SCIENCE CHINA Mathematics . ARTICLES . April 2012 Vol. 55 No. 4: 851–868 doi: 10.1007/s11425-011-4287-5 c Science China Press and Springer-Verlag Berlin Heidelberg 2011 math.scichina.com www.springerlink.com Image space analysis for variational inequalities with cone constraints and applications to traffic equilibria LI Jun 1, ∗ & HUANG NanJing 2 1 College of Mathematics and Information, China West Normal University, Nanchong 637009, China; 2 Department of Mathematics, Sichuan University, Chengdu 610064, China Email: junli1026@163.com, [email protected]Received July 25, 2010; accepted May 2, 2011; published online September 6, 2011 Abstract In this paper, the image space analysis (for short, ISA) is employed to investigate variational in- equalities (for short, VI) with cone constraints. Linear separation for VI with cone constraints is characterized by using the normal cone to a regularization of the image, and saddle points of the generalized Lagrangian func- tion. Lagrangian-type necessary and sufficient optimality conditions for VI with cone constraints are presented by using a separation theorem. Gap functions and weak sharpness for VI with cone constraints are also investi- gated. Finally, the obtained results are applied to standard and time-dependent traffic equilibria introduced by Daniele, Maugeri and Oettli. Keywords variational inequality, image space analysis, linear separation, necessary and sufficient optimality condition, traffic equilibrium MSC(2010) 90C33, 49J40 Citation: Li J, Huang N J. Image space analysis for variational inequalities with cone constraints and applications to traffic equilibria. Sci China Math, 2012, 55(4): 851–868, doi: 10.1007/s11425-011-4287-5 1 Introduction The image of a constrained extremum problem was developed in [15] by exploiting previous results on theorems of the alternative [14]. The image space analysis (for short, ISA) is a powerful tool and a unifying scheme for studying both variational inequalities and optimization problems (for short, VI and OP, respectively). This approach can be applied to any kind of problem that can be expressed under the form of the impossibility of a parametric system. The impossibility of such a system is reduced to the disjunction of two suitable subsets of the image space. The disjunction between the two suitable subsets is proved by showing that they lie in two disjoint level sets of a separating functional. Recently, there has been an increasing interest in the ISA of vector variational inequalities and vector optimization problems (for short, VVI and VOP, respectively) [16, 17, 26, 31, 32]. The first step in this direction was made by Giannessi, Mastroeni and Pellegrini [19]. In [34, 35], Moldovan and Pellegrini employed the ISA to present necessary and sufficient optimality conditions for constrained extremum problems in finite-dimensional spaces. In [29, 30], Madani, Mastroeni and Moldovan dealt with ISA for constrained extremum problems having infinite-dimensional image and established sufficient and neces- sary optimality conditions for these problems. Some other related works on the ISA for VI, OP, VVI and VOP, can be found in [18,21]. ∗ Corresponding author
Image space analysis for variational inequalities withcone constraints and applications to traffic equilibria
LI Jun1,∗ & HUANG NanJing2
1College of Mathematics and Information, China West Normal University, Nanchong 637009, China;2Department of Mathematics, Sichuan University, Chengdu 610064, China
= (f(x) + α,−(1− t)h(x1)− th(x2) + d) (from the affinity of f)
= (f(x),−(1 − t)h(x1)− th(x2)) + (α, d)
∈ (f(x),−h(x)) + R+ ×D (since D is a convex cone)
= (f,−h)(x) + R+ ×D
⊆ (f,−h)(X) + R+ ×D,
which establishes the convexity of (f,−h)(X) + R+ ×D.
3 ISA for VI with cone constraints
In this section, we shall carry on the ISA for VI with cone constraints.
854 Li J et al. Sci China Math April 2012 Vol. 55 No. 4
Observe that, x ∈ K solves VI with cone constraints iff the system (in the unknown y):
⎧⎪⎪⎨
⎪⎪⎩
〈F (x), x − y〉 > 0,
g(y) �D 0,
y ∈ X
(3.1)
is impossible. We can associate VI with cone constraints with the following sets:
H := {(u, v) ∈ R× Rn : u > 0, v �D 0},
K(x) := {(u, v) ∈ R× Rn : u = 〈F (x), x− y〉, v = g(y), y ∈ X}.
The set K(x) is called the image of VI with cone constraints at x ∈ K, while the space R × Rn is the
image space associated with VI with cone constraints at x ∈ K.
