Contents lists available at SciVerse ScienceDirect
Journal of Mathematical Analysis andApplications
journal homepage: www.elsevier.com/locate/jmaa
Second-order optimality conditions with the envelope-likeeffect in nonsmooth multiobjective mathematicalprogramming II: Optimality conditionsPhan Quoc Khanh a,∗, Nguyen Dinh Tuan b
a Department of Mathematics, International University of Hochiminh City, Linh Trung, Thu Duc, Hochiminh City, Viet Namb Department of Mathematics and Statistics, University of Economics of Hochiminh City, 59C Nguyen Dinh Chieu, D.3, Hochiminh City, Viet Nam
a r t i c l e i n f o
Article history:Available online 16 February 2013Submitted by Heinz Bauschke
Second-order necessary conditions and sufficient conditions, with the envelope-like effect,for optimality in nonsmooth multiobjective mathematical programming are established.We use set-valued second-order directional derivatives and impose strict differentiabilityfor necessary conditions and l-stability for sufficient conditions, avoiding continuousdifferentiability. The results improve and sharpen several recent existing ones. Examplesare provided to show advantages of our theorems over some known ones in the literature.In Part I, we consider l-stability and second-order set-valued directional derivatives ofvector functions. Part II is devoted to second-order necessary optimality conditions andsufficient ones.
This is Part II, which continues Part I appeared in [13]. In Part I, first we discussed l-stability of vector functions (Definition2.5) and provided basic properties (Propositions 2.6 and 2.7). Then, properties of second-order set-valued directionalderivatives of vector functions (defined in Definition 3.1) were presented in Propositions 3.3–3.5. These facts will be appliedin this Part II to investigate second-order optimality conditions for the followingmultiobjectivemathematical programmingproblem. Let f : Rn
→ Rm, g : Rn→ Rp and h : Rn
→ Rr be given. Let C be a closed convex cone in Rm and K a convex setin Rp. The problem under question is
min f (x), s.t. g(x) ∈ −K , h(x) = 0. (P)
If K = Rn+, the constraint g(x) ∈ −K collapses to the usual inequality constraint.
Section 2 is devoted to necessary conditions and Section 3 for sufficient ones.
2. Second-order necessary optimality conditions
For problem (P), we denote G = g−1(−K) and H = h−1(0). Then, the feasible set is
Recall that x0 ∈ M is said to be a local weak solution of (P) if there exists a neighborhood U of x0 such that, for all x ∈ U ∩M ,
f (x) − f (x0) ∈ −int C .
x0 ∈ M is called a local firm solution of order 2 of (P) if there are γ > 0 and a neighborhood U of x0 such that, for allx ∈ U ∩ M \ {x0},
(f (x) + C) ∩ Bm(f (x0), γ ∥x − x0∥2) = ∅.
Note that, in the literature, a firm solution is also known as a strict solution or an isolated solution. A firm solution is a Paretoone and hence a weak solution as well. Therefore, a necessary condition for the last one is also for the others, and a sufficientcondition for the first one is for the others too.
Set K(g(x0)) = cone(K + g(x0)). Then, [K(g(x0))]∗ = N(−K , g(x0)), the normal cone of −K at g(x0). Notice that, if K isa convex cone, then
[K(g(x0))]∗ = {k∗∈ K ∗
: ⟨k∗, g(x0)⟩ = 0}.
To establish second-order necessary optimality conditions, we use the following metric subregularity notion.
Definition 2.1 ([14]). Let x0, u ∈ Rn, u = 0, S ⊂ Rp and h : Rn→ Rp. One says that h is directionally metrically subregular
at (x0, u) wrt S if there exist µ > 0, ρ > 0 such that, for every t ∈ (0, ρ) and v ∈ Bn(u, ρ), one has
d(x0 + tv, h−1(S)) ≤ µd(h(x0 + tv), S). (DMSRu)
One says that h is metrically subregular at x0 wrt S if there exist µ > 0, ρ > 0 such that, for every x ∈ Bn(x0, ρ),
d(x, h−1(S)) ≤ µd(h(x), S). (MSR)
Let us observe that many notions related to metric regularity have been studied and applied, but the terminology stillvaries from paper to paper. Here we adapt the terminology of [5]. Condition (MSR) implies the condition (DMSRu) for allu = 0. But, (DMSR0) (i.e., (DMSRu) with u = 0) is equivalent to (MSR). Furthermore, when S is closed and convex, and h iscontinuously differentiable at x0, condition (MSR) is implied by the Mangasarian–Fromovitz qualification:
h′(x0)Rn− cone(S − h(x0)) = Rp. (MF)
We first establish a second-order necessary condition for (P) in a primal form, i.e., a condition in terms of the primalspaces.
Theorem 2.2. Let the interiors of C and K be nonempty and x0 a local weak solution of (P). Then, for all u ∈ Rn, the followingassertions hold.
