, λ) + 〈x∗k, xk〉 = infK∩{x0+rB}

f (·, λ) + x∗k ≤ f (x0, λ) + 〈x∗k, x0〉,

then by (3.15) we will have

‖x − x0‖[c‖x − x0‖ − γ] ≤ ‖x∗k‖‖x − x0‖ ≤ εk‖x − x0‖ < γ‖x − x0‖,

or

‖x − x0‖ ≤2γc< r,

which means that x ∈ x0 + rB and clearly entails x = xk.However, in both cases for any k we found x∗k ∈ X∗ with ‖x∗k‖ ≤ εk and unique

xk ∈ K satisfying

f (x) + 〈λ + x∗k, x〉 ≥ f (xk) + 〈λ + x∗k, xk〉, ∀x ∈ K.

This means that S (λ + x∗k) = {xk} and since λ + x∗k ∈ V , (3.14) reads

(3.16) f (x) + 〈λ + x∗k, x〉 ≥ f (xk) + 〈λ + x∗k, xk〉 + c‖x − xk‖2, ∀x ∈ K.


Substitute x = xn and rearrange to obtain

f (xn) − f (xk) ≥ 〈λ + x∗k, xk − xn〉 + c‖xn − xk‖2.

Also, swapping k and n we get

f (xk) − f (xn) ≥ 〈λ + x∗n, xn − xk〉 + c‖xn − xk‖2.

Adding the above two, we obtain 2c‖xn−xk‖2 ≤ 〈x∗k−x∗n, xn−xk〉 ≤ (‖x∗k‖+‖x

∗n‖)‖xn−xk‖.

That is, 2c‖xn − xk‖ ≤ εk + εn, which means that xk is a Cauchy sequence. Let xλ ∈ Kbe its limit. Passing to limit in (3.16) we see that xλ ∈ S (λ).

A straightforward application of Corollary 3.1.7 completes the proof. �

3.1.2 Parameterized minimax problem

In this subsection we study the behavior of the saddle points set of a parameterizedfamily of minimax problems.

Preliminaries and statement of the problem

Let X and Y be Banach spaces, and let { f (·, ·, λ) : X × Y → R, λ ∈ Λ} be a familyof functions defined on the product space X × Y , parameterized by λ ∈ Λ ⊂ Z.

Let us consider the parameterized family of minimax problems

M(λ) infx∈K

supy∈L

f (x, y, λ),

where the constraints are nonempty closed sets K ⊂ X and L ⊂ Y . Denote the optimalvalue of M(λ) by m(λ) and recall that the (possibly empty) set of saddle points off (·, ·, λ) on K × L is given by (3.1).

For a set C ⊂ X × Y we denote by πXC and πYC the canonical projections of Con the spaces X and Y , respectively. More precisely, x ∈ πXC whenever there existssome y ∈ Y with (x, y) ∈ C and y ∈ πYC whenever there exists some x ∈ X with(x, y) ∈ C.

It is well known that the saddle point set is a product set; that is,

(3.17) S(λ) = πXS(λ) × πYS(λ).

To the parameterized family of functions { f (·, ·, λ), λ ∈ Λ} one naturally associatesthree difference functions:

f1(x, x′, λ, y) := f (x, y, λ) − f (x′, y, λ),f2(y, y′, λ, x) := f (x, y, λ) − f (x, y′, λ),f3(λ, λ′, x, y) := f (x, y, λ) − f (x, y, λ′).


By analogy with Definition 3.1.1 we write f1 ∈ Lα,βW (U; V) whenever the functions

f y1 (x, x′, λ) := f1(x, x′, λ, y) are such that for all y ∈ W, f y

1 ∈ Lα,β(U; V), and supy∈W k f y

1is finite. We set kf1 := supy∈W k f y

1.

Easy computations as those done in Proposition 3.1.2 show that f1 ∈ Lα,βW (U; V)

exactly when f3 ∈ Lβ,αW (V; U) and that f2 ∈ L

α,βU (W; V) exactly when f3 ∈ L

β,αU (V; W).

Now we are ready to state the sufficient condition for Aubin continuity of thesaddle points map S : Λ⇒ X × Y .

Let (λ, x, y) ∈ gphS. We set the following local assumption A at (λ, x, y):

(A)

there exist neighborhoods U of x, W of y, and V of λ, such that1. S(λ) ∩ [U ×W] , ∅ for all λ ∈ Λ ∩ V;and there exist constants c > 0 and α ∈ [0, 1] such that2. f (x, y′, λ′)≥m(λ′)+cd1+α(x, πXS(λ′)),

f (x′, y, λ′)≤m(λ′)−cd1+α(y, πYS(λ′)),∀λ, λ′ ∈ Λ ∩ V,∀(x, y) ∈ S(λ) ∩ [U ×W],∀(x′, y′) ∈ S(λ′);

3. f1 ∈ Lα,1L∩W(K; Λ ∩ V) and f2 ∈ L

α,1K∩U(L; Λ ∩ V).

Clearly, condition A3 could be replaced by

f3 ∈ L1,αL∩W(Λ ∩ V; K) ∩ L1,α

K∩U(Λ ∩ V; L).

A1 implies that m(λ) is finite for λ ∈ Λ ∩ V .We would show the consistency of our main hypothesis by giving some examples

of parameterized families of functions { f (·, ·, λ), λ ∈ Λ} for which A is satisfied.One gets a parameterized family of functions { f (·, ·, λ), λ∈Λ} satisfying A2, for

example, by assuming that K ⊂ X and L ⊂ Y are nonempty closed convex sets; thefunction f (·, y, λ) is lower semicontinuous and uniformly on (y, λ) ∈ L × Λ stronglyconvex on K, i.e., such that for some constant c > 0 the inequality

f (tx + (1 − t)x′, y, λ) ≤ t f (x, y, λ) + (1 − t) f (x′, y, λ) − ct(1 − t)‖x − x′‖2

holds for every t ∈ [0, 1], every x, x′ ∈ K, and every (y, λ) ∈ L × Λ; the functionf (x, ·, λ) is upper semicontinuous and uniformly on (x, λ) ∈ K × Λ strongly concaveon L, i.e., such that the inequality

f (x, ty + (1 − t)y′, λ) ≥ t f (x, y, λ) + (1 − t) f (x, y′, λ) + ct(1 − t)‖y − y′‖2

holds for every t ∈ [0, 1], every y, y′ ∈ L, and every (x, λ) ∈ K × Λ.Then it is routine to see that A2 holds at any (λ, x, y) ∈ gphS with α = 1, V = Z,

U = X, and W = Y .Examples of parameterized families of functions satisfying A3 are given by the

following.


Lemma 3.1.9. Let (λ, x, y) ∈ gphS and let U ⊂ X, W ⊂ Y and V ⊂ Z be convexneighborhoods of K, L, and λ, respectively. Consider the conditions:

(F 1)

for any (y, λ) ∈ L × [Λ ∩ V], f (·, y, λ) is Lipschitz and regularfunction on U and ∂x f (x, y, ·) : Λ ∩ V → X∗ is a k–Lipschitz map onΛ ∩ V with k that does not depend on (x, y) ∈ K × L,

(F 2)

for any (x, λ) ∈ K × [Λ ∩ V], f (x, ·, λ) is Lipschitz and regularfunction on W and ∂y f (x, y, ·) : Λ ∩ V → Y∗ is a k–Lipschitz map onΛ ∩ V with k that does not depend on (x, y) ∈ K × L,

(F 3)

for any (x, y) ∈ K × L, f (x, y, ·) is Lipschitz and regularfunction on V and ∂λ f (·, y, λ) : K → Λ∗ is a k–Lipschitz map onK with k that does not depend on (y, λ) ∈ L × [Λ ∩ V],

(F 4)

for any (x, y) ∈ K × L, f (x, y, ·) is Lipschitz and regularfunction on V and ∂λ f (x, ·, λ) : L→ Λ∗ is a k–Lipschitz map onL with k that does not depend on (x, λ) ∈ K × [Λ ∩ V].

If f satisfies F 1 − F 2 or F 3 − F 4, then A3 holds with α = 1.

Proof. We follow the same steps as in the proof of Lemma 3.1.3.If f satisfies F 1, then f1 ∈ L

1,1L (K; Λ ∩ V).

If f satisfies F 2, then f2 ∈ L1,1K (L; Λ ∩ V).

If f satisfies F 3, then f3 ∈ L1,1L (Λ ∩ V; K).

If f satisfies F 4, then f3 ∈ L1,1K (Λ ∩ V; L). �

Obviously, if f ∈ C1,1(U ×W × V), then F 1 to F 4 hold.

Lipschitz-like continuity of the saddle point map

Here we will prove that assumption A is sufficient for Aubin continuity of thesaddle point map S. Let us note that the result cannot be derived (or at least not inan obvious manner) from the case of minimization only. Indeed, if f (x, y, λ) satisfiesassumption A, then the function f (x, λ) := supy∈L f (x, y, λ) satisfies assumption A2but A3 for this f (x, λ) cannot be derived from A3 since the differences of supremainvolved do not yield themselves to rearrangement.

Theorem 3.1.10. Assume that for the parameterized family of minimax problemsM(λ) the assumption A holds at some (λ, x, y) ∈ gphS. Then for all λ, µ ∈ Λ ∩ V

(3.18) e(S(λ) ∩ [U ×W],S(µ)) ≤2kc‖λ − µ‖,

where k := max{kf1 , kf2}, hence the saddle point map S : Λ ⇒ X × Y is Aubincontinuous near (λ, x, y) ∈ gphS.


Proof. By A1 for all λ ∈ Λ ∩ V the set S(λ) ∩ [U ×W] is nonempty.Fix λ ∈ Λ ∩ V and take some (xλ, yλ) ∈ S(λ) ∩ [U ×W].Pick any other µ ∈ Λ ∩ V .Since S(µ) is a nonempty set we find some xεµ ∈ πXS(µ) such that

‖xλ − xεµ‖ ≤ d(xλ, πXS(µ)) + ε.

Similarly, there is yεµ ∈ πYS(µ) such that

‖yλ − yεµ‖ ≤ d(yλ, πYS(µ)) + ε.

By the product form of the saddle point set, (xεµ, yεµ) ∈ S(µ). The first inequality

of A2 for (xλ, yλ) ∈ S(λ) ∩ [U ×W] and (xεµ, yεµ) ∈ S(µ) reads

(3.19) f(xλ, yεµ, µ

)≥ m(µ) + cd1+α(xλ, πXS(µ)),

in particular,

(3.20) f(xλ, yεµ, µ

)≥ m(µ),

while the second inequality of A2 states

(3.21) m(µ) ≥ f(xεµ, yλ, µ

)+ cd1+α(yλ, πYS(µ)),

in particular,

(3.22) m(µ) ≥ f(xεµ, yλ, µ

).

Combining (3.19) with (3.22) and (3.20) with (3.21), we get

f(xλ, yεµ, µ

)≥ f

(xεµ, yλ, µ

)+ cd1+α(xλ, πXS(µ)),

f(xλ, yεµ, µ

)≥ f

(xεµ, yλ, µ

)+ cd1+α(yλ, πYS(µ)),

which yields

f(xλ, yεµ, µ

)− f

(xεµ, yλ, µ

)≥ c[max{d(xλ, πXS(µ)), d(yλ, πYS(µ))}]1+α.