It is easily seen that the following holds.
Proposition 3.1. Let x ∈ K. System (3.1) is impossible iff
H ∩K(x) = ∅. (3.2)
Consequently, x ∈ K is a solution to VI with cone constraints iff (3.2) is true.
Since D ⊆ Rn is a closed convex cone, we have 0 ∈ D. Set Hu := {(u, v) ∈ R× R
n : u > 0, v = 0}.As pointed out in [17–19, 32], to prove directly whether or not (3.2) holds is generally too difficult.
Therefore, in order to show such a disjunction, one must prove that the two sets, or the set H and
an extension of the image depending on H, admit a separating hyperplane. In general, the image of a
generalized system is not convex even when the functions involved enjoy some convexity properties. To
overcome this difficulty, similar to that in [18, 19, 32], we introduce a regularization of the image K(x),
namely, the extension with respect to the cone clH, denoted by E :
E := K(x)− clH = {(u, v) ∈ R× Rn : u � 〈F (x), x− y〉, v �D g(y), y ∈ X}.
Remark 3.1. As a direct consequence of the definition of E , we have E = −[(f,−g)(X) + R+ ×D],
where f(x) := 〈F (x), x− x〉. In [8,25,33], a constraint qualification involving the tangent cone to the set
E was employed to investigate infinite-dimensional duality.
The following proposition is useful in deriving the linear separation and the Lagrangian-type necessary
and sufficient optimality conditions for VI with cone constraints.
Proposition 3.2. Let x ∈ K. The following statements are true :
(i) If either (f,−g) is convex-like on X with respect to R+ × D or −g is D-convex on X, where
f(x) := 〈F (x), x − x〉, then the set E is convex ; if g is continuous and X is compact, then the set E is
closed ;
(ii) System (3.1) is impossible, or (3.2) holds iff
H ∩ E = ∅, (3.3)
iff
Hu ∩ E = ∅. (3.4)
Proof. (i) If (f,−g) is convex-like on X with respect to R+ ×D, where f(x) := 〈F (x), x − x〉, then it
follows from Remark 3.1 that the set E is convex.
Suppose that −g isD-convex onX . Since f is affine, from Lemma 2.3 one has that (f,−g) is convex-likeon X with respect to R+ ×D. Therefore, the set E is convex.
Notice that f(x) = 〈F (x), x − x〉 is continuous and R+ ×D is compact. If g is continuous and X is
compact, then from Remark 3.1 we know that the set E is closed.
Li J et al. Sci China Math April 2012 Vol. 55 No. 4 855
(ii) Since K(x) ⊆ E , it suffices to prove that (3.2) implies (3.3) holds. Let System (3.2) hold. Suppose
to the contrary that H ∩ E �= ∅. Then there is (u, v) ∈ H ∩ E , i.e., there exists y ∈ X such that
u � 〈F (x), x− y〉, v �D g(y), and u > 0, v �D 0.
Therefore, one has
0 < 〈F (x), x− y〉, 0 �D g(y),
which contradicts the assumption H ∩K(x) = ∅.We now establish the equivalence between (3.3) and (3.4). We only need to prove that (3.4) implies
(3.3) since Hu ⊆ H. The proof is similar to that in [18]. For the completeness, we include it here. Assume
that (3.4) holds. Suppose to the contrary that there is (u, v) ∈ H ∩ E . Then (0, v) ∈ clH. Since
E − clH = K(x)− (clH+ clH) = K(x)− clH = E ,
one has (u, v)− (0, v) = (u, 0) ∈ E , which is a contradiction since (u, 0) ∈ Hu.
4 Linear separation and Lagrangian-type optimality conditions for VI with
cone constraints
As it is well-known, linear separation [7, 11, 18, 31] is closely related to the Lagrangian type optimality
conditions. In this section, by following the analysis developed in [31], we shall characterize the linear
separation in terms of the normal cone to E , a saddle point condition of generalized Lagrangian function
associated with the system (3.1) and present the Lagrangian-type necessary and sufficient optimality
conditions for VI with cone constraints.