(i) Let f , g and h be Fréchet differentiable at x0. Then, (f , g, h)′(x0)u ∈ −int[C × K(g(x0))] × {0}, provided (DMSRu) of h wrtS = {0} for u = 0.
(ii) Let f , g and h be strictly differentiable at x0. If (f , g, h)′(x0)u ∈ −[C × clK(g(x0)) \ int(C × K(g(x0)))] × {0}, then for all(y0, z0, w0) ∈ D2(f , g, h)(x0, u) and w ∈ Rn, under (DMSRu) of h wrt S = {0} one has
(f , g, h)′(x0)w + (y0, z0, w0) ∈ −intcone[C + f ′(x0)u] × IT 2(−K , g(x0), g ′(x0)u) × {0}.
(iii) Let f be l-stable at x0 (and int K not necessarily nonempty). Then, f ′(x0)w ∈ −intcone[C+f ′(x0)u] for allw ∈ T ′′(M, x0, u).
Proof. (i) If u = 0, then there is nothing to prove. Suppose to the contrary that, for some nonzero u ∈ Rn,
(f , g, h)′(x0)u ∈ −int[C × K(g(x0))] × {0}.
Then,
h(x0 + tku)tk
=h(x0 + tku) − h(x0)
tk→ h′(x0)u = 0.
By themetric subregularity of h, there areµ > 0, ρ > 0 such that, for every t ∈ (0, ρ) and v ∈ Bn(u, ρ), d(x0 + tv,H) ≤
µ∥h(x0 + tv)∥. Therefore, for large k, there exists yk ∈ H such that ∥x0 + tku − yk∥ ≤ µ∥h(x0 + tku)∥ + o(tk), and hence(x0 + tku − yk)/tk → 0. Setting uk := (yk − x0)/tk, one gets x0 + tkuk ∈ H . By the differentiability of f and g at x0, it followsthat
(f (x0 + tkuk) − f (x0))/tk → f ′(x0)u ∈ −int C,
and so, for large k, f (x0 + tkuk) − f (x0) ∈ −int C , and similarly
Similarly, by (2.4) and the definition of IT 2, for sufficiently large k we have
g(x0) + tkg ′(x0)u +12t2k
gx0 + tkuk +
12 t
2k wk
− g(x0) − tkg ′(x0)u
t2k /2∈ −K .
Therefore,
gx0 + tkuk +
12t2k wk
∈ −K . (2.6)
The formulas (2.2), (2.5) and (2.6) contradict the assumption about x0.(iii) Forw ∈ T ′′(M, x0, u), there are (tk, rk) → (0+, 0+) : tk/rk → 0, and xk ∈ M such thatwk := (xk−x0− tku)/ 1
2 tkrk →
w. By Proposition 3.4(iii) in Part I, one has
f (xk) − f (x0) − tkf ′(x0)utkrk/2
→ f ′(x0)w.
If f ′(x0)w ∈ −intcone[C + f ′(x0)u], then, for k large enough,
f ′(x0)u +12rkf (xk) − f (x0) − tkf ′(x0)u
tkrk/2∈ −int C,
which implies that f (xk) − f (x0) ∈ −int C , a contradiction. �
Note that the conclusion of Theorem 2.2 is about all points ofD2(f , g, h)(x0, u)which is larger than d2(f , g, h)(x0, u) usedin Theorem 1 of [6], hence our conclusion is stronger. Moreover, Theorem 2.2 extends Propositions 3.5 of [2] and 4.1 of [9],which consider only the case h = 0 and g ′(x0)u ∈ −K(g(x0)).
To get from Theorem 2.2 a dual form in terms of Lagrange multipliers, let us denote the set of Fritz John multipliers by
Lemma 2.3 ([15, Theorem 20.2]). Let C1, C2 ⊂ Rn be convex subsets such that C1 is polyhedral. Then, there exists a hyperplaneseparating C1 and C2 properly and not containing C2 if and only if C1 ∩ riC2 = ∅.
Theorem 2.4. Let int C and int K be nonempty and the assumptions (for (i), (ii) and (iii)) of Theorem 2.2 satisfied, and x0 alocal weak solution of (P). Then,
(i) there exists (c∗, k∗, h∗) ∈ Λ(x0) such that (c∗, k∗) = (0, 0);(ii) for every u ∈ Rn with (f , g, h)′(x0)u ∈ −[C ×clK(g(x0))\ int(C ×K(g(x0)))]×{0}, and (y0, z0, w0) ∈ D2(f , g, h)(x0, u),
does not contain C2. Since (f , g, h)′(x0)Rn and C×K(g(x0)) are cones,α = 0. Then, (2.7) implies that c∗◦f ′(x0)+k∗
◦g ′(x0)+h∗
◦ h′(x0) = 0. Letting k = 0 in (2.8), one obtains c∗∈ C∗. Setting c = 0 in (2.8) gives k∗
∈ K(g(x0))∗ = N(−K , g(x0)).Because the hyperplane H does not contain C2, (c∗, k∗) = (0, 0).