By the definition of the supremum norm and since S(µ) is a product set, it isobvious that

(3.23)d((xλ, yλ),S(µ)) = d((xλ, yλ), πXS(µ) × πYS(µ))

= max{d(xλ, πXS(µ)), d(yλ, πYS(µ))},


and the above inequality can be rewritten as

f(xλ, yεµ, µ

)− f

(xεµ, yλ, µ

)≥ cd1+α((xλ, yλ),S(µ)).

We transform the left-hand side to get

f(xλ, yεµ, µ

)− f (xλ, yλ, µ) + f (xλ, yλ, µ) − f

(xεµ, yλ, µ

)≥ cd1+α((xλ, yλ),S(µ)),

which is

(3.24) −f2

(xλ, yλ, yεµ, µ

)− f1

(xεµ, xλ, yλ, µ

)≥ cd1+α((xλ, yλ),S(µ)).

On the other hand, since (xλ, yλ) ∈ S(λ), the saddle point inequalities give

f (x, yλ, λ) ≥ f (xλ, yλ, λ) ≥ f (xλ, y, λ), ∀x ∈ K,∀y ∈ L.

In particular, for x = xεµ ∈ K we have

f (xεµ, yλ, λ) ≥ f (xλ, yλ, λ),

which is

(3.25) f1(xεµ, xλ, yλ, λ) ≥ 0,

and for y = yεµ ∈ L we get

f (xλ, yλ, λ) ≥ f (xλ, yεµ, λ),

which is

(3.26) f2(xλ, yλ, yεµ, λ) ≥ 0.

Adding the inequalities (3.24), (3.25), and (3.26) and rearranging we obtain

(3.27)[f1

(xεµ, xλ, yλ, λ

)−f1


)]+

[f2

(xλ, yλ, yεµ, λ

)−f2


)]≥ cd1+α((xλ, yλ),S(µ)).

Since by A3, f1 ∈ Lα,1L∩W(K; Λ∩V), the term in first brackets in (3.27) is estimated

by

(3.28) f1


)− f1

(xεµ, xλ, yλ, λ

)≤ kf1‖x

εµ − xλ‖α‖λ − µ‖,

and since f2 ∈ Lα,1K∩U(L; Λ ∩ V) the term in second brackets in (3.27) is estimated by

(3.29) f2

(xλ, yλ, yεµ, λ

)− f2


)≤ kf2‖yλ − yεµ‖

α‖λ − µ‖.


Using (3.29) and (3.28) in (3.27) and setting k := max{kf1 , kf2}, we get

k‖λ − µ‖[‖xλ − xεµ‖α + ‖yλ − yεµ‖

α] ≥ cd1+α((xλ, yλ),S(µ)).

By the choice of xεµ and yεµ, we have that

k‖λ − µ‖[(d(xλ, πXS(µ)) + ε)α + (d(yλ, πYS(µ)) + ε)α]

≥ cd1+α((xλ, yλ),S(µ)).

Passing to limit ε ↓ 0 we obtain

(3.30) k‖λ − µ‖[dα(xλ, πXS(µ)) + dα(yλ, πYS(µ))] ≥ cd1+α((xλ, yλ),S(µ)).

By (3.23) we get

dα(yλ, πYS(µ)) + dα(xλ, πXS(µ)) ≤ 2[max{d(yλ, πYS(µ)), d(xλ, πXS(µ))}

]α= 2dα((xλ, yλ),S(µ)),

and from (3.30) we obtain

2k‖λ − µ‖dα((xλ, yλ),S(µ)) ≥ cd1+α((xλ, yλ),S(µ)).

This yields2kc‖λ − µ‖ ≥ d((xλ, yλ),S(µ)),

and since (xλ, yλ) was an arbitrary element of S(λ) ∩ [U ×W] the latter implies

e(S(λ) ∩ [U ×W],S(µ)) ≤2kc‖λ − µ‖.

The proof is completed. �

As an immediate consequence of Theorem 3.1.10 and Lemma 3.1.9 one deducesthe following.

Corollary 3.1.11. Let for the parameterized family of minimax problems M(λ) thefollowing assumption hold:

(A′)

1. S(λ) , ∅ for any λ ∈ Λ;

2. for some constant c > 0 and all λ∈Λ, (x, y)∈S(λ), (x′, y′)∈K×L :

f (x′, y, λ) ≥ m(λ) + cd2(x′, πXS(λ)),

f (x, y′, λ) ≤ m(λ) − cd2(y′, πYS(λ));

3. f ∈ C1,1(X × Y × Z).

Then the saddle point map S : Λ→ X × Y is single-valued and Lipschitz continuous.


As we pointed out after Corollary 3.1.7 we could deduce the single-valuednessand Lipschitz continuity of the saddle point map S when X and Y has separableduals, the sets K and L are convex, f ∈ C1,1(X × Y × Z), and there exists a constantc > 0 such that for all λ ∈ Λ, x, z ∈ K, y,w ∈ L,

〈M(z − x), z − x〉 ≥ c‖z − x‖2, 〈N(w − y),w − y〉 ≤ −c‖w − y‖2

for all M ∈ ∂2 f (·, y, λ)(x) and all N ∈ ∂2 f (x, ·, λ)(y).

3.1.3 Lipschitz continuity of the saddle points map in context oftwo-player zero sum differential games

In this subsection we briefly consider a differential game for which our resultmight be of relevance.

In differential games, open-loop strategies are of low interest in many examples.One major reason is that differential games with open-loop strategies do not satisfy,in general, the dynamic programming principle (see Cardaliaguet, Quincampoix andSaint-Pierre [39, 41], Plaskacz and Quincampoix [134]). It is well known now thatto solve many problems in differential games (existence of a value, characterizationof the game through Hamilton–Jacobi equations), one needs a more general classof strategies which contains the feedback strategies.1 Such class of strategies is theclass of nonanticipative strategies introduced by Elliot–Roxin–Varaiya–Kalton (cf. forinstance Cardaliaguet, Quincampoix and Saint-Pierre [40]); another possible class ofstrategies are the positional strategies discussed in Krasovskii and Subbotin [104].The class of nonanticipative strategies is nice enough to prove the existence of thevalue, but it is hard to implement for the players. So it is important to know when thenonanticipative strategies giving the value of the game can be reduced to feedbackstrategies. We will explain in this part how the main result of the paper can lead to apartial answer to this question.

We consider the following differential game with dynamic described by the diffe-rential equation:

(3.31)

x′(t) = f (x(t), u(t)), y′(t) ∈ g(y(t), v(t)),

u(t) ∈ U, v(t) ∈ V,

where f : Rn×U → Rn and g : Rn×V → Rn are (globally) Lipschitz, U ⊂ Rm, V ⊂ Rp

being the control sets of the players. The first player—Ursula, playing with u—wants

1It has been shown in Cardaliaguet [38] that the class of regular feedback is not rich enough tosolve differential games at a satisfactory level of generality.


to minimize a given cost. The goal of the second player—Victor, playing with v—is tomaximize the cost

J(x0, y0, u(·), v(·)) :=∫ ∞

0e−rtl(x(t), y(t)) dt,

where (x(·), y(·)) is the unique solution starting at t = 0 from (x0, y0) and r > 0 isfixed. Observe that the game is in a separable form, i.e., each player acts in his owndynamics. This is the case, for instance, for pursuit games. Moreover, the integralcost does not depend directly on the control but only on the trajectories.

We work here in the framework of the nonanticipative strategies (also calledVaraiya–Roxin–Elliot–Kalton strategies). Let

(3.32) U = L1([0,+∞[,U), V = L1([0,+∞[,V)

be the sets of time-measurable controls of the first (Ursula) and the second (Victor)player, respectively. We denote t 7→ (x(t, x0, u(t)), y(t, y0, v(t))) the solution of (3.31)starting at t = 0 from (x0, y0).

Definition 3.1.12. A map α : V → U is a nonanticipative strategy (for Ursula) if itsatisfies the following condition: For any s ≥ 0, for any v1(·) and v2(·) belonging toV such that v1(·) and v2(·) coincide almost everywhere on [0, s], the images α(v1(·))and α(v2(·)) coincide almost everywhere on [0, s].

Nonanticipative strategies β : U → V (for Victor) are defined in the symmetricway.

Assume now that f and g are continuous and Lipschitz with respect to x andy. Then, we know that the game has a value (cf. Cardaliaguet, Quincampoix andSaint-Pierre [41]), namely,

V(x0, y0) = infα

supv∈V

J(x0, y0, α(v(·)), v(·)) = supβ

infu∈U

J(x0, y0, u(·), β(u(·))).

Let us denote by R(t) the attainable set of the dynamics (3.31) at moment t; i.e.,

R(t) = {(x(t), y(t)) ∈ Rn × Rn : ∃ u ∈ U, v ∈ V such that

(x(·), y(·)) is the solution of (3.31) starting at t = 0 from (x0, y0)}.

Now, suppose that U and V are convex and compact. Saddle point of the functionl(·, ·) on R(t) will be any point (x, y) ∈ R(t) that satisfies

l(x, y) ≤ l(x, y) ≤ l(x, y), ∀(x, y) ∈ R(t),

and, because of e−rt > 0, the saddle points of l(·, ·) on R(t) will be the same as thesaddle points of e−rtl(·, ·) on R(t).


Let us denote the (possibly empty) set of all saddle points of the function l(·, ·)on R(t) by S(t) := {(x, y) ∈ R(t) : l(x, y) ≤ l(x, y) ≤ l(x, y), ∀(x, y) ∈ R(t)}.

Let us suppose that the parameterized by t family of functions {e−rtl(·, ·), t ∈ [0,∞)}satisfies an assumption slightly stronger than assumption A, namely:

1. S(t) , ∅, ∀t ≥ 0;and there exist constants k, c > 0 and α ∈ [0, 1] such that∀t, t′ ≥ 0,∀(x, y)∈S(t),∀(x′, y′)∈S(t′) it holds :

2. l(x′, y) ≥ l(x, y) + cert‖x′ − x‖1+α,

l(x, y′) ≤ l(x, y) − cert‖y′ − y‖1+α;

3. |l(x, y) − l(x′, y)| ≤ k‖x − x′‖α,

|l(x, y) − l(x, y′)| ≤ k‖y − y′‖α.

This assumption guarantees that for any t ∈ [0,∞) the saddle point mapping S(t) issingle-valued and Lipschitz continuous; i.e., for all positive t, the function e−rtl(·, ·)has a saddle point (x(t), y(t)) on the attainable set R(t) of the dynamics (3.31), whichdepends in a Lipschitz way on t.

Therefore, if it turns out that so-obtained single valued saddle point mapping is atrajectory (x(·), y(·)) of (3.31), then it is an optimal feedback strategy of the game.

For example, under the above assumptions in the case when m = p = n andf (x, u) = u, g(y, v) = v, the Lipschitz continuity on t of the saddle point map impliesthat the corresponding controls u and v belong to U and V, respectively, and, hence,they generate a trajectory of the differential game.