Definition 4.1. Let x ∈ K. The sets H and K(x) admit a linear separation iff there exists (λ, θ) ∈R+ ×D∗, with (λ, θ) �= (0, 0), such that
λ〈F (x), x − y〉+ 〈θ, g(y)〉 � 0, ∀ y ∈ X. (4.1)
Let
B := {(λ, θ) ∈ R+ ×D∗ : ‖(λ, θ)‖ = 1}.Then B is a compact subset of R+ ×D∗.
It is easy to see that the following conclusion holds.
Proposition 4.1. Let x ∈ K. Then the sets H and K(x) admit a linear separation iff there exists
(λ, θ) ∈ B, such that (4.1) holds.
We first characterize the linear separation of the sets H and E by using the normal cone to E at zero.
Theorem 4.1. Let x ∈ K. Then the sets H and K(x) admit a linear separation iff there exists
(λ, θ) ∈ NE(0, 0), or equivalently, (λ, θ) ∈ −E∗, with (λ, θ) �= (0, 0).
Proof. It is obvious that (0, 0) ∈ E and so (λ, θ) ∈ NE(0, 0) iff (λ, θ) ∈ −E∗.Necessity. Suppose that the sets H and K(x) admit a linear separation. Then there exists (λ, θ) ∈
856 Li J et al. Sci China Math April 2012 Vol. 55 No. 4
From Remark 3.1, the above inequality implies that (λ, θ) ∈ NE(0, 0).Sufficiency. Let (λ, θ) ∈ NE(0, 0), with (λ, θ) �= (0, 0). Then it follows from Remark 3.1 that
Since D is closed and convex, D∗∗ = D. We declare that g(x) �D 0, i.e., g(x) ∈ D. In fact, if g(x) �∈ D,
then there is θ∗ ∈ D∗ such that 〈θ∗, g(x)〉 < 0. Since D∗ is a cone, we have tθ∗ ∈ D∗ for all t � 0 and
thus
−〈tθ∗, g(x)〉 = −t〈θ∗, g(x)〉 → +∞, as t→ +∞,
858 Li J et al. Sci China Math April 2012 Vol. 55 No. 4
which is a contradiction with the first inequality in (4.5).
Since θ ∈ D∗ and g(x) �D 0, one has 〈θ, g(x)〉 � 0. Setting θ := 0 in the first inequality in (4.5) leads
to 〈θ, g(x)〉 � 0 and therefore 〈θ, g(x)〉 = 0.
From the second inequality in (4.5), one has that x ∈ X is a global minimum point of L(x; λ, θ, y) onX . Since g is differentiable, each component of g, gi (i = 1, 2, . . . , n), is differentiable. Since X is an open
convex set in Rm, it follows that
0 = ∇yL(x; λ, θ, x) = λF (x)−n∑
i=1
θi∇gi(x),
i.e.,
λF (x) =
n∑
i=1
θi∇gi(x).
Sufficiency. Let (λ, θ, x) be a solution to the system (4.4). Since −g isD-convex onX , from Lemma 2.1,
one has that y �→ 〈λ,−g(y)〉 is convex. Since g is differentiable, the function y �→ L(x; λ, θ, y) is convexand differentiable. Now, the equality λF (x) =
∑ni=1 θi∇gi(x) implies that ∇yL(x; λ, θ, x) = 0 and thus
L(x; λ, θ, x) � L(x; λ, θ, y), ∀ y ∈ X.
Then, the complementarity condition 〈θ, g(x)〉 = 0 allows to
Remark 4.2. Let X be an open convex set in Rm, g be differentiable and −g be D-convex on X .
Suppose that the sets H and K(x) admit a linear separation, or equivalently, there exists (λ, θ) ∈ B such
that the point (λ, θ, x) is a saddle point for L(x;λ, θ, y) on R+ ×D∗ ×X .
(i) If λ = 0, then θ �= 0 and
n∑
i=1
(θi∇gi(x)) = 0,
which implies that ∇gi (x) (i = 1, 2, . . . , n) are linearly dependent.
(ii) If θ �= 0 and λ �= 0, then
F (x) =
n∑
i=1
(θi
λ∇gi(x)
)
.
As a consequence, Lagrangian duality for VI with cone constraints can be derived. For further details,
please see [2, 3].
Next, we shall present the Lagrangian-type necessary and sufficient optimality conditions for VI with
cone constraints.
First, we obtain the following necessary optimality condition for VI with cone constraints via ISA.