(ii) Suppose A2(−K , g(x0), g ′(x0)u) = ∅ (otherwise, the result is trivial). According to Theorem 2.2(ii), applyingLemma 2.3 with C1 = (f , g, h)′(x0)Rn
+ (y0, z0, w0) and C2 = −intcone[C + f ′(x0)u] × IT 2(−K , g(x0), g ′(x0)u) × {0},we obtain (c∗, k∗, h∗) ∈ (Rm)∗ × (Rp)∗ × (Rr)∗ with (c∗, k∗, h∗) = (0, 0, 0) and α ∈ R such that, for all (y, z, t) ∈ (f , g, h)′(x0)Rn, w ∈ −intcone[C + f ′(x0)u] and k ∈ IT 2(−K , g(x0), g ′(x0)u),
and the hyperplane H defined above does not contain C2. Since (f , g, h)′(x0)Rn is a subspace, from (2.9) one has, for all(y, z, t) ∈ (f , g, h)′(x0)Rn,
⟨c∗, y⟩ + ⟨k∗, z⟩ + ⟨h∗, t⟩ = 0,
and hence c∗◦ f ′(x0) + k∗
◦ g ′(x0) + h∗◦ h′(x0) = 0 and
⟨c∗, y0⟩ + ⟨k∗, z0⟩ + ⟨h∗, w0⟩ ≥ α. (2.11)
Since −intcone[C + f ′(x0)u] is a cone, (2.10) yields that ⟨c∗, w⟩ ≤ 0, for all w ∈ −intcone[C + f ′(x0)u], which implies thatc∗
∈ C∗ and ⟨c∗, f ′(x0)u⟩ = 0. Letting w tend to the origin in (2.10), we have ⟨k∗, k⟩ ≤ α, for all k ∈ IT 2(−K , g(x0), g ′(x0)u),and thus, for all k ∈ A2(−K , g(x0), g ′(x0)u) (cf. Proposition 1.3(iv) in Part I).
To see that (c∗, k∗, h∗) ∈ Λ(x0) as required, we observe from Proposition 3.1 [4] that
A2(−K , g(x0), g ′(x0)u) + T (T (−K , g(x0)), g ′(x0)u) ⊂ A2(−K , g(x0), g ′(x0)u).
Therefore, ⟨k∗, k + k1⟩ ≤ α, for k ∈ A2(−K , g(x0), g ′(x0)u) and k1 ∈ T (T (−K , g(x0)), g ′(x0)u). Because T (T (−K , g(x0)),g ′(x0)u) is a cone, it follows from formula (2.110) in [3] that
k∗∈ −[T (T (−K , g(x0)), g ′(x0)u)]∗ = {k∗
∈ N(−K , g(x0)) : ⟨k∗, g ′(x0)u⟩ = 0}. (2.12)
Now, suppose h = 0. If (c∗, k∗) = (0, 0), then by (2.9) and (2.10), α = 0 and consequently the hyperplane H containsC2, a contradiction.
Suppose condition (TRu) is satisfied, i.e., for every (y, z) ∈ Rp× Rr , there are x ∈ Rn and k ∈ T (T (−K , g(x0)), g ′(x0)u)
such that (g, h)′(x0)x − (k, 0) = (y, z). Therefore,
(ii) for every u ∈ Rn with (f , g)′(x0)u ∈ −[C × clK(g(x0)) \ int(C × K(g(x0)))] and (y0, z0) ∈ D2(f , g)(x0, u), there exists(c∗, k∗) ∈ Λ1(x0) such that
⟨c∗, y0⟩ + ⟨k∗, z0⟩ ≥ supk∈A2(−K ,g(x0),g ′(x0)u)
⟨k∗, k⟩,
and c∗= 0 if, in addition, the following constraint qualification holds
g ′(x0)Rn− T (T (−K , g(x0)), g ′(x0)u) = Rp
; (TR1u)
(iii) if w ∈ T ′′(M, x0, u), then there exists c∗∈ C∗
\ {0} with ⟨c∗, f ′(x0)u⟩ = 0 such that ⟨c∗, f ′(x0)w⟩ ≥ 0.