3.2 Aubin criterion for metric regularity

In this section we investigate metric regularity of set-valued mappings. We presenta derivative criterion for metric regularity of set-valued mappings that is based onworks of J.-P. Aubin and co-authors. A related implicit mapping theorem is alsoobtained and several applications are given.

Throughout this section, X and Y are Banach spaces. The norms of both X andY are denoted by ‖ · ‖; the closed ball centered at x with radius r by B[x, r] andthe open ball by B(x, r); the closed unit ball is simply B and the open one B◦. Aneighborhood of a point x is any open set containing x. The distance from a pointx to a set A is denoted by d(x, A). By a mapping F from X to Y we generallymean a set-valued mapping and write F : X ⇒ Y , having its inverse F−1 defined asF−1(y) = {x | y ∈ F(x)} and graph gph F = {(x, y) | y ∈ F(x)}. When F is single-valued(a function) we write F : X → Y .

3.2. Aubin criterion for metric regularity 181

A mapping F : X ⇒ Y is said to be metrically regular at x for y if (x, y) ∈ gph Fand there exist a constant κ > 0 and neighborhoods U of x and V of y such that

(3.33) d(x, F−1(y)) ≤ κd(y, F(x)) for all (x, y) ∈ U × V.

The metric regularity can be identified with the finiteness of the regularity mod-ulus defined as

reg F(x | y) = inf{ κ | there exist neighborhoods U and V such that (3.33) holds}.

The absence of metric regularity is indicated by reg F(x | y) = ∞.The concept of metric regularity goes back to classical results by Banach, Lyus-

ternik and Graves. More recently, its central role has been recognized in variationalanalysis for both theoretical developments such as obtaining necessary optimality con-ditions and also in numerically oriented studies, e.g., when deriving error bounds forsolution approximations. Discussions of the property of metric regularity, its relationsto other properties and characterizations by various approximations are presented inRockafellar and Wets [152] and Ioffe [90].

Given a mapping F : X ⇒ Y , the graphical (contingent ) derivative of F at(x, y) ∈ gph F is the mapping DF(x |y) : X ⇒ Y whose graph is the tangent coneTgph F(x, y) to gph F at (x, y):

v ∈ DF(x |y)(u) ⇐⇒ (u, v) ∈ Tgph F(x, y).

Recall that the tangent cone is defined as follows: (u, v) ∈ Tgph F(x, y) when there existsequences tn ↓ 0, un → u and vn → v such that y + tnvn ∈ F(x + tnun) for all n.

The mapping DF(x |y) is positively homogeneous since its graph is a cone; specifi-cally, one has DF(x |y)(0) 3 0 and DF(x |y)(λu) = λDF(x |y)(u) for all u ∈ X for λ > 0.The convexified graphical derivative D??F(x |y) of F at x for y is defined in a similarway:

v ∈ D??F(x |y)(u) ⇐⇒ (u, v) ∈ co Tgph F(x, y)

where co stands for the closed convex hull. We also use the inner and the outer“norms” (see Rockafellar and Wets [152, Section 9D]) of a mapping H : X ⇒ Y:

‖H‖− = supx∈B

infy∈H(x)

‖y‖ and ‖H‖+ = supx∈B

supy∈H(x)

‖y‖.

Outer and inner norms can be related through adjoints. For a positively homogeneousmapping F : X ⇒ Y the upper adjoint F∗+ : Y∗ ⇒ X∗ is defined by

(y∗, x∗) ∈ gph F∗+ ⇐⇒ 〈x∗, x〉 ≤ 〈y∗, y〉 for all (x, y) ∈ gph F,

while the lower adjoint F∗− : Y∗ ⇒ X∗ is

(y∗, x∗) ∈ gph F∗− ⇐⇒ 〈x∗, x〉 ≥ 〈y∗, y〉 for all (x, y) ∈ gph F,


where X∗ and Y∗ are the dual spaces of X and Y . Borwein derived in [23] thefollowing duality relations between outer and inner norms for a sublinear mapping Fwith closed graph:

‖F‖+ = ‖F∗+‖− = ‖F∗−‖− and ‖F‖− = ‖F∗+‖+ = ‖F∗−‖+.

Recall that a mapping F : X ⇒ Y is said to have a locally closed graph at (x, y) whengph F ∩ (B[x, r] × B[y, r]) is a closed set for some r > 0.

The central result in this section is the following theorem:

Theorem 3.2.1 (Aubin criterion). Consider two Banach spaces X and Y , and amapping F : X ⇒ Y which graph is locally closed at (x, y) ∈ gph F. Then

(3.34) reg F(x|y) ≤ lim sup(x,y)→(x,y)(x,y)∈gph F

‖DF(x |y)−1‖−,

and hence F is metrically regular at x for y provided that

(3.35) lim sup(x,y)→(x,y)(x,y)∈gph F

‖DF(x |y)−1‖− < ∞.

If X is finite dimensional, then (3.34) becomes an equality,

(3.36) reg F(x|y) = lim sup(x,y)→(x,y)(x,y)∈gph F

‖DF(x |y)−1‖−,

and hence F is metrically regular at x for y if and only if (3.35) holds. Moreover,when both spaces X and Y are finite dimensional one has

(3.37) reg F(x | y) = lim sup(x,y)→(x,y)(x,y)∈gph F

‖D??F(x |y)−1‖−.

Theorem 3.2.1 can be viewed as a partial extension of Theorem 5.4.3 in Aubinand Frankowska book [10] where a sufficient condition for the Aubin property ofthe inverse F−1 is shown, for predecessors of this result see Aubin [3] and Aubinand Frankowska [9]. That condition is in general weaker than (3.35) but, as we seehere, for a finite-dimensional X is actually equivalent to it. Recall that a mappingS : Y ⇒ X has the Aubin property at y for x when (y, x) ∈ gph S and there existneighborhoods V of y and U of x such that

(3.38) e(S (y) ∩ U, S (y′)) ≤ κ‖y − y′‖ for all y, y′ ∈ V,

where e(A, B) = supx∈A d(x, B) is the excess from A to B. The Aubin property ofa mapping S is known to be equivalent to the metric regularity of S −1 and was


introduced in Aubin [5] under the name “pseudo-Lipschitz” continuity, it was studiedin Aubin and Frankowska [9] in the infinite dimensional case, for more bibliographicaldetails see Rockafellar and Wets [152]. Moreover, the infimum of the constant κ in(3.38) is equal to reg S −1(x | y).

The equality (3.37) was stated recently in Aubin, Bayen, Bonneuil, and Saint-Pierre [7] with a proof based on viability theory. The given here proof of (3.37) isinspired by the proof of Theorem 3.2.4 in Aubin [6] due to Frankowska.

The characterization of metric regularity exhibited in Theorem 3.2.1 complements,and in some sense also completes, results previously displayed by J.-P. Aubin andco-authors; therefore, we call it the Aubin criterion for metric regularity.

In a sense “dual” to the Aubin criterion is the known Mordukhovich criterion infinite dimensions, see Mordukhovich [122] and Rockafellar and Wets [152] whichuses the coderivative D∗F(x |y) defined as

v ∈ D∗F(x |y)(u) ⇐⇒ (v,−u) ∈ Ngph F(x, y),

where NC(x) is the (nonconvex, limiting) normal cone to the set C at x. The Mor-dukhovich criterion says that F is metrically regular at x for y if and only if

‖D∗F−1(y | x)‖+ < ∞.

One would expect that each of these criteria could be directly derived from theother one, and this is clearly so when gph F is Clarke regular, see Rockafellar andWets [152, 8.40 and 11.29]. In infinite dimensions, the characterizations of the metricregularity via coderivatives assume some more from the spaces, e.g., to be Asplund,see Mordukhovich [123], Mordukhovich and Shao [125]. When the domain space Xis finite dimensional and Y is any Banach space, a necessary and sufficient conditionfor metric regularity in terms of the Ioffe approximate coderivative is given in Jouraniand Thibault [99]. This latter result is the closest to Theorem 3.2.1 from the literatureknown to the author. We also refer to the book Klatte and Kummer [102] as a sourceof information on criteria for metric regularity.

If F : X → Y is a bounded linear mapping, denoted F ∈ L(X,Y), then theAubin criterion (3.36) is also valid for X and Y Banach spaces and reduces toreg F(x | y) = ‖F−1‖−. Equivalently, F is metrically regular (at any point) if and only ifF is surjective; this covers the classical case of the Banach open mapping principle.The equivalence among metric regularity at the origin, the finiteness of the innernorm, and the surjectivity holds also for mappings acting in Banach spaces whosegraphs are closed and convex cones. Specifically, we have the following result provedin Dontchev, Lewis and Rockafellar [62, Example 2.1] :

Proposition 3.2.2. Let X and Y be Banach spaces and let F : X ⇒ Y be such thatgph F is a closed and convex cone. Then the modulus of regularity of F at the originsatisfies

reg F(0 |0) = ‖DF(0 |0)−1‖− = ‖F−1‖−.


Moreover, reg F(0 |0) < ∞ if and only if F is surjective and then F is metricallyregular at any point in its graph.

In Subsection 3.2.2 we give a proof of Theorem 3.2.1 by first obtaining thesufficiency part of the Aubin criterion as a corollary from a more general “implicitmapping theorem” (Theorem 3.2.3) in Subsection 3.2.1, which is about the solutionmapping of a generalized equation of the form

(3.39) 0 ∈ G(p, x),

where p is a parameter. We show that if the partial graphical derivative with respectto x of the mapping G is bounded in the sense of (3.35), then G has a property of“partial metric regularity”.

In a related paper [110], Ledyaev and Zhu obtained an implicit mapping theo-rem for a general inclusion of the form (3.39) in terms of coderivatives in Banachspaces assumed to have Frechet-smooth Lipschitz bump functions. Putting aside thederivative condition in our Theorem 3.2.3 and the coderivative condition in Ledyaevand Zhu [110, Theorem 3.7] which are independent from each other and can not becompared, we impose weaker conditions on the mapping G and allow for arbitraryBanach spaces X and Y .

In Subsection 3.2.3 we present applications of the Aubin criterion to systems ofinequalities and to variational inequalities, obtaining a new characterization of strongregularity of variational inequalities over polyhedral sets. We also give a new proofof the radius (Eckart-Young) theorem first proven in Dontchev, Lewis and Rockafel-lar [62] with the help of Mordukhovich criterion; for history and recent develop-ments, see Dontchev, Lewis and Rockafellar [62], Dontchev and Rockafellar [66] andDontchev and Lewis [61].

In addition to the Aubin criterion, in Subsection 3.2.3 we use a fundamentalresult in the modern nonlinear analysis, commonly known as the Lyusternik-Gravestheorem, for more see, e.g., Aubin [5], Aubin and Frankowska [9], Dontchev, Lewisand Rockafellar [62] and Ioffe [90]. First, we need some terminology. For a functiong : X → Y and a point x ∈ int dom g, we introduce Lipschitz modulus of g at x asfollows:

lip g(x) = lim supx,x′→x

x,x′

‖g(x) − g(x′)‖‖x − x′‖

.

Recall that a function g : X → Y is strictly differentiable at x ∈ int dom g with a strictderivative mapping ∇g(x) ∈ L(X,Y), the space of linear bounded mappings from X toY , if

lip(g − ∇g(x))(x) = 0.