Theorem 4.5. Assume that either (f,−g) is convex-like on X with respect to R+ ×D or −g is D-
convex on X, where f(x) := 〈F (x), x − x〉. If x ∈ K is a solution to VI with cone constraints, then the
sets H and K(x) admit a linear separation, or equivalently, there is a vector (λ, θ) ∈ B such that :
(i) λ〈F (x), x− y〉+ 〈θ, g(y)〉 � 0, ∀ y ∈ X ;
(ii) 〈θ, g(x)〉 = 0,
where B = {(λ, θ) ∈ R+ ×D∗ : ‖(λ, θ)‖ = 1}.
Li J et al. Sci China Math April 2012 Vol. 55 No. 4 859
Proof. Let x ∈ K. Clearly, the set H is convex and from (i) of Proposition 3.2, E is convex. If x ∈ K
is a solution of VI with cone constraints, then (ii) of Proposition 3.2 implies that
H ∩ E = ∅.
Therefore, by a separation theorem [38], there exists a vector (λ, θ) ∈ R × Rn, with (λ, θ) �= (0, 0), such
that
λ〈F (x), x− y〉+ 〈θ, g(y)〉 � 0, ∀ y ∈ X, (4.6)
and
λu+ 〈θ, v〉 � 0, ∀ (u, v) ∈ H. (4.7)
It follows immediately from (4.7) that (λ, θ) ∈ R+ ×D∗. Since (λ, θ) �= (0, 0), without loss of generality,
we can suppose that (λ, θ) ∈ B. From Proposition 4.1, Inequality (4.6) implies that the sets H and K(x)
admit a linear separation. The left proof follows directly from Theorem 4.3.
If X is an open convex set in Rm, then from Theorems 4.2, 4.4 and 4.5 one has the following classical
Lagrangian-type necessary optimality condition for VI with cone constraints.
Corollary 4.1. Let X be an open convex set in Rm and let x ∈ K be a solution of VI with cone
constraints. Suppose that g is differentiable and −g is D-convex on X. Then the sets H and K(x) admit
a linear separation, or equivalently, there exists (λ, θ) ∈ B such that the following hold :
(i) λF (x) =∑n
i=1(θi∇gi(x));(ii) 〈θ, g(x)〉 = 0.
Under certain assumptions, we obtain the following sufficient optimality condition for VI with cone
constraints.
Theorem 4.6. Let x ∈ K. Assume that there is a vector (λ, θ) ∈ B such that λ〈F (x), x−y〉+〈θ, g(y)〉 �0, ∀ y ∈ X. If either λ > 0 or θ >D∗ 0 and g(y) �= 0 for all y ∈ K, then x solves VI with cone constraints.
Proof. Suppose that x is not a solution of VI with cone constraints. Then from Proposition 3.1, there
is a y0 ∈ X such that
〈F (x), x− y0〉 > 0 and g(y0) �D 0.
Since (λ, θ) ∈ B, it follows from the assumption that λ〈F (x), x − y0〉+ 〈θ, g(y0)〉 � 0. If either λ > 0 or
θ >D∗ 0 and g(y) �= 0 for all y ∈ K, then from Lemma 2.2(i) the above inequality is strict, contradicting
the assumption.
5 Gap functions and weak sharpness for VI with cone constraints
In this section, we investigate the gap functions for VI with cone constraints by following similar idea
to that in [31], and prove that the solutions set of VI with cone constraints is weakly sharp for certain
functions under strong monotonicity.
Definition 5.1. Let K be the domain of VI with cone constraints. A function ψ : K → R ∪ {+∞} is
said to be a gap function for VI with cone constraints iff it satisfies the following properties :
(1) ψ(x) � 0, ∀x ∈ K;
(2) ψ(x) = 0, iff x solves VI with cone constraints.
Definition 5.2. Let K be the domain of VI with cone constraints and ψ : K → R∪{+∞} be a function
derived from VI with cone constraints. Denote by S the solutions set of VI with cone constraints. We
say that the set S is weakly sharp for the function ψ on K iff there exists μ > 0 such that
d(x, S) � μψ(x), ∀x ∈ K.
860 Li J et al. Sci China Math April 2012 Vol. 55 No. 4
The notion of weak sharpness is closely related to that of error bounds. If S = {x ∈ K : ψ(x) = 0}and there exists μ > 0 such that
d(x, S) � μψ(x), ∀x ∈ K,
then we say that an error bound holds for the set S with ψ and with respect to the set K. For more
details, please see [12, 37].