Remark 1. (i) Observe, regarding Theorems 2.2 and 2.4 (and Corollary 2.5) that, since g and h do not appear explicitly andthere is no restriction on M in assertion (iii), this assertion holds true for arbitrary g and h. Moreover, in strengthening theassumption on (g, h) as: (g, h) is l-stable at x0 and directionally metrically subregular at (x0, u) wrt −K ×{0}, this assertionbecomes stronger and expressed in terms of g and h as follows: if g ′(x0)w ∈ T ′′(−K , g(x0), g ′(x0)u) and h′(x0)w = 0, thenthere exists c∗
∈ C∗\ {0} with ⟨c∗, f ′(x0)u⟩ = 0 such that ⟨c∗, f ′(x0)w⟩ ≥ 0. Indeed, applying Proposition 2.6 below with
(g, h) instead of g and −K × {0} instead of −K , forM = (g, h)−1(−K × {0}) = G ∩ H and by the definition of T ′′, we have
T ′′(M, x0, u) = {w ∈ Rn: (g, h)′(x0)w ∈ T ′′(−K × {0}, (g, h)(x0), (g, h)′(x0)u)}
= {w ∈ Rn: g ′(x0)w ∈ T ′′(−K , g(x0), g ′(x0)u), h′(x0)w = 0}.
(ii) Though it is known that the following expression appearing in Theorem 2.4 is nonpositive:
supk∈A2(−K ,g(x0),g ′(x0)u)
⟨k∗, k⟩ ≤ 0, (2.13)
we give a simple explanation because of its importance. By Proposition 1.3(i) in Part I, one has
A2(−K , g(x0), g ′(x0)u) ⊂ clcone[cone(−K − g(x0)) − g ′(x0)u].
On the other hand (see the last part of the proof of Theorem 2.4(ii) here and Proposition 2.5(ii) [7]),
k∗∈ −[T (T (−K , g(x0)), g ′(x0)u)]∗ = −[clcone(cone(−K − g(x0)) − g ′(x0)u)]∗.
This implies (2.13). This formula reflects the envelop-like effect, because it can hold as a strict inequality. The supremumin (2.13) vanishes (and it looks classical) if 0 ∈ A2(−K , g(x0), g ′(x0)u). In particular, this is the case if one considers second-order necessary conditions only for direction u ∈ Rn with g ′(x0)u ∈ cone(−K − g(x0)) = −K(g(x0)) (see Proposition 1.3(v)(b) in Part I), as many authors do. However, Theorem 2.4 deals with g ′(x0)u ∈ clcone(−K − g(x0)) = −clK(g(x0)). Namely,the envelop-like effect occurs for points u in this seemingly small gap of the closure. Let us emphasize that no envelop-likeeffect happens for g ′(x0)u ∈ −K(g(x0)), even when this point lies on the boundary. Paper [6] is a careful and detailed studyconcerning the envelop-like effect. In Remark 3 there, the authors observed errors related to this effect in several earlierpapers, including an assertion that Proposition 4.1 and Theorem 4.1(ii) of [9] is false because of lacking the envelop-likeeffect. However, they did not pay attention to the fact that no envelope-like effect occurs here, since these results deal withthe case g ′(x0)u ∈ −K(g(x0)).
Observe also that, if K is polyhedral, −K(g(x0)) is closed, and hence we do not meet the envelop-like effect either.Corollary 2.5 sharpens Theorem 4.1 of [9] and Theorem 3.1 of [2], by considering additionally points u in the mentioned
gap and thus facing the envelop-like effect. Moreover, if h = (h1, . . . , hr) is twice differentiable at x0 with h′
1(x0), . . . , h′r(x0)
linearly independent, then condition (MF) holds with S = {0} and, consequently, condition (DMSRu) is satisfied. Hence,Theorem 2.4 (resp, Theorem 2.2) improves Theorem 2 (resp, Theorem 1) of [6], in which the assumptions are morerestrictive as: f ′ and g ′ are continuous at x0, d2(f , g)(x0, u) is used, and h = (h1, . . . , hr) is twice Fréchet differentiableat x0 with h′
1(x0), . . . , h′r(x0) linearly independent. Furthermore, D2(f , g, h)(x0, u) used in Theorems 2.2 and 2.4 is larger
than d2(f , g, h)(x0, u) in [6], and then their conclusions are stronger.The following result gives a characterization of the asymptotic second-order tangent cone to the feasible set M =
g−1(−K).
Proposition 2.6. Let g be l-stable at x0 and directionally metrically subregular at (x0, u) wrt −K. Then, for M = g−1(−K) wehave
T ′′(M, x0, u) = {w ∈ Rn: g ′(x0)w ∈ T ′′(−K , g(x0), g ′(x0)u)}.
Proof. Forw ∈ T ′′(M, x0, u), there are (tk, rk) → (0+, 0+) : tk/rk → 0 andwk → w such that xk := x0+tku+12 tkrkwk ∈ M .
By Proposition 3.4(iii) in Part I, one has
g(xk) − g(x0) − tkg ′(x0)utkrk/2
→ g ′(x0)w.