We use the following form of the well-known


Lyusternik-Graves theorem. Let X and Y be Banach spaces and consider a mappingF : X ⇒ Y and a point (x, y) ∈ gphF at which gphF is locally closed. Then for anyfunction g : X → Y which is strictly differentiable at x one has

reg(g + F )(x | y + g(x)) = reg(∇g(x) + F )(x | y + ∇g(x)x).

3.2.1 An implicit mapping theorem

In this subsection we study the inclusion (generalized equation)

0 ∈ G(p, x),

where G : P × X ⇒ Y , X and Y are Banach spaces, P is a metric space, x ∈ X is thevariable we are solving for and p ∈ P is a parameter. Let us denote by S : P⇒ X thesolution mapping which associates to a value p the set of solutions

(3.40) S (p) := {x ∈ X | G(p, x) 3 0}.

We will show that the local boundedness of the partial graphical derivative of themapping G in x, of the kind displayed in (3.35), implies partial metric regularityof G. The partial graphical derivative DxG(p, x |y) of G is defined as the graphicalderivative of the mapping x 7→ G(p, x) with p fixed.

Theorem 3.2.3 (Implicit mapping theorem). Let X and Y be Banach spaces, andlet P be a metric space. Consider a mapping G : P × X ⇒ Y and a point ( p, x, 0) ∈gph G such that the graph of G is locally closed near ( p, x, 0) and the functionp → d(0,G(p, x)) is upper semicontinuous at p. Then for every positive scalar csatisfying

(3.41) lim sup(p,x,y)→( p,x,0)(p,x,y)∈gph G

‖DxG(p, x |y)−1‖− < c

there exist neighborhoods V of p and U of x such that one has

(3.42) d(x, S (p)) ≤ cd(0,G(p, x)) for x ∈ U and p ∈ V.

Proof. On the product space Z := X × Y we consider the norm

|||(x, y)||| := max{‖x‖, c‖y‖},

which makes (Z, ||| · |||) a Banach space, and on the space P × Z we introduce themetric

σ((p, z), (q,w)) := max{ρ(p, q), |||z − w|||} for p, q ∈ P, z,w ∈ Z,


where ρ stands for the metric of P.A constant c satisfies (3.41) if and only if there exists η > 0 such that

(3.43)for every (p, x, y) ∈ gph G with σ((p, x, y), ( p, x, 0)) ≤ 3η,

and for every v ∈ Y there exists u ∈ DxG(p, x |y)−1(v) with ‖u‖ ≤ c‖v‖.

We can always choose η smaller so that the set gph G∩ ( p, x, 0)+3ηB is closed. Next,let us pick ε > 0 such that

(3.44) cε < 1.

In the proof we use the following lemma:

Lemma 3.2.4. For η and ε as above, choose any (p, ω, ν) ∈ gph G with (p, ω, ν) ∈( p, x, 0) + ηB and any s, 0 < s ≤ εη. Then for every y′ ∈ ν + sB◦ there exists x with(p, x, y′) ∈ gph G such that

(3.45) ‖x − ω‖ ≤1ε‖y′ − ν‖.

Proof of Lemma 3.2.4. Pick (p, ω, ν) ∈ gph G and s as required. The set Ep :={(x, y)|(p, x, y) ∈ gph G ∩ (( p, x, 0) + 3ηB)} ⊂ X × Y equipped with the metric inducedby the norm ||| · ||| is a complete metric space. The function Vp : Ep → IR defined as

Vp(x, y) := ‖y′ − y‖ for(x, y) ∈ Ep

is continuous in its domain Ep. We apply the Ekeland variational principle to Vp for(x, y) near (ω, ν) and the ε chosen in (3.44) to obtain the existence of (x, y) ∈ Ep suchthat

(3.46) Vp(x, y) + ε|||(ω, ν) − (x, y)||| ≤ Vp(ω, ν)

and

(3.47) Vp(x, y) ≤ Vp(x, y) + ε|||(x, y) − (x, y)||| for all (x, y) ∈ Ep.

The relations (3.46) and (3.47) come down as

(3.48) ‖y′ − y‖ + ε|||(ω, ν) − (x, y)||| ≤ ‖y′ − ν‖

and

(3.49) ‖y′ − y‖ ≤ ‖y′ − y‖ + ε|||(x, y) − (x, y)||| for all(x, y) ∈ Ep.

From (3.48) we obtain

(3.50) |||(ω, ν) − (x, y)||| ≤1ε‖y′ − ν‖.


Since y′ ∈ ν + sB◦, we then have

|||(ω, ν) − (x, y)||| <sε,

and hence, from the choice of (p, ω, ν),

σ((p, x, y), ( p, x, 0)) ≤ σ((p, x, y), (p, ω, ν)) + σ((p, ω, ν), ( p, x, 0))

≤ η + |||(ω, ν) − (x, y)||| < η +sε≤ 2η.

Thus, (p, x, y) ∈ gph G with σ(( p, x, 0), (p, x, y)) ≤ 2η, and then (3.43) implies thatthere exists u ∈ X such that

(3.51) y′ − y ∈ DxG(p, x | y)(u) and ‖u‖ ≤ c‖y′ − y‖.

By the definition of the partial graphical derivative, there exist sequences tn ↓ 0,un → u, and vn → y′ − y such that

y + tnvn ∈ G(p, x + tnun) for all n,

meaning that, for sufficiently large n, (x + tnun, y + tnvn) ∈ Ep. Now, if we plug thepoint (x + tnun, y + tnvn) into (3.49), we obtain

‖y′ − y‖ ≤ ‖y′ − (y + tnvn)‖ + ε|||(x + tnun, y + tnvn) − (x, y)|||

resulting in

‖y′ − y‖ ≤ (1 − tn)‖y′ − y‖ + tn‖vn − (y′ − y)‖ + εtn|||(un, vn)|||.

After obvious simplifications, this gives

‖y′ − y‖ ≤ ε|||(un, vn)||| + ‖vn − (y′ − y)‖.

Passing to the limit with n→ ∞ we obtain

‖y′ − y‖ ≤ ε|||(u, y′ − y)|||

and hence, taking into account the second relation in (3.51) we conclude that

‖y′ − y‖ ≤ εc‖y′ − y‖.

Since by (3.44) εc < 1, we finally obtain that y′ = y. Then (3.50) yields (3.45) andthe proof of the lemma is complete. �

We continue with the proof of Theorem 3.2.3.


Fix s ∈ (0, εη/2]. Since the function p → d(0,G(p, x)) is upper semicontinuousat p, there exists δ > 0 such that d(0,G(p, x)) ≤ s/2 for all p with ρ(p, p) < δ. Ofcourse, we can take smaller δ, e.g., δ ≤ s/ε. For such p we can find y such thaty ∈ G(p, x) with ‖y‖ ≤ d(0,G(p, x)) + s/3 < s. Then we apply Lemma 3.2.4 with s,y′ = 0 and (p, ω, ν) = (p, x, y) inasmuch as

σ((p, x, y), ( p, x, 0)) = max{ρ(p, p), c‖y‖} ≤ max{δ, cs} ≤ max{ sε, cs

}=

sε≤εη

ε= η,

obtaining the existence of x such that (p, x, 0) ∈ gph G; that is, x ∈ S (p). Also, fromthe estimate (3.45) with ω = x we have that x ∈ x + s

εB◦.

Set V := B( p, δ), U := B(x, s

ε

)and pick p ∈ V and x ∈ U. We consider two cases.

Case 1. d(0,G(p, x)) ≥ 2s.We just proved that there exists x ∈ S (p) with x ∈ B

(x, s

ε

); then

(3.52)d(x, S (p)) ≤ d(x, S (p)) + ‖x − x‖ ≤ ‖x − x‖ + ‖x − x‖

≤sε

+sε

=2sε≤

1ε

d(0,G(p, x)).

Case 2. d(0,G(p, x)) < 2s.In this case, for a sufficiently small γ > 0 we can find yγ ∈ G(p, x) such that

‖yγ‖ ≤ d(0,G(p, x)) + γ < 2s.

Thenc‖yγ‖ < 2cs ≤ 2c

εη

2< η

and, hence, (p, x, yγ) ∈ gph G is such that σ((p, x, yγ), ( p, x, 0)) ≤ η.Applying Lemma 3.2.4 for (p, ω, ν) = (p, x, yγ), y′ = 0 and 2s in place of s, we

find xγ ∈ S (p) such that

‖x − xγ‖ ≤1ε‖yγ‖.

Then, by the choice of yγ,

d(x, S (p)) ≤ ‖x − xγ‖ ≤1ε‖yγ‖ ≤

1ε

(d(0,G(p, x)) + γ),

thus

d(x, S (p)) ≤1ε

(d(0,G(p, x)) + γ).

The left-hand side of this inequality does not depend on γ, hence letting γ ↓ 0 leadsto

(3.53) d(x, S (p)) ≤1ε

d(0,G(p, x)).


We obtained this inequality also in the Case 1 in (3.52), hence it holds for any pin V and x ∈ U. Since 1/ε can be arbitrarily close to c, this gives us (3.42) whichcompletes the proof of the theorem. �

The relation (3.42), obtained in Theorem 3.2.3 can be considered as metric reg-ularity of G with respect to x at (p, x) for 0. Parallel to the partial metric regularityof G in x, we can define the partial Aubin property for G in p in the following way:G : P × X ⇒ Y is said to have the Aubin property with respect to p uniformly in x at( p, x) for 0 if 0 ∈ G(p, x) and there exist a constant κ > 0 and neighborhoods O of 0,Q for p and U of x such that

e(G(p, x) ∩ O,G(p′, x)) ≤ κρ(p, p′) for all p, p′ ∈ Q and x ∈ U.

By combining this definition with (3.42) one obtains (see also Ledyaev and Zhu [110,Corollary 3.9])

Proposition 3.2.5. Let G : P × X ⇒ Y be both metrically regular with respect to xand have the Aubin property with respect to p uniformly in x at ( p, x) for 0. Thenthe solution mapping S has the Aubin property at p for x.

Proof. Take p, p′ near p and x ∈ S (p) near x. Then we have

d(x, S (p′)) ≤ κ′d(0,G(p′, x)) ≤ κ′κρ(p, p′),

where κ′ and κ are the constants of the assumed metric regularity and Aubin property,respectively. Since x is arbitrarily chosen in S (p) near x, we are done. �

3.2.2 Proof of Aubin criterion

This subsection contains the proof of Theorem 3.2.1.For short, denote

d−DF(x | y) := lim sup(x,y)→(x,y)(x,y)∈gph F

‖DF(x |y)−1‖−.

Step 1. Proof of the inequality reg F(x | y) ≤ d−DF(x | y).If d−DF(x | y) = +∞ there is nothing to prove. Let d−DF(x | y) < ∞. Applying Theo-

rem 3.2.3 with P = Y and G(p, x) = F(x) − p, for y in the place of p and y = p,we have that S (y) = F−1(y) and d(0,G(y, x)) = d(y, F(x)). Then for any c > d−DF(x | y)from (3.42) we obtain that F is metrically regular at x for y with a constant c. Thus,reg F(x | y) ≤ c and therefore reg F(x | y) ≤ d−DF(x|y) which gives us (3.34).