Let
ϕ(λ, θ, x) := supy∈X
[λ〈F (x), x − y〉+ 〈θ, g(y)〉], ∀ (λ, θ, x) ∈ B ×K.
Define φ : K → R ∪ {+∞} by
φ(x) := min(λ,θ)∈B
ϕ(λ, θ, x) = min(λ,θ)∈B
supy∈X
[λ〈F (x), x − y〉+ 〈θ, g(y)〉], ∀x ∈ K,
where B = {(λ, θ) ∈ R+ ×D∗ : ‖(λ, θ)‖ = 1} is as that in Section 4.
It is clear that the following holds.
Remark 5.1. The function (λ, θ) �→ ϕ(λ, θ, x), being the supremum of a collection of linear functions,
is a convex function. As a consequence, φ(x) = min(λ,θ)∈B ϕ(λ, θ, x) is the optimal value of a parametric
problem on a compact set, with a convex objective function.
Lemma 5.1. Let x ∈ K. Assume that the sets H and K(x) admit a linear separation.
(i) If the following system is possible :⎧⎪⎪⎨
⎪⎪⎩
〈F (x), x − y〉 > 0,
g(y) >D 0,
y ∈ X,
then we can suppose that λ > 0 in (4.1).
(ii) If there exists y ∈ X such that g(y) >D 0 (Slater condition), then we can suppose that λ > 0 in
(4.1).
Proof. Clearly, Statement (i) follows from (ii). We only need to prove Statement (ii). Assume that
the sets H and K(x) admit a linear separation. From Proposition 4.1, there is (λ, θ) ∈ B such that (4.1)
holds and hence
λ〈F (x), x− y〉+ 〈θ, g(y)〉 � 0. (5.1)
Suppose to the contrary that λ = 0 in (4.1). Then θ �= 0. Since g(y) >D 0, again from Lemma 2.2(ii) it
holds 0 < 〈θ, g(y)〉 = λ〈F (x), x− y〉+ 〈θ, g(y)〉, a contradiction with (5.1). �
Definition 5.3. A mapping T : X → X is said to be strongly monotone on X with modulus δ > 0 iff
for any x, y ∈ X,
〈T (x)− T (y), x− y〉 � δ‖x− y‖2.Under suitable assumptions, we prove that φ is a gap function for VI with cone constraints, and show
that the solutions set of VI with cone constraints is weakly sharp for the function
x �→ (ϕ(λ, θ, x)− 〈θ, g(x)〉) 12 ,
where (λ, θ) is some vector in B.
Theorem 5.1. Let S := {x ∈ K : φ(x) = 0} �= ∅. Assume that either (f,−g) is convex-like on X
with respect to R+ ×D or −g is D-convex on X, where f(x) = 〈F (x), x − x〉, and the following system
is possible :⎧⎪⎪⎨
⎪⎪⎩
〈F (x), x − y〉 > 0,
g(y) >D 0,
y ∈ X.
Li J et al. Sci China Math April 2012 Vol. 55 No. 4 861
Then φ is a gap function for VI with cone constraints. Furthermore, if F is strongly monotone on X
with modulus δ > 0, then there is (λ, θ) ∈ B, with λ > 0, such that S = {x}, the solutions set of VI with
cone constraints, is weakly sharp for the function x �→ (ϕ(λ, θ, x)− 〈θ, g(x)〉) 12 on K.
Proof. We first show that φ is a gap function for VI with cone constraints. Let x ∈ K be given. Then
Remark 5.2. Let r(x) := (ϕ(λ, θ, x) − 〈θ, g(x)〉) 12 . From (5.3), we have 〈θ, g(x)〉 = 0 and from (5.2)
and (5.4), we have ϕ(λ, θ, x) = supy∈X [λ〈F (x), x − y〉 + 〈θ, g(y)〉] = 0. Then it follows that r(x) = 0.
From (5.5) one has
d(x, S) = ‖x− x‖ � r(x)
(λδ)12
, ∀x ∈ K,
which implies that x ∈ S = {x} if r(x) = 0. As a consequence, x ∈ S = {x} iff r(x) = 0 and so the above
inequality yields that an error bound holds for S with r(x) and with respect to K.