As g(xk) ∈ −K , one has g ′(x0)w ∈ T ′′(−K , g(x0), g ′(x0)u).Now assume that g ′(x0)w ∈ T ′′(−K , g(x0), g ′(x0)u). Then, there are (tk, rk) → (0+, 0+) : tk/rk → 0 and zk → g ′(x0)w
such that g(x0) + tkg ′(x0)u +12 tkrkzk ∈ −K for all k. By the assumed metric subregularity, for k large enough, we have
tk ∈ (0, ρ) and uk := u +12 rkw ∈ Bn(u, ρ) such that
(the last inequality follows from Lemma 3.2 in Part I). As zk → g ′(x0)w, this inequality yields xk ∈ M such that ∥x0 + tkuk− xk∥/ 1
2 tkrk → 0. Consequently, w ∈ T ′′(M, x0, u) since
wk :=xk − x0 − tku
tkrk/2=
xk − x0 − tkuk
tkrk/2+ w → w. �
The following example gives a case where Theorem 2.4 rejects a candidate for a local weak solution, while some recentresults do not.
Example 2.1. Let ϕ : [0, +∞) → R be defined by
ϕ(s) =
1 if s > 1,1/(q + 1) if 1/(q + 1) < s ≤ 1/q, q ∈ N,0 if s = 0.
Since ϕ in nondecreasing on [0, +∞), the function
θ(x) =
|x|
0ϕ(s)ds
is well-defined for x ∈ R (ϕ and θ are taken from [2]). We notice that θ is even and convex, and have the following right andleft derivatives, for q ∈ N,
θ ′
+(1/q) = 1/q, θ ′
−(1/q) = 1/(q + 1).
Let C = K = R+, x0 = (0, 0) and f , g, h : R2→ R be defined by
f ′(x0) = (0, 1), g ′(x0) = (1, 0), h′(x0) = (0, −1).
Since condition (MF) wrt S = {0} is satisfied, h is directionally metrically subregular at (x0, u) wrt this set for all u ∈ R2. Theset of Fritz John multipliers of (P) is
Λ(x0) = {(c∗, k∗, h∗) ∈ R3: c∗
= h∗ > 0, k∗= 0}.
The function f is not differentiable in U \ {x0}, for any neighborhood U of x0. Therefore, we cannot use Theorem 2 of [6].We claim that f is l-stable at x0. Indeed, let U be a neighborhood of x0, x = (x1, x2) ∈ U , and u = (u1, u2) ∈ S2. If x1 > 0,
then
|f l(x, u) − f l(x0, u)| = |2x1u1| ≤ 2∥x − x0∥.
If x1 < 0, then
|f l(x, u) − f l(x0, u)| = |θ ′
±(x1)u1| ≤ lim
s→x±1
ϕ(s) ≤ −x1 ≤ ∥x − x0∥.
If x1 = 0, then |f l(x, u) − f l(x0, u)| = 0. Thus, f is l-stable at x0, since |f l(x, u) − f l(x0, u)| ≤ 2∥x − x0∥ for all x ∈ U andu ∈ S2.
With α > 0, given (c∗, k∗) = (α, 0, 0, −α) ∈ Λ1(x0), we have
supk∈A2(−K ,g(x0),g ′(x0)u)
⟨k∗, k⟩ = −4α,
and, for (y0, z0) ∈ D2(f , g)(x0, u),
⟨c∗, y0⟩ + ⟨k∗, z0⟩ = α(−6 + β) − αβ = −6α < −4α.
By Corollary 2.5(ii), x0 is not a local weak solution of problem (P).
The next example illustrates the usefulness of assertion (iii) of Corollary 2.5 in a case, where assertion (ii) is out of use.
Example 2.3. Let C = R+, K = {0}, x0 = (0, 0), f be as in Example 2.1 and g : R2→ R be defined by g(x1, x2) = x31 + x22.
Then, f is l-stable at x0 and f ′(x0) = (0, 1). Since int K = ∅, Corollary 2.5(ii) cannot be applied. Trying with assertion (iii),we have
M = {(x1, x2) ∈ R2: x31 + x22 = 0}, T (M, x0) = {(u1, u2) ∈ R2
: u1 ≤ 0, u2 = 0}.
For u = (−1, 0) ∈ T (M, x0), T ′′(M, x0, u) = R2. Take w = (w1, w2) with w2 < 0. Then, for every c∗∈ C∗
\ {0} with⟨c∗, f ′(x0)u⟩ = 0, i.e., c∗ > 0, one has ⟨c∗, f ′(x0)w⟩ = c∗w < 0. By (iii), x0 is not a local weak solution.
Note that approximations (defined in [1,8]) are very general derivatives and useful in studies of optimality conditions(see [10–12]). The following example shows that, even in a problem without constraint, Corollary 2.5 rejects a candidatewhile the necessary conditions in terms of approximations in Theorem 4.1 of [11] does not.
Example 2.4. Let C = R+, K = R, x0 = 0, and f , g : R → R be defined by g = 0 and
f (x) =
−x2/2 if x ≥ 0,x2/2 − (2/3)x
√−x if x < 0,
(f is taken from Example 4.1 of [9]). Then, f is continuously differentiable at x0 and f ′(x0) = 0. In using Corollary 2.5, wekeep its notations, though no constraint imposed. The set of Fritz John multipliers of (P) is
Λ1(x0) = {(c∗, k∗) ∈ R2: c∗ > 0, k∗
= 0}.