Step 2. Proof of reg F(x | y) = d−DF(x | y) when X is finite dimensional.


If reg F(x | y) = +∞ we are done. Let reg F(x | y) < κ < ∞. Then there are neigh-borhoods U of x and V of y such that

(3.54) d(x, F−1(y)) ≤ κd(y, F(x)) whenever x ∈ U, y ∈ V.

It is obvious that when F satisfies (3.54) one can choose V so small that F−1(y)∩U,∅for all y ∈ V . Pick any y ∈ V and x ∈ F−1(y) ∩ U, and let v ∈ B. Take a sequencetn ↓ 0 such that yn := y + tnv ∈ V for all n. By (3.54) there exists xn ∈ F−1(y + tnv)such that

‖x − xn‖ = d(x, F−1(yn)) ≤ κd(yn, F(x)) ≤ κ‖yn − y‖ = κtn‖v‖.

For un := (xn − x)/tn we obtain

(3.55) ‖un‖ ≤ κ‖v‖;

thus the sequence un is bounded and hence un → u for a subsequence. Since (xn, y +

tnv) ∈ gph F, by the definition of the tangent cone, we obtain (u, v) ∈ Tgph F(x, y) andhence, by the definition of the graphical derivative, we have u ∈ DF(x |y)−1(v). From(3.55) it follows

‖DF(x |y)−1‖− ≤ κ.

Since (x, y) ∈ gph F is arbitrarily chosen near (x, y), we conclude that d−DF(x | y) ≤ κ.Finally, since κ can be arbitrarily close to reg F(x | y) we obtain d−DF(x | y) ≤ reg F(x | y).This, combined with (3.34), gives us (3.36) and Step 2 of the proof is complete.

Step 3. Proof oflim sup

(x,y)→(x,y)(x,y)∈gph F

‖D??F(x |y)−1‖− = d−DF(x | y)

when both X and Y are finite dimensional.Since gph D??F(x |y) = co gph DF(x |y), we have D??F(x |y)−1(v) ⊃ DF(x |y)−1(v)

for any v, which implies

infu∈D??F(x |y)−1(v)

‖u‖ ≤ infu∈DF(x |y)−1(v)

‖u‖,

consequently‖D??F(x |y)−1‖− ≤ ‖DF(x |y)−1‖−,

and thenlim sup

(x,y)→(x,y)(x,y)∈gph F

‖D??F(x |y)−1‖− ≤ d−DF(x | y).

Therefore, we only need to prove the opposite inequality

(3.56) d−DF(x | y) ≤ lim sup(x,y)→(x,y)(x,y)∈gph F

‖D??F(x |y)−1‖−


If the right hand side in (3.56) is finite, pick λ such that

lim sup(x,y)→(x,y)(x,y)∈gph F

‖D??F(x |y)−1‖− < λ < +∞.

Let X × Y be equipped with the Euclidian norm, and let r > 0 be small enough toensure that

(3.57) maxv∈B

minu∈D??F(x |y)−1(v)

‖u‖ ≤ λ for all (x, y) ∈ gph F ∩ B[(x, y), r],

and that gph F ∩ B[(x, y), r] is a closed set. We will prove that

(3.58) maxv∈B

minu∈DF(x |y)−1(v)

‖u‖ ≤ λ for all (x, y) ∈ gph F ∩ B((x, y), r).

Fix v ∈ B. For any sets A,C denote by d(A,C) := inf{‖a − c‖ | a ∈ A, c ∈ C}. Let usfix (x, y) ∈ gph F ∩ ((x, y) + rB◦). Let (u∗, v∗) ∈ gph DF(x |y) and w ∈ λB be such that

‖(w, v) − (u∗, v∗)‖ = d(λB × {v}, gph DF(x |y)).

Observe that the point (u∗, v∗) is the unique projection of any point in the opensegment ((u∗, v∗), (w, v)) on gph DF(x |y). We will prove that (u∗, v∗) = (w, v) and thiswill be enough to have (3.58) and hence (3.56).

By the definition of the graphical derivative, there exist sequences tn ↓ 0, un → u∗,and vn → v∗ such that y + tnvn ∈ F(x + tnun) for all n. Let (xn, yn) be a point in cl gph Fwhich is closest to (x, y) + tn

2 (u∗ + w, v∗ + v) (a projection, not necessarily unique, ofthe latter point on the closure of gph F). Since (x, y) ∈ gph F we have∥∥∥∥∥(x, y) +

tn

2(u∗ + w, v∗ + v) − (xn, yn)

∥∥∥∥∥ ≤ tn

2‖(u∗ + w, v∗ + v)‖ ,

and hence

‖(x, y) − (xn, yn)‖ ≤∥∥∥∥∥(x, y) +

tn

2(u∗ + w, v∗ + v) − (xn, yn)

∥∥∥∥∥+

tn

2‖(u∗ + w, v∗ + v)‖ ≤ tn ‖(u∗ + w, v∗ + v)‖

Thus, for n sufficiently large, we have (xn, yn) ∈ ((x, y) + rB◦) and hence (xn, yn) ∈gph F ∩ ((x, y) + rB◦). Setting (un, vn) = (xn−x, yn−y)/tn, we deduce by the usualproperty of a projection that

12

(u∗ + w, v∗ + v) − (un, vn) ∈ [Tgph F(xn, yn)]0 = [gph D??F(xn |yn)]0,

where K0 stands for the negative polar cone of a set K. Then, by (3.57), there existswn ∈ λB such that v ∈ D??F(xn |yn)(wn) and from the above relation

(3.59)

⟨u∗ + w

2− un,wn

⟩+

⟨v∗ + v

2− vn, v

⟩≤ 0.


We claim that (un, vn) converges to (u∗, v∗) as n→ ∞. Indeed,∥∥∥∥∥(u∗ + w2

,v∗ + v

2

)− (un, vn)

∥∥∥∥∥ =1tn

∥∥∥∥∥(x, y) + tn

(u∗ + w2

,v∗ + v

2

)− (xn, yn)

∥∥∥∥∥≤

1tn

∥∥∥∥∥(x, y) + tn

(u∗ + w2

,v∗ + v

2

)− (x, y) − tn(un, vn)

∥∥∥∥∥=

∥∥∥∥∥(u∗ + w2

,v∗ + v

2

)− (un, vn)

∥∥∥∥∥.Therefore, (un, vn) is a bounded sequence and then, since yn = y + tnvn ∈ F(xn) =

F(x + tnun), every cluster point (u, v) of it belongs to gph DF(x |y). Moreover, (u, v)satisfies ∥∥∥∥∥(u∗ + w

2,

v∗ + v2

)− (u, v)

∥∥∥∥∥ ≤ ∥∥∥∥∥(u∗ + w2

,v∗ + v

2

)− (u∗, v∗)

∥∥∥∥∥.The above inequality together with the fact that (u∗, v∗) is the unique closest point to12 (u∗ + w, v∗ + v) in gph DF(x |y) implies that (u, v) = (u∗, v∗). Our claim is proved.

Up to a subsequence, wn satisfying (3.59) converges to some w ∈ λB. Passing tothe limit in (3.59) one obtains

(3.60) 〈w − u∗, w〉 + 〈v − v∗, v〉 ≤ 0.

Since (w, v) is the unique closest point of (u∗, v∗) to the closed convex set λB × {v},we have

(3.61) 〈w − u∗,w − w〉 ≤ 0.

Finally, since (u∗, v∗) is the unique closest point to 12 (u∗ + w, v∗ + v) in gph DF(x |y)

which is a closed cone, we get

(3.62) 〈w − u∗, u∗〉 + 〈v − v∗, v∗〉 = 0.

n view of (3.60), (3.61) and (3.62), we obtain

‖(w, v) − (u∗, v∗)‖2 = 〈w − u∗,w − w〉+ (〈w − u∗, w〉 + 〈v − v∗, v〉) − (〈w − u∗, u∗〉 + 〈v − v∗, v∗〉) ≤ 0.

Hence w = u∗ and v = v∗ and the proof is complete. �

3.2.3 Applications of the Aubin criterion

As a first specific application of the Aubin criterion we consider the constraintsystem

(3.63) Find x ∈ IRn such that fi(x){

= 0 for i = 1, . . . , r,≤ 0 for i = r + 1, . . . ,m,


where fi : Rn → R, i = 1, . . . ,m. This system can also be written as the inclusion0 ∈ F(x) with F : Rn ⇒ Rm given by

(3.64) F(x) = f (x) + K,

where f = ( f1, . . . , fm) and K = {0}r × IRm−r+ . Let x be a solution of (3.63) and f be

strictly differentiable at x. We denote the index set of active inequality constraints atx as

J = {i ∈ {r + 1, . . . ,m} | fi(x) = 0}.

We will now show that Aubin criterion directly leads to the following well-knownresult:

Theorem 3.2.6. The mapping F in (3.64) is metrically regular at x for 0 if and onlyif the Mangasarian-Fromovitz condition holds: the vectors ∇ fi(x), i = 1, . . . , r arelinearly independent and also there exists w ∈ Rn such that

(3.65)

{∇ fi(x)w = 0 for i = 1, . . . , r,∇ fi(x)w < 0 for i ∈ J.

Proof. By Lyusternik-Graves theorem with F = K and g = f , the metric regularityof the mapping F at x for 0 is equivalent to the metric regularity at x for 0 of its“partial linearization”

F0(x) = f (x) + A(x − x) + K where A = ∇ f (x).

Also, by the specific form of K,

v ∈ DK(x |y)(u) ⇐⇒{

vi = 0 for i ∈ I(y),vi ≥ 0 for i ∈ J(y),

where

(3.66) I(y) = {i ∈ {1, . . . , r} | yi = 0} and J(y) = {i ∈ {r + 1, . . . ,m} | yi = 0}.

Then, of course, J = J( f (x)). Since fi(x) < 0 for i ∈ {r + 1, . . . ,m} \ J, we have thatyi− fi(x) > 0 for all such i and for (x, y) close to (x, 0). This means that for such (x, y)the set J(y) in (3.66) is always a subset of J. Then the Aubin criterion for metricregularity of F0 becomes the following condition: for every I ⊂ {1, . . . , r} and forevery J ⊂ J we have:

(3.67) ∀v ∈ RI∪J ∃ u ∈ Rn such that (v−Au)i=0 for i∈I and (v−Au)i≥0 for i∈J.

Assume that Mangasarian-Fromovitz condition holds and let I ⊂ {1, . . . , r} andJ ⊂ J. If either I = ∅ or J = ∅ we skip the corresponding step of the proof. Let


I , ∅. Then the matrix H = [∇ fi(x)]i∈I is onto and hence, by the metric regularity ofH there exists a constant κ such that

(3.68) ∀v ∈ RI ∃ u ∈ Rn such that v − Hu = 0 and ‖u‖ ≤ κ‖v‖.

This means in particular that taking v with a norm small enough we can have thecorresponding u in (3.68) with arbitrarily small norm. Then, since ∇ fi(x)w < −α forall i ∈ J and some α > 0, we end up having that for any v ∈ RI∪J with sufficientlysmall norm

(3.69) vi − ∇ fi(x)(u + w){

= 0 for i ∈ I,≥ 0 for i ∈ J.