862 Li J et al. Sci China Math April 2012 Vol. 55 No. 4
6 Applications to traffic equilibria
In [9, 10], Daniele, Maugeri and Oettli introduced traffic equilibria in the case where the numerical data
of the traffic network depend on time and characterized it by a suitably generalized form of Wardrop
principle in terms of a VI. In this section, we shall apply the results obtained in Sections 3 and 4 to
investigate the Lagrangian-type necessary and sufficient optimality conditions for traffic networks. The
notation that we employ is, for the most part, the same as that in [9, 10].
6.1 Applications to standard traffic networks
In the traffic network, one has a set W of origin-destination pairs and a set R of routes. Each route links
exactly one origin-destination pair w ∈ W . The set of all r ∈ R which links a given w ∈ W is denoted
by R(w).
We consider the flow vectors H = (Hr)�r∈R ∈ R
R, where Hr, r ∈ R, denotes the flow in route r. A
feasible flow has to satisfy the capacity restrictions,
λr � Hr � μr, ∀ r ∈ R,
and demand requirements,
∑
r∈R(w)
Hr = ρw, ∀w ∈ W ,
where λ <R
R+μ are given in R
R and ρ �R
W+
0 is given in RW .
Thus, the set of all feasible flows is given by
K :=
{
H ∈ RR : λ �
RR+H �
RR+μ,
∑
r∈R(w)
Hr = ρw, ∀w ∈ W
}
.
Furthermore, we are given a cost function C : K → RR. Then, to every feasible flow H ∈ K, there
corresponds a cost vector C(H) ∈ RR; Cr(H) gives the marginal cost of sending one additional unit of
flow through route r, when the flow H is already present.
Definition 6.1. A flow H ∈ RR is called an equilibrium flow iff
H ∈ K and 〈C(H), H − H〉 � 0, ∀H ∈ K.
Let
X := {H ∈ RR : λ �
RR+H �
RR+μ},
G1(H) :=
(∑
r∈R(w)
Hr − ρw
)�
w∈W
, ∀H ∈ RR,
G2(H) := −G1(H) =
(
ρw −∑
r∈R(w)
Hr
)�
w∈W
, ∀H ∈ RR,
G(H) :=
(G1(H)
G2(H)
)
, ∀H ∈ RR.
Clearly, X is compact and convex, the function G is affine. Then we can rewrite the set of all feasible
flows K as follows: K = {H ∈ X : G(H) �R
2W+
0}.Let H be an equilibrium flow. Set
H := {(u, v) ∈ R× R2W : u > 0, v �
R2W+
0},K(H) := {(u, v) ∈ R× R
2W : u = 〈C(H), H −H〉, v = G(H), H ∈ X},
Li J et al. Sci China Math April 2012 Vol. 55 No. 4 863
Hu := {(u, v) ∈ R× R2W : u > 0, v = 0},
E := K(H)− clH= {(u, v) ∈ R× R
2W : u � 〈C(H), H −H〉, v �R
2W+
G(H), H ∈ X}= −[(f,−G)(X) + R+ × R
2W+ ],
where f(H) := 〈C(H), H − H〉. The set K(H) is called the image of the above traffic network at H ∈ K,
while the space R× R2W is the image space associated with the above traffic network at H ∈ K.
Based on the results presented in Sections 3 and 4, one can easily check the following results hold.
Proposition 6.1. That H ∈ K is an equilibrium flow iff
H ∩K(H) = ∅,iff
H ∩ E = ∅,iff
Hu ∩ E = ∅.Theorem 6.1. If H ∈ K is an equilibrium flow, then the sets H and K(H) admit a linear separation,
or equivalently, there is a vector (u, v) ∈ B such that :
(i) u〈C(H), H −H〉+ 〈v, G(H)〉 � 0, ∀H ∈ X ;
(ii) 〈v, G(H)〉 = 0,
where B := {(u, v) ∈ R+ × R2W+ : ‖(u, v)‖ = 1}.
Proof. Since the function G ia affine, −G is R2W+ -convex on X . Thus the desired conclusion follows
directly from Theorem 4.5.
Similarly, we have the following:
Theorem 6.2. Let H ∈ K. Assume that there is a vector (u, v) ∈ B such that u〈C(H), H − H〉 +〈v, G(H)〉 � 0, ∀H ∈ X, where B is as that in Theorem 6.1. If u > 0, then H is an equilibrium flow.