For u = 1, A2(−K , g(x0), g ′(x0)u) = R,D2(f , g)(x0, u) = {(−1, 0)}, and for all (c∗, k∗) ∈ Λ1(x0), one has
By Corollary 2.5(ii), x0 is not a local weak solution of problem (P).Now we try with Theorem 4.1 of [11]. It is obvious that for all u ∈ Rn, T 2(M, x0, u) = T ′′(M, x0, u) = R. We calculate a
second-order approximation of f at x0 and related sets as follows: Bf (x0) = {−1/2} ∪ (α, +∞) for some α > 0, clBf (x0) =
{−1/2} ∪ [α, +∞) and Bf (x0)∞ = [0, +∞) (for the definitions of approximations, see [10,11]). Hence, x0 satisfies thenecessary conditions in Theorem 4.1 of [11] and is not rejected.
3. Second-order sufficient optimality conditions
In this section, we consider problem (P) with h = 0, and C, K may be nonconvex with empty interior. The followinglemma is used to prove second-order sufficient conditions.
Lemma 3.1. Assume that x0 ∈ M ⊆ Rn. If xk ∈ M \ {x0} tends to x0, then there exists u ∈ T (M, x0) \ {0} and a subsequence,denoted again by xk, such that
(i) (classic) (xk − x0)/tk → u, where tk = ∥xk − x0∥;(ii) [7, Lemma 3.4] either z ∈ T 2(M, x0, u) ∩ u⊥ exists such that (xk − x0 − tku)/ 1
2 t2k → z or z ∈ T ′′(M, x0, u) ∩ u⊥
\ {0} andrk → 0+ exist such that tk/rk → 0+ and (xk − x0 − tku)/ 1
2 tkrk → z, where u⊥ is the orthogonal complement of u ∈ Rn.
Theorem 3.2. For problem (P) with h = 0, let f and g be l-stable at x0 ∈ M. Then, either of the following conditions is sufficientfor x0 be a local firm solution of order 2.
(i) ∀u ∈ Sn, ∃(c∗, k∗) ∈ C∗× K(g(x0))∗ such that
⟨c∗, f ′(x0)u⟩ + ⟨k∗, g ′(x0)u⟩ > 0.
(ii) ∀u ∈ Sn with u ∈ T (M, x0) and f ′(x0)u ∈ −C, one has(a) ∀w ∈ T 2(M, x0, u) ∩ u⊥, ∀(y0, z0) ∈ d2(f , g)(x0, u) with g ′(x0)w + z0 ∈ T 2(−K , g(x0), g ′(x0)u), ∃(c∗, k∗) ∈ Λ1(x0)
(defined in Corollary 2.5) satisfying⟨c∗, y0⟩ + ⟨k∗, z0⟩ > ⟨k∗, g ′(x0)w + z0⟩; (3.1)
Proof. (i) See Theorem 4.2(i) in [9].(ii) Suppose to the contrary that there are xk ∈ M ∩ Bn
x0, 1
k
\ {x0} and ck ∈ C such that
f (xk) − f (x0) + ck ∈ Bm
0,
1kt2k
, (3.2)
where tk = ∥xk − x0∥. We can assume that (xk − x0)/tk → u ∈ T (M, x0) ∩ Sn. Dividing (3.2) by tk and passing to limit, onegets f ′(x0)u ∈ −C . By Lemma 3.1, it suffices to consider the following two cases (using subsequences).
First case: There is w ∈ T 2(M, x0, u) ∩ u⊥ such that wk := (xk − x0 − tku)/ 12 t
for some (y, z) ∈ Dp2(f , g)(x0, u, w). By Proposition 3.5 in Part I,
(y, z) = (f , g)′(x0)w + (y0, z0)
with (y0, z0) ∈ d2(f , g)(x0, u). As g(xk) ∈ −K and
zk = (g(xk) − g(x0) − tkg ′(x0)u)/12t2k → z = g ′(x0)w + z0,
it follows that g ′(x0)w + z0 ∈ T 2(−K , g(x0), g ′(x0)u), and so g ′(x0)u ∈ T (−K , g(x0)). By assumption (ii) (a), there exists(c∗, k∗) ∈ Λ1(x0) satisfying (3.1). Adding ⟨c∗, f ′(x0)w⟩ to both sides of (3.1) and taking into account that c∗
(ii) Condition (ii) (b) in Theorem 3.2 is derived by either of the following conditions
(b′) ∀w ∈ u⊥\ {0} with g ′(x0)w ∈ T ′′(−K , g(x0), g ′(x0)u), ∃(c∗, k∗) ∈ Λ1(x0), ⟨k∗, g ′(x0)w⟩ < 0.