By the positive homogeneity, from (3.68) and (3.69) we obtain (3.67).Conversely, if (3.67) holds, then taking I = {1, . . . , r} and J = ∅ we conclude

that ∇ fi(x), i = 1, . . . , r must be linearly independent. Next, taking I = {1, . . . , r} andJ = J, for

vi =

{0 for i = 1, . . . , r,−ε for i ∈ J

with some ε > 0 we obtain (3.65). �

Our second application is for a mapping describing the variational inequality

(3.70) 〈 f (x), u − x〉 ≥ 0 for all u ∈ C,

where f : Rn → Rn and C a nonempty convex closed set in Rn that is polyhedral. Interms of the normal cone mapping

NC(x) =

{{y | 〈y, u − x〉 ≤ 0 for all u ∈ C} for x ∈ C,∅ otherwise,

we can write the variational inequality (3.70) as the inclusion 0 ∈ F(x) where

(3.71) F(x) = f (x) + NC(x).

We assume that x is a solution of (3.70) and f is strictly differentiable at x. Then,again, the Lyusternik-Graves theorem, this time with F = NC and g = f , allows us torestrict our attention to the linearized mapping

F0(x) = f (x) + A(x − x) + NC(x) where A = ∇ f (x).

Let [v] be the subspace of dimension one (or zero for v = 0) spanned on a vectorv ∈ Rn, that is, [v] = {τv | τ ∈ R}, and let [v]⊥ be its orthogonal complement. The


form of the graphical derivative of F0 will be obtained by introducing the criticalcone K(x, v) to the set C at x ∈ C for v ∈ NC(x),

K(x, v) = TC(x) ∩ [v]⊥,

via the following

Reduction lemma in Dontchev and Rockafellar [64]. Let C be a convex polyhedralset in Rn. For any (x, v) ∈ gph NC there is a neighborhood O of the origin in Rn × Rn

such that for (x′, v′) ∈ O one has

v + v′ ∈ NC(x + x′) ⇐⇒ v′ ∈ NK(x,v)(x′).

Consequently,

(x′, v′) ∈ Tgph NC (x, v) ⇐⇒ v′ ∈ NK(x,v)(x′),

and hence, for (x, y) ∈ gph F0 and v = y − f (x) − A(x − x), where A = ∇ f (x), we have

(3.72) DF0(x |y)(u) = Au + NK(x,v)(u).

For any cone K, a set of the form

F = K ∩ [v]⊥ for some v ∈ K0,

where K0 is the polar to K, is said to be a face of K. The largest of the faces is Kitself while the smallest is the set K ∩ (−K) which is the largest subspace containedin K. Every polyhedral cone has finitely many faces.

It was proved in Dontchev and Rockafellar [64, Theorem 1] that the metricregularity of a mapping F of the form (3.71) with a polyhedral set C implies a sharperproperty called strong regularity. A mapping F : X ⇒ Y is said to be strongly regularat x for y is it is metrically regular there and, in addition, the graphical localizationof its inverse F−1 near (y, x) is single-valued. In other words, F is strongly regularat x for y when there are neighborhoods U of x and V of y such that the mappingV 3 y 7→ F−1(y) ∩ U is a Lipschitz continuous function.

We are now ready to apply the Aubin criterion to obtain a new necessary and suf-ficient condition for strong regularity of variational inequalities over polyhedral sets,which complements the criterion given in Dontchev and Rockafellar [64, Theorem 2]:

Theorem 3.2.7. The variational inequality mapping (3.71) is strongly regular at x fory if and only if for all choices of faces F1 and F2 of the critical cone K to the set Cat x for v = y − f (x), with F1 ⊃ F2, the following condition holds:

∀ v ∈ IRn ∃ u ∈ F1 − F2 such that (v − Au) ∈ (F1 − F2)0 and v − Au ⊥ u.


Proof. According to Aubin criterion given in Theorem 2.74, the mapping F0 (and,hence F) is metrically regular, and hence strongly regular, if and only if the limsupof the inner norms of the graphical derivatives is finite. The form of the graphicalderivative of F0 is given in (3.72).

On the other hand, next result gives the form of the of critical cones in a neigh-borhood of a fixed reference point. It is extracted from the proof of Dontchev andRockafellar [64, Theorem 2]: Let C be a convex polyhedral set, let v ∈ NC(x) and letK be the critical cone to C at x for v. Then there exists an open neighborhood O of(x, v) such that for every choice of (x, v) ∈ gph NC ∩O the corresponding critical coneK(x, v) has the form

K(x, v) = F1 − F2,

for some faces F1, F2 of K with F1 ⊃ F2. And conversely, for every two faces F1, F2

of K with F1 ⊃ F2 and every neighborhood O of (x, v) there exists (x, v) ∈ O suchthat K(x, v) = F1 − F2.

So, there are finitely many critical cones near the reference point (x, v) that are tobe taken into account, and these cones are given by faces of K in a way described inthis result. Hence, for any choice of faces F1 and F2 of K with F1 ⊃ F2 it is enoughto ensure that ‖A + NF1−F2‖

− is finite. For any cone K

v ∈ NK(x) ⇐⇒ x ∈ K, v ∈ K0, x ⊥ v.

It remains to observe that the inner norm of the mapping A + NF1−F2 will be finite ifand only if the condition claimed in the theorem holds. �

Our last application of Aubin criterion is a new proof of the radius theorem firstproved by Dontchev, Lewis and Rockafellar in [62].

Theorem 3.2.8. Let X and Y be finite-dimensional linear normed spaces and letF : X ⇒ Y has closed graph locally near (x, y) ∈ gph F. Then

infG∈L(X,Y)

{‖G‖ | F + G is not metrically regular at x for y + G(x)} =1

reg F(x|y).

Moreover, the infimum is unchanged if taken with respect to linear mappings G ofrank 1, but also remains unchanged when the perturbations G are locally Lipschitzcontinuous functions with ‖G‖ replaced by the Lipschitz modulus lip G(x) of G at x.

Proof. The general perturbation inequality derived in Dontchev, Lewis and Rockafel-lar [62, Corollary 3.4] yields (also in infinite dimensions) the estimate

(3.73) infG:X→Y

{lip G(x) | F+G is not metrically regular at x for y+G(x)

}≥

1reg F(x | y)

.

It remains to show the opposite inequality. The limit cases are easy to handle, since ifreg F(x | y) = ∞ we have nothing to prove, and if reg F(x | y) = 0, then by the general


perturbation inequality (3.73), which also holds in this case, we obtain the claimedequality.

Let now 0 < reg F(x | y) < ∞. By Theorem 3.2.1 we have reg F(x | y) = d−DF(x | y) =

d−D??F(x | y) where d−DF(x | y) is defined in the beginning of Subsection 3.2.2 whiled−D??F(x | y) is defined in the same way with DF replaced by D??F.

Take a sequence of positive reals εk → 0. Then for any k there exists (xk, yk) ∈gph F with (xk, yk)→ (x, y) and

d−D??F(x | y) + εk ≥ ‖D??F(xk|yk)−1‖− ≥ d−D??F(x | y) − εk > 0.

For short, set Hk := D??F(xk|yk); then Hk is a sublinear mapping with closed graph.For S k := H∗+k the norm duality gives us ‖H−1

k ‖− = ‖S −1

k ‖+.

For each k choose a positive real rk which satisfies ‖S −1k ‖

+ − εk < 1/rk < ‖S −1k ‖

+.From the last inequality there must exist (yk, xk) ∈ gph S k with ‖xk‖ = 1 and ‖S −1

k ‖+ ≥

‖yk‖ > 1/rk. Pick y∗k ∈ Y with 〈yk, y∗k〉 = ‖yk‖ and ‖y∗k‖ = 1 and define the rank-onemapping Gk ∈ L(Y, X) as

Gk(y) := −〈y, y∗k〉‖yk‖

xk.

Then Gk(yk) = −xk and hence (S k + Gk)(yk) = S k(yk) + Gk(yk) = S k(yk)− xk 3 0. There-fore, yk ∈ (S k + Gk)−1(0) and since yk , 0, by Dontchev, Lewis and Rockafellar [62,Proposition 2.5],

(3.74) ‖(S k + Gk)−1‖+ = ∞.

Note that ‖Gk‖ = ‖xk‖/‖yk‖ = 1/‖yk‖ < rk.Since the sequences yk, xk and y∗k are bounded, we can extract from them subse-

quences converging respectively to y, x and y∗; the limits then satisfy ‖y‖ = d−D??F(x | y),‖x‖ = 1 and ‖y∗‖ = 1. Define the rank-one mapping G ∈ L(Y, X) as

G(y) := −〈y, y∗〉‖y‖

x.

Then we have ‖G‖ ≤ 1/d−D??F(x | y) and ‖Gk − G‖ → 0.Denote G := (G)∗ and suppose that F + G is metrically regular at x for y +

G(x). Then Theorem 3.2.1 yields that for some finite positive constant c and for ksufficiently large we have

c > ‖D??(F + G)(xk|yk + G(xk))−1‖− = ‖(D??F(xk|yk) + G)−1‖−,

which, by norm duality and the equality G∗ = ((G)∗)∗ = G, is equivalent to

(3.75) c > ‖([D??F(xk|yk) + G]∗+)−1‖+ = ‖[D??F(xk|yk)∗+ + G∗]−1‖+ = ‖(S k + G)−1‖+.

We apply the following lemma, which is a reformulation of a result by Robin-son [141], see also Dontchev and Lewis [61]:


Lemma 3.2.9. For a sublinear mapping H : X → Y with closed graph and forB ∈ L(X,Y), if [‖H−1‖+]−1 ≥ ‖B‖, then

‖(H + B)−1‖+ ≤ [[‖H−1‖+]−1 − ‖B‖]−1.

Now we are ready to complete the proof of Theorem 3.2.8. Take k sufficientlylarge such that ‖G−Gk‖ ≤ 1/(2c) and (xk, yk) that satisfies (3.75). Setting Pk := S k +Gand Bk := Gk − G we have that [‖P−1

k ‖+]−1 ≥ 1/c > 1/(2c) ≥ ‖Bk‖. By Lemma 3.2.9

we obtain

‖(S k + Gk)−1‖+ = ‖(Pk + Bk)−1‖+ ≤ [[‖P−1k ‖

+]−1 − ‖Bk‖]−1 ≤ 2c < ∞,

which contradicts (3.74). Hence, F + G is not metrically regular at x for y + G(x).Remembering that ‖G‖ = ‖G‖ ≤ 1/ reg F(x | y) we complete the proof. �

3.3 Long orbit or empty value principle, fixed pointand surjectivity theorems

It is well known that there exist close relations between iteration schemes re-lated to dissipative mechanical systems, Ekeland perturbed minimization principle,fixed point theorems and inverse and implicit function theorems of all kinds, see forexample Aubin and Ekeland [8].

In the present section we explore some of these relations in a new light. General-izing from Ivanov [92] (see also e.g. Fabian and Priess [72], Penot [132, p.62]), weobtain a flexible Long Orbit or Empty Value (LOEV) principle, see Theorem 3.3.2and Corollary 3.3.3.