6.2 Applications to time-dependent traffic networks
In [9, 10], Daniele, Maugeri and Oettli considered the dynamic case of traffic networks. The traffic
network, whose geometry remains fixed, is considered at all times t ∈ T , where T := [0, T ].
For each time t ∈ T , we have a route-flow vector H(t) ∈ RR. H : T → R
R is the flow (trajectory)
over time. For technical reasons, let the functional setting for the flow trajectories be the reflexive Banach
space Lp(T ,RR) with p > 1. For simplicity, let L := Lp(T ,RR). The dual space of Lp(T ,RR), for
instance, Lq(T ,RR), will be denoted by L ∗, where 1q +
1p = 1. The canonical bilinear form on L ∗ ×L
is given by
� G,H �:=
∫
T
〈G(t), H(t)〉dt, ∀ (G,H) ∈ L ∗ × L .
The feasible flows have to satisfy the time-dependent capacity constraints and demand requirements;
namely, almost everywhere (a.e.) on T ,
λ(t) �R
R+H(t) �
RR+μ(t) and
∑
r∈R(w)
Hr(t) = ρw(t), ∀w ∈ W ,
where λ < μ (i.e., λ(t) <R
R+μ(t) for all t ∈ T ) are given in L and ρ � 0 (i.e., ρ(t) �
RW+
0 for all t ∈ T )
is given in Lp(T ,RW ). The set of feasible flows is given by
K :=
{
H ∈ L : λ(t) �R
R+H(t) �
RR+μ(t) a.e. onT ;
864 Li J et al. Sci China Math April 2012 Vol. 55 No. 4
∑
r∈R(w)
Hr(t) = ρw(t) a.e. onT , ∀w ∈ W
}
.
We assume that∑
r∈R(w)
λr(t) � ρw(t) �∑
r∈R(w)
μr(t), a.e. onT , ∀w ∈ W .
Then, K is nonempty. It is easy to see that K is closed, convex and bounded, hence weakly compact.
Furthermore, we are given a mapping C : K → L ∗, which assigns to each flow trajectory H ∈ K the
cost trajectory C(H) ∈ L ∗.
Definition 6.2. A flow H ∈ L is called an equilibrium flow iff
H ∈ K and � C(H), H − H �� 0, ∀H ∈ K.
In [9, 10], Daniele, Maugeri and Oettli presented Wardrop principle for the dynamic traffic networks.
We will derive the Lagrangian-type necessary and sufficient optimality conditions for the dynamic traffic
networks.
To begin with, let
X := {H ∈ L : λ(t) �R
R+H(t) �
RR+μ(t) a.e. onT },
G1(H) :=
(∑
r∈R(w)
Hr − ρw
)�
w∈W
, ∀H ∈ L ,
G2(H) := −G1(H) =
(
ρw −∑
r∈R(w)
Hr
)�
w∈W
, ∀H ∈ L ,
G(H) :=
(G1(H)
G2(H)
)
=
(∑
r∈R(w)
Hr,−∑
r∈R(w)
Hr
)�
w∈W
+ (−ρw, ρw)�w∈W , ∀H ∈ L .
Clearly, X is also closed, convex and bounded, hence weakly compact, the function G is affine. Denote
by Z := Lp(T ,R2W ) and by Z∗ := Lq(T ,R2W ) the dual space of Lp(T ,R2W ), where 1q + 1
p = 1.
Let
D := {I ∈ Z : I(t) �R
2W+
0, a.e. onT }
and
D+ := {I ∈ Z : I(t) >R
2W+
0, a.e. onT }.
Then D is a closed convex cone in Z. The dual cone (or positive polar cone) of D is given by
D∗ := {V ∗ ∈ Z∗ :� V ∗, I �� 0, ∀ I ∈ D}.Clearly, D∗ = {V ∗ ∈ Z∗ : V ∗(t) �
R2W+
0, a.e. onT }.We will use the following notations:
I �D J ⇔ I − J ∈ D, ∀ I, J ∈ Z ,and
I >D+ J ⇔ I − J ∈ D+, ∀ I, J ∈ Z .