(b′′) ∀w ∈ u⊥\ {0} with g ′(x0)w ∈ clcone[cone(−K − g(x0)) − g ′(x0)u], ∃(c∗, k∗) ∈ Λ1(x0), ⟨k∗, g ′(x0)w⟩ < 0.
In fact, by Proposition 3.4(iii) in Part I, if w ∈ T ′′(M, x0, u), then
g ′(x0)w ∈ T ′′(−K , g(x0), g ′(x0)u) ⊂ clcone[cone(−K − g(x0)) − g ′(x0)u].
Therefore, Theorem 3.2 improves Theorem 3 of [6], in which f ′ and g ′ are assumed stable at x0 and conditions (a′) and(b′′) are used. Observe also that, instead of D2(f , g)(x0, u) as in the necessary conditions, in Theorem 3.2 we use the smallerset d2(f , g)(x0, u) to get a stronger conclusion.
(iii) Nowwe discuss the gap between the necessary condition in Theorem 2.4 and the sufficient one in Theorem 3.2. Notefirst that, if u ∈ T (M, x0), then g ′(x0)u ∈ T (−K , g(x0)) = −clK(g(x0)). Hence, Theorem 3.2(ii) is also true if directionsu ∈ Sn for which (f , g)′(x0)u ∈ −[C × clK(g(x0))] are considered (as in Theorem 2.4). Furthermore, Theorem 2.4 (resp,Theorem 3.2) is also true (in fact is weakened) if we replace D2 (resp, d2) by d2 (resp, D2). Even with this weakening (for aclear comparison), when −K is parabolically derivable at g(x0), i.e., T 2(−K , g(x0), u) = A2(−K , g(x0), u) for all u ∈ Rp, thementioned gap is small: the inequalities in the necessary condition is replaced by the strict ones in the sufficient condition.
In the following example, Theorem 3.2 confirms a firm efficiency, while recent papers do not.
Example 3.1. Consider Example 2.2 but with f defined by (θ as in Example 2.1)
f (x1, x2) =
−x21 + x2 if x1 ≥ 0,θ(x1) + x2 if x1 < 0.
Similarly as in Example 2.2, we can prove that f is l-stable at x0 and f ′(x0) = (0, 1). Then, for u = (1, 0) ∈ S2 and(c∗, k∗) ∈ C∗
× K(g(x0))∗, one has
⟨c∗, f ′(x0)u⟩ + ⟨k∗, g ′(x0)u⟩ = 0.
Therefore, condition (i) in Theorem 3.2 is not satisfied. Let u = (u1, u2) ∈ S2 such that (f , g)′(x0)u ∈ −[C × clK(g(x0))].Then, u = (u1, 0) with u1 = ±1. One has
T 2(−K , g(x0), g ′(x0)u) = A2(−K , g(x0), g ′(x0)u),
and so, for k∗= (0, 0, −1) ∈ N(−K , g(x0)), supk∈T2(−K ,g(x0),g ′(x0)u)⟨k
If u1 = −1, then for (y0, z0) ∈ d2(f , g)(x0, u), there is tk → 0+ such that y0 = limk→+∞θ(−tk)t2k /2
and z0 = (0, 2, 0). Hence,
for (c∗, k∗) = (1, 0, 0, −1) ∈ Λ1(x0), one has
⟨c∗, y0⟩ + ⟨k∗, z0⟩ = limk→+∞
θ(−tk)t2k /2
≥ 0 > supk∈T2(−K ,g(x0),g ′(x0)u)
⟨k∗, k⟩.
Therefore, condition (a′′) is satisfied, and so is condition (a) of Theorem 3.2.Given w = (w1, w2) ∈ v⊥
\ {(0, 0)}, i.e., w1 = 0 and w2 = 0, if g ′(x0)w = (0, 0, w2) ∈ clcone[cone(−K − g(x0)) −
g ′(x0)u] = {(k1, k2, k3) ∈ R3: k3 ≥ 0}, then w2 > 0. Choosing (c∗, k∗) = (1, 0, 0, −1) ∈ Λ1(x0), one has ⟨k∗, g ′(x0)w⟩
= −w2 < 0. Hence, condition (b′′) is fulfilled, and so is condition (b) of Theorem 3.2. In consequence, by Theorem 3.2, x0 isa local firm solution of order 2 of problem (P).
Since f ′ is not stable at x0, Theorem 4.2 of [9] and Theorem 3 of [6] cannot be applied. Moreover, because the left-handside of (3.4) can be negative, Theorem 4.1 of [2] cannot be used.
Acknowledgments
This research was supported by the grant 101.01-2011-10 of the National Foundation for Science and TechnologyDevelopment of Vietnam (NAFOSTED). The final part of working on the paper was completed during a stay of the authorsas research visitors at the Vietnam Institute for Advanced Study in Mathematics (VIASM), whose hospitality is gratefullyacknowledged. The authors are indebted to the Referees for their valuable remarks and suggestions.