The reader may notice certain similarity between the conditions imposed on themap S in LOEV principle and that in Caristi-Kirk Fixed Point Theorem. Indeed, thelatter readily follows from the former, see Theorem 3.3.4.

The rest of this section is devoted to surjectivity results. These are derived froma novel Theorem 3.3.8 which may be regarded as a kind of interpolation betweenclassical Graves Theorem and a recent result of Ekeland [69]. However, we do notrequire differentiability in any usual sense, exploring instead a generalization of socalled contingent derivative, see e.g. Aubin and Frankowska [10].

Theorem 3.3.8 can also be considered as generalization of Aubin derivative crite-rion for metric regularity of a single-valued map.

3.3. Long orbit or empty value principle, fixed point and surjectivity theorems 199

3.3.1 Long Orbit or Empty Value (LOEV) principle

Everywhere in this subsection (M, ρ) denotes complete metric space.

Definition 3.3.1. Let S : M ⇒ M be a multivalued map. We say that S satisfies thecondition (∗) if x < S (x), ∀x ∈ M, and whenever y ∈ S (x) and limn xn = x, there areinfinitely many xn’s such that

y ∈ S (xn).

The following results, which constitute LOEV principle, suggest why maps satis-fying (∗) could be useful.

Theorem 3.3.2 (LOEV principle). Let S : M ⇒ M satisfy (∗) and let x0 ∈ M bearbitrary.

Then at least one of (a) and (b) below is true:(a) There are xi ∈ M, i = 1, 2, . . ., such that

xi+1 ∈ S (xi), i = 0, 1, . . . ,

and∞∑

i=0

ρ(xi, xi+1) = ∞;

(b) There is x ∈ M such thatS (x) = ∅.

Proof. Assume (a) was not true, that is, each S -orbit, starting at x0, is of finite length.We can construct finite or infinite orbit (xi)i≥0 ⊂ M by the following procedure.If x0, x1, . . . , xi are already chosen, theneither S (xi) = ∅ and we are done;or si := min

{1, sup {ρ(xi, y) : y ∈ S (xi)}

}> 0. Take xi+1 ∈ S (xi) such that

(3.76) ρ(xi, xi+1) >si

2.

If we end up with infinite orbit then, since (a) was assumed false,

∞∑i=0

ρ(xi, xi+1) < ∞ ⇒ limn→∞

xn =: x.

In particular, limi ρ(xi, xi+1) = 0 and from (3.76) it follows that si → 0.Assume that S (x) , ∅.Take y ∈ S (x). By (∗) we have ρ(y, x) > 0 and y ∈ S (xi) for infinitely many i’s.

By the definition of si we have that si ≥ ρ(y, xi) for infinitely many i’s. Passing tolimit over the latter subsequence we get 0 ≥ ρ(y, x) > 0. Contradiction. �


Corollary 3.3.3. Let S : M ⇒ M satisfy (∗) and let x0 ∈ M and K > 0 be arbitrary.Then at least one of the two conditions below is true:(a) There are xi ∈ M, i = 1, 2, . . . , n + 1, such that

xi+1 ∈ S (xi), i = 0, 1, . . . , n,

andn∑

i=0

ρ(xi, xi+1) > K;

(b) There is x ∈ M such that ρ(x0, x) ≤ K and

S (x) = ∅.

Proof. Assume that (b) is not true, that is S (x) , ∅ for all x ∈ M such that ρ(x0, x)≤K.If (a) from Theorem 3.3.2 is true, we have nothing to prove.If not, from the proof of Theorem 3.3.2 we get a x ∈ M such that S (x) = ∅

and x is either the end point of finite orbit or the limit of infinite orbit (xi)i≥0. Sinceρ(x0, x) > K, by triangle inequality we have that∑

i≥0

ρ(xi, xi+1) > K. �

3.3.2 Caristi-Kirk fixed point theorem

As first application of LOEV principle we prove the following Theorem due toCaristi and Kirk (see Kirk [100] and Caristi [42]).

Theorem 3.3.4 (Caristi-Kirk Fixed Point Theorem). Let (M, ρ) be complete metricspace. Let T : M ⇒ M satisfy T (x) , ∅ for all x ∈ M. Let, moreover, the functionf : M → [0,∞) be lower semicontinuous. Assume that for any x ∈ M there existy ∈ T (x) such that

(3.77) ρ(x, y) ≤ f (x) − f (y).

Then T has a fixed point, that is, there exists x such that x ∈ T (x).

Proof. Assume that x < T (x) for all x ∈ M.Define for x ∈ M

S (x) := {y ∈ M : f (y) < f (x) − 2−1ρ(x, y)}.

We are given that for each x ∈ M there is a y ∈ T (x) satisfying (3.77). By assumptionρ(x, y) > 0 and then (3.77) ensures that S (x) , ∅.


The lower semicontinuity of f implies that S satisfies (∗).From Theorem 3.3.2 we get an orbit (xn)∞0 , xi+1 ∈ S (xi), with

∞∑i=0

ρ(xi, xi+1) = ∞.

But the definition of S implies f (xi+1) − f (xi) < −ρ(xi+1, xi)/2 and summing thelatter we get f (xi)→ −∞. Contradiction with f ≥ 0. �

3.3.3 Surjectivity theorems

Our second application of LOEV principle are the following two Lemmas, whichcan be regarded as a kind of quantitative version of Theorem 3.3.4. Later on we willapply these results for obtaining several surjectivity results for functions.

Our notation is standard. (X, ‖ · ‖) and (Y, ‖ · ‖) are Banach spaces with dual spacesX∗ and Y∗, respectively. BX is the closed unit ball of the Banach space X, respectivelyB◦X is the open unit ball. The set U is a neighbourhood of the point x0 ∈ X ifx0 + εBX ⊂ U for some ε > 0, that is, x0 is interior point to U, and so on.

Lemma 3.3.5. Let α, β, γ > 0. Let X and Y be Banach spaces. Let f : X → Y be acontinuous on βBX function such that f (0) = 0.

Suppose that for all y ∈ Y such that ‖y‖ < α and all x ∈ X such that ‖x‖ < β andf (x) , y, there is z ∈ X with ‖z‖ < β and

(3.78) ‖ f (z) − y‖ < ‖ f (x) − y‖ − γ‖z − x‖.

Then for every y ∈ Y satisfying ‖y‖ < min{α, βγ} there exists x ∈ X with f (x) = y and

‖x‖ ≤‖y‖γ.

Proof. Obviously, if y = 0 we can take x = 0.Fix y ∈ Y such that y , 0 and ‖y‖ < α.For x ∈ βBX define

S (x) := {z ∈ βBX : ‖ f (z) − y‖ < ‖ f (x) − y‖ − γ‖z − x‖}

and note that S satisfies (∗) on βBX due to continuity of f .It is given that

(3.79) S (x) , ∅, ∀x ∈ X such that ‖x‖ < β and f (x) , y.


Now we apply Corollary 3.3.3 for the space M = βBX, the map S , the initial point

x0 = 0 and the constant K =‖y‖γ

.

So, if we assume that there is finite orbit (xi)n+1i=0 such that xi+1 ∈ S (xi) for i =

0, . . . , n, andn∑

i=0

‖xi+1 − xi‖ >‖y‖γ⇐⇒ γ

n∑i=0

‖xi+1 − xi‖ > ‖y‖,

then by the definition of S and f (0) = 0

‖ f (xn+1) − y‖ − ‖y‖ < −γn∑

i=0

‖xi+1 − xi‖ < −‖y‖ ⇒ ‖ f (xn+1) − y‖ < 0,

contradiction. Thus from Corollary 3.3.3 it follows that there is x ∈ KBX such thatS (x) = ∅. Since K < β, (3.79) implies that f (x) = y. �

Let us recall that the function f : X → Y is metrically regular at x ∈ X if thereexists κ > 0 together with neighbourhoods U 3 x and V 3 f (x) such that

d(x, f −1(y)) ≤ κd(y, f (x)), ∀x ∈ U, ∀y ∈ V,

see Dontchev and Rockafellar [67, p.253 (2)]. The infimum of κ over all such combi-nations of κ, U and V is called the regularity modulus for f at x for f (x) and denotedby reg( f ; x). The absence of metric regularity is signaled by reg( f ; x) = ∞.

Recall also that metric regularity is equivalent to openness at linear rate, seeDontchev and Rockafellar [67, p.254 (5)].

Lemma 3.3.6. Let the assumptions of Lemma 3.3.5 be satisfied. Then there exista neighbourhood U 3 0 in X and a neighbourhood V 3 0 in Y such that for everyx′ ∈ U and every y ∈ V there exists x ∈ X with f (x) = y and

‖x′ − x‖ ≤‖ f (x′) − y‖

γ.

That is, f is metrically regular at 0 with reg( f ; 0) ≤ γ−1.

Proof. From continuity of f at 0 there exist δ ≤ β/4 and η < α such that for allx′ ∈ δBX and all y ∈ ηBY it holds ‖ f (x′) − y‖ ≤ βγ/4. Set U = δBX, V = ηBY .

Fix x′ ∈ U and y ∈ V . If f (x′) = y, take x = x′. Let now f (x′) , y.For x ∈ βBX define

S (x) := {z ∈ βBX : ‖ f (z) − y‖ < ‖ f (x) − y‖ − γ‖z − x‖}

and note that S satisfies (∗) on βBX due to continuity of f .


It is given that

(3.80) S (x) , ∅, ∀x ∈ X such that ‖x‖ < β and f (x) , y.

Now we apply Corollary 3.3.3 for the space M = βBX, the map S , the initial point

x0 = x′ and the constant K =‖ f (x′) − y‖

γ.

So, if we assume that there is finite orbit (xi)n+1i=0 such that xi+1 ∈ S (xi) for i =

0, . . . , n, and

n∑i=0

‖xi+1 − xi‖ >‖ f (x′) − y‖

γ⇐⇒ γ

n∑i=0

‖xi+1 − xi‖ > ‖ f (x′) − y‖,

then by the definition of S

‖ f (xn+1) − y‖ − ‖ f (x′) − y‖ < −γn∑

i=0

‖xi+1 − xi‖ < −‖ f (x′) − y‖,

hence ‖ f (xn+1)− y‖ < 0, contradiction. Thus from Corollary 3.3.3 it follows that thereis x ∈ x′+KBX such that S (x) = ∅. Since ‖x‖ ≤ ‖x−x′‖+‖x′‖ ≤ K+β/4 ≤ β/4+β/4 < β,(3.80) implies that f (x) = y. �

Before proceeding further we have to recall several notions.The multivalued map H : X ⇒ Y is called positively homogeneous if its graph

gph H := {(x, y) ∈ X × Y : y ∈ H(x)}

is a cone. That is, y ∈ H(x) ⇐⇒ ty ∈ H(tx) for all t ≥ 0.The inner norm of such H, see Dontchev and Rockafellar [67, p.256], is

‖H‖− = sup‖x‖≤1

infy∈H(x)

‖y‖.

The inverse of H is the positively homogeneous map H−1 : Y ⇒ X such that H−1(y)is the set of solutions of y ∈ H(x).