Then, we can rewrite the set of all feasible flows K as follows: K = {H ∈ X : G(H) �D 0}.Let H be an equilibrium flow. Set
H := {(u, V ) ∈ R×Z : u > 0, V �D 0},
Li J et al. Sci China Math April 2012 Vol. 55 No. 4 865
K(H) := {(u, V ) ∈ R×Z : u =� C(H), H −H �, V = G(H), H ∈ X},Hu := {(u, V ) ∈ R×Z : u > 0, V = 0},E := K(H)− clH
= {(u, V ) ∈ R×Z : u �� C(H), H −H �, V �D G(H), H ∈ X}= −[(f,−G)(X) + R+ × D],
where f(H) :=� C(H), H − H �. The set K(H) is called the image of the above time-dependent traffic
network at H ∈ K, while the space R× Z is the image space associated with the above time-dependent
traffic network at H ∈ K.
Similarly, one has the following:
Proposition 6.2. That H ∈ K is an equilibrium flow iff
H ∩K(H) = ∅,
iff
H ∩ E = ∅,
iff
Hu ∩ E = ∅.
Since the sets H and E have empty interior, the classic separation theorem for convex sets cannot be
applied. To this aim, we introduce the notion of quasi-relative interior and apply separation theorems
related to this notion.
Let M and N be two subsets of the topological vector space Y . Denote by coneN := {y ∈ Y : y =
λx, λ � 0, x ∈ N} the cone generated by N and by NM (x) := {x∗ ∈ Y ∗ : 〈x∗, y − x〉 � 0, ∀ y ∈ M}the normal cone to M at x, where Y ∗ is the topological dual space of Y . We say that x ∈ M is a quasi
interior point of M , denoted by x ∈ qiM , iff cl cone(M − x) = Y , or equivalently, NM (x) = {0}; we say
that x ∈ M is a quasi relative interior point of M , denoted by x ∈ qriM , iff cl cone(M − x) is a linear
subspace of Y , or equivalently, NM (x) is a linear subspace of Y ∗. For more details, see [4, 40].
It follows from the definitions above that qri{x} = {x}, ∀x ∈ Y . For any convex set M , we have that
Proof. (i) Clearly, clH = R+ × D, qriR+ = qri{u ∈ R : u > 0} = {u ∈ R : u > 0}. From [4, 20, 27], we
have qriD = D+. Since {u ∈ R : u > 0} − {u ∈ R : u > 0} = R+ − R+ = R and D − D = Z, from (ii) in
Lemma 6.1 one has qiR+ = qi{u ∈ R : u > 0} = {u ∈ R : u > 0} and qiD = D+. As a consequence, (i)
in Lemma 6.1 allows that qiH =qi (clH) = {(u, V ) ∈ R×Z : u > 0, V >D+ 0}. Since E = K(H)− clH,
it follows from (iii) in Lemma 6.1 that qiE �= ∅.(ii) Since H ∈ K is an equilibrium flow, (0, 0) ∈ E . Suppose to the contrary that (0, 0) ∈ qiE , i.e.,
cl coneE = R×Z , or equivalently, NE(0, 0) = {(0, 0)}. Let us consider (0, V ∗) ∈ R×Z∗ with
V ∗ =
(V ∗0
V ∗0
)
such that V ∗0 ∈ D∗
0 with V ∗0 �= 0, where D∗
0 = {I ∈ Lq(T ,RW ) : I(t) �R
W+
0, a.e. onT }. Then V ∗ ∈ D∗.Since
V ∗ =
(V ∗0
V ∗0
)
and G(H) =
(G1(H)
−G1(H)
)
,
for any H ∈ X, t ∈ R+, V ∈ D, we have
〈(0, V ∗), (−f(H)− t, G(H)− V )〉= 〈V ∗, G(H)− V 〉 = 〈V ∗, G(H)〉 − 〈V ∗, V 〉= 〈V ∗
0 , G1(H)〉 − 〈V ∗0 , G1(H)〉 − 〈V ∗, V 〉
� 0,
which implies that (0, V ∗) ∈ NE(0, 0), a contradiction with the assumption since (0, V ∗) �= (0, 0).
Now, we apply separation theorems related to quasi relative interior point to derive the Lagrangian-type
necessary and sufficient optimality conditions for time-dependent traffic networks.
Theorem 6.3. If H ∈ K is an equilibrium flow, then the sets H and K(H) admit a linear separation,
or equivalently, there is a vector (u, V ) ∈ B such that :
(i) u� C(H), H −H � + � V , G(H) �� 0, ∀H ∈ X ;
(ii) � V , G(H) �= 0,
where B := {(u, V ∗) ∈ R+ ×