References
[1] K. Allali, T. Amahroq, Second-order approximations and primal and dual necessary optimality conditions, Optimization 40 (1997) 229–246.[2] D. Bednařík, K. Pastor, On second-order optimality conditions in constrained multiobjective optimization, Nonlinear Anal. 74 (2011) 1372–1382.[3] J.F. Bonnans, A. Shapiro, Perturbation Analysis of Optimization Problems, Springer, New York, 2000.
[4] R. Cominetti, Metric regularity, tangent sets and second order optimality conditions, Appl. Math. Optim. 21 (1990) 265–287.[5] A.L. Donchev, R.T. Rockafellar, Implicit Functions and Solution Mappings, Springer, Dordrecht, 2009.[6] C. Gutiérrez, B. Jiménez, V. Novo, On second order Fritz John type optimality conditions in nonsmooth multiobjective programming, Math. Program.
B 123 (2010) 199–223.[7] B. Jiménez, V. Novo, Optimality conditions in differentiable vector optimization via second-order tangent sets, Appl. Math. Optim. 49 (2004) 123–144.[8] A. Jourani, L. Thibault, Approximations and metric regularity in mathematical programming in Banach spaces, Math. Oper. Res. 18 (1992) 390–400.[9] P.Q. Khanh, N.D. Tuan, Optimality conditions for nonsmooth multiobjective optimization using Hadamard directional derivatives, J. Optim. Theory
Appl. 133 (2007) 341–357.[10] P.Q. Khanh, N.D. Tuan, First and second-order optimality conditions using approximations for nonsmooth vector optimization in Banach spaces,
J. Optim. Theory Appl. 130 (2006) 289–308.[11] P.Q. Khanh, N.D. Tuan, First and second-order approximations as derivatives of mappings in optimality conditions for nonsmooth vector optimization,
Appl. Math. Optim. 58 (2008) 147–166.[12] P.Q. Khanh, N.D. Tuan, Second-order optimality conditions using approximations for nonsmooth vector optimization problems under inclusion
constraints, Nonlinear Anal. 74 (2011) 4338–4351.[13] L. Liu, P. Neittaanmäki, M. Křížek, Second-order optimality conditions for nondominated solutions of multiobjective programming with C1,1 data,
Appl. Math. 45 (2000) 381–397.[14] R.T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, New Jersey, 1970.[15] D.E. Ward, Calculus for parabolic second-order derivatives, Set Valued Anal. 1 (1993) 213–246.
Further reading
[1] D. Bednařík, K. Pastor, On second-order conditions in unconstrained optimization, Math. Program. A 113 (2008) 283–298.[2] D. Bednařík, K. Pastor, Decrease of C1,1 property in vector optimization, RAIRO Oper. Res. 43 (2009) 359–372.[3] D. Bednařík, K. Pastor, l-stable functions are continuous, Nonlinear Anal. 70 (2009) 2317–2324.[4] F.H. Clarke, Optimization and Nonsmooth Analysis, Wiley Interscience, New York, 1983.[5] I. Ginchev, On scalar and vector l-stable functions, Nonlinear Anal. 74 (2011) 182–194.[6] I. Ginchev, A. Guerraggio, M. Rocca, Second-order conditions for C1,1 constrained vector optimization, Math. Program. B 104 (2005) 389–405.[7] I. Ginchev, A. Guerraggio, M. Rocca, From scalar to vector optimization, Appl. Math. 51 (2006) 5–36.[8] B. Jiménez, V. Novo, Second order necessary conditions in set constrained differentiable vector optimization, Math. Methods Oper. Res. 58 (2003)
299–317.[9] H. Kawasaki, An envelope-like effect of infinitelymany inequality constraints on second-order necessary conditions forminimization problems,Math.
Program. 41 (1988) 73–96.[10] P.Q. Khanh, N.D. Tuan, Second-order optimality conditions with envelope-like effect for nonsmooth vector optimization in infinite dimensions,
Nonlinear Anal. 77 (2013) 130–148.[11] P.Q. Khanh, N.D. Tuan, Second-order optimality conditions with the envelope-like effect in nonsmooth multiobjective mathematical programming I:
l-stability and set-valued directional derivatives, J. Math. Anal. Appl. online first 2013 http://dx.doi.org/10.1016/j.jmaa.2012.12.076.[12] Y. Maruyama, Second-order necessary conditions for nonlinear optimization problems in Banach spaces and their applications to an optimal control
problem, Math. Oper. Res. 15 (1990) 467–482.[13] J.P. Penot, Second order conditions for optimization problems with constraints, SIAM J. Control Optim. 37 (1999) 303–318.[14] J.P. Penot, Optimality conditions in mathematical programming and composite optimization, Math. Program. 67 (1994) 225–245.