Obviously, if ‖H‖− < ∞ then H is surjective and H−1 is everywhere defined.Moreover, if ‖H−1‖− < κ then

(3.81) ∀v ∈ Y \ {0} ∃u ∈ H−1(v) : ‖u‖ < κ‖v‖.

Below we will generalize the contingent cone for the case of single valued map,that is function. But in order to make more sense, we will first recall the notion ofcontingent derivative of a function, see Aubin and Ekeland [8].


If f : X → Y is a function then the contingent derivative

d f (x) : X ⇒ Y

of f at x is the contingent cone to gph f = {(x, y) : y = f (x)} at (x, f (x)). That is,v ∈ d f (x)(u) if and only if there are un → u, vn → v and tn > 0 with tn → 0, such that

f (x + tnun) = f (x) + tnvn.

The graph of d f (x) is closed cone. So, d f (x) is positively homogeneous map.

Definition 3.3.7. Let X and Y be Banach spaces. Let θ > 0 and µ ≥ 0. We considerthe following equivalent norm on X × Y:

‖(x, y)‖θ := θ‖x‖ + ‖y‖.

The approximate contingent derivative dµθ f (x) of f at x is defined by: v ∈ dµθ f (x)(u)if there are (un, vn)→ (u, v) and tn > 0 with tn → 0 such that

(3.82) dist θ((x + tnun, f (x) + tnvn), gph f

)≤ µtn‖un‖,

where dist θ means the distance measured in norm ‖(·, ·)‖θ.

It is clear that dµθ f ⊃ d f . Obviously, d0θ f = d f for all θ > 0.

Roughly speaking, the following general result states that if the approximatecontingent derivative is linearly open at uniform rate in a neighbourhood, then so isthe function itself. Several known results follow from this. They are presented at theend of the section.

Theorem 3.3.8. Let X and Y be Banach spaces. Let f : X → Y be continuous onx0 + εBX for some ε > 0. Suppose that θ > 0, and µ ≥ 0, m > 0 with mµ < 1 are suchthat

‖dµθ f (x)−1‖− ≤ m, ∀x ∈ x0 + εB◦X.

Then for any κ > m for c := κ−1 − µ it holds that for any y ∈ Y such that ‖y− f (x0)‖ <εmin{c, θ} there exists x ∈ X such that

‖x − x0‖ ≤‖y − f (x0)‖min{c, θ}

< ε,

and f (x) = y.

Moreover, f is metrically regular at x0 with reg( f ; x0) ≤ max{

m1 − mµ

, θ−1}.


Proof. We assume without loss of generality that x0 = 0 and f (0) = 0.Take κ > m.Let x ∈ εB◦X and y , f (x).Apply (3.81) to dµθ f (x) and v = y − f (x) , 0 to get u ∈ X such that

‖u‖ < κ‖v‖ s.t. dµθ f (x)(u) 3 v.

By Definition 3.3.7 here are (un, vn) → (u, v) and tn > 0 with tn → 0 such that (3.82)holds. Thus, there are xn ∈ X with

θ‖x + tnun − xn‖ + ‖ f (x) + tnvn − f (xn)‖ ≤ µtn‖un‖ + t2n.

If we omit the second addend at left hand side, we will get θ‖x + tnun − xn‖ ≤ O(tn)and, therefore, xn → x. So, we may assume that ‖xn‖ < ε.

We now add and subtract y within the second addend at the left hand side above,use v = y − f (x) and rearrange to get

‖ − v + tnvn + y − f (xn)‖ ≤ µtn‖un‖ − θ‖x + tnun − xn‖ + t2n.

We add and subtract tnv within left hand side and use triangle inequality to get

‖v − tnv + f (xn) − y‖ − tn‖v − vn‖ ≤ µtn‖un‖ − θ‖x + tnun − xn‖ + t2n.

Applying once again triangle inequality:

‖ f (xn) − y‖ − (1 − tn)‖v‖ ≤ tn‖v − vn‖ + µtn‖un‖ − θ‖x + tnun − xn‖ + t2n.

That is,

(3.83) ‖ f (xn) − y‖ − ‖v‖ ≤ −tn‖v‖ + µtn‖un‖ − θ‖x + tnun − xn‖ + δntn,

where δn → 0. We consider two cases.Case 1. u = 0. Then un → 0 and

‖ f (xn) − y‖ − ‖v‖ ≤ −tn‖v‖ − θ‖x − xn‖ + δ′ntn, where δ′n → 0< −θ‖x − xn‖,

for all n large enough, since ‖v‖ > 0. Finally, in this case we get

(3.84) ‖ f (xn) − y‖ − ‖ f (x) − y‖ < −θ‖x − xn‖, n > N1.

Case 2. u , 0. Since tn‖un‖ = tn‖u‖ + o(tn), (3.83) becomes

‖ f (xn) − y‖ − ‖v‖ ≤ −tn(‖v‖ − µ‖u‖) − θ‖x + tnu − xn‖ + δ′′n tn,


where δ′′n → 0. Since ‖u‖ < κ‖v‖, we have ‖v‖ − µ‖u‖ > (κ−1 − µ)‖u‖. Recall thatc = κ−1 − µ > 0 to get from the above:

‖ f (xn) − y‖ − ‖v‖ < −ctn‖u‖ − θ‖x + tnu − xn‖

for n large enough. Obviously, −ctn‖u‖−θ‖x+tnu−xn‖ ≤ −min{c, θ}(tn‖u‖+‖x+tnu−xn‖)and by triangle inequality

(3.85) ‖ f (xn) − y‖ − ‖ f (x) − y‖ < −min{c, θ}‖x − xn‖, n > N2.

From (3.84) and (3.85) we conclude that there is z ∈ X with ‖z‖ < ε such that

‖ f (z) − y‖ − ‖ f (x) − y‖ < −min{c, θ}‖x − z‖.

With α = ∞, β = ε and γ = min{c, θ} we apply Lemma 3.3.5 to get the first conclusionand Lemma 3.3.6 to get that

reg( f ; x0) ≤ (min{c, θ})−1 = max{c−1, θ−1} = max{

κ

1 − κµ,

1θ

}and since the latter holds for all κ > m, we finally obtain that

reg( f ; x0) ≤ max{

m1 − mµ

,1θ

}. �

With the help of Theorem 3.3.8 we can proof with an unified approach a lot ofsurjectivity results. First of them will be the classical Graves theorem.

Let X and Y be Banach spaces and A be bounded linear surjection from X onto Y .Recall, e.g. Dontchev and Rockafellar [67, p.254], that

reg A = sup‖y‖≤1

d(0, A−1(y)),

where A−1(y) = {x ∈ X : Ax = y}. Note that reg A < ∞ by Banach Open MappingTheorem. In the sequel we use that if κ > reg A then for any y ∈ Y , y , 0 there isx ∈ X with ‖x‖ < κ‖y‖ and Ax = y.

Theorem 3.3.9 (Graves, e.g. p. 276–278 in [67]). Let X and Y be Banach spaces.Consider a function f : X → Y continuous on x + εBX for some x ∈ X and ε > 0. LetA be bounded linear and surjective from X onto Y and let m = reg A. Suppose thatthere is µ ≥ 0 such that mµ < 1 and

(3.86) ‖ f (x) − f (x′) − A(x − x′)‖ ≤ µ‖x − x′‖, ∀x, x′ ∈ x + εBX.


Set y := f (x). For any κ > m and c := κ−1 − µ, if y is such that ‖y − y‖ < cε, then theequation y = f (x) has a solution x such that

‖x − x‖ ≤‖y − y‖

c,

in particular ‖x − x‖ < ε.Moreover, f is metrically regular at x with reg( f ; x) ≤

m1 − mµ

.

Proof. We will show that (3.86) implies that

(3.87)∥∥∥dµθ f (x)−1

∥∥∥− ≤ m, ∀x ∈ x + εB◦X, ∀θ > 0.

Fix arbitrary x ∈ x + εB◦X.Take κ > m. Let v ∈ Y \ {0} be arbitrary. Since κ > reg A, there is u ∈ X such that

Au = v (thus u , 0) and

(3.88) ‖u‖ < κ‖v‖.

For all t ∈ (0, 1) such that ‖x + tu‖ < ε from (3.86) it follows that

‖ f (x + tu) − f (x) − A(tu)‖ ≤ µ‖tu‖.

Hence for d := dist θ((x + tu, f (x) + tv), gph f

)we have

d ≤ θ‖x + tu − (x + tu)‖ + ‖ f (x) + tv − f (x + tu)‖= ‖ f (x) + tv − f (x + tu)‖ ≤ µt‖u‖.

By Definition 3.3.7 we have v ∈ dµθ f (x)u for all θ > 0.From this and (3.88) it follows that ‖dµθ f (x)−1‖− ≤ κ. Since this holds for all κ > m,

(3.87) is verified.

Finally, we apply Theorem 3.3.8 with κ and θ =1 − mµ

mto conclude. �

We can also derive Aubin derivative criterion for metric regularity, see Section 3.2,in the partial case of continuous function.

Theorem 3.3.10. Let X and Y be Banach spaces. Let f : X → Y be continuous onx + RBX for some R > 0. Suppose that m > 0 is such that

‖d f (x)−1‖− ≤ m, ∀x ∈ x + RBX.

Then for every κ > m and any y ∈ Y such that ‖y − f (x)‖ < R/κ there exists x ∈ Xsuch that ‖x − x‖ ≤ κ‖y − f (x)‖ and f (x) = y.

Moreover, f is metrically regular at x with reg( f ; x) ≤ m.


Proof. Fix κ > m.We apply Theorem 3.3.8 with κ, ε = R, µ = 0 and θ = 1/m. �

By a similar way we can obtain a slight generalization of a result proved byEkeland in [69]:

Theorem 3.3.11. Let X and Y be Banach spaces. Let f : X → Y be continuousand Gateaux-differentiable with f (0) = 0. Assume that the derivative D f (x) has aright-inverse L(x), linear and uniformly bounded in a neighbourhood of 0. That is,there are R,m > 0 such that

∀x ∈ RBX, ∀v ∈ Y ⇒ D f (x)L(x)v = v and ‖L(x)‖ ≤ m.

Then for each y such that ‖y‖ < R/m and each ν > m there is some x such that

‖x‖ < R, ‖x‖ ≤ ν‖y‖, and f (x) = y.

Moreover, f is metrically regular at 0 with reg( f ; 0) ≤ m.

Proof. Let x ∈ RBX and v ∈ Y . Since D f (x) ∈ d f (x), the contingent derivative d f (x)is surjective. If u = L(x)v, that is D f (x)u = v, then v ∈ d f (x)u. Since

‖u‖ ≤ ‖L(x)‖‖v‖ ≤ m‖v‖,

we have that

(3.89) ‖d f (x)−1‖− ≤ m.

Using (3.89), we apply Theorem 3.3.10 with R, m and ν > m to conclude. �

Let us note that the results from this section obtained for single-valued map canbe easily extended to set-valued maps using an approach of A. Ioffe.

Also, Long Orbit or Empty Value principle could be applied for obtaining Nash-Mozer-Ekeland type surjectivity result in Frechet spaces but this will be a subject offuture research.

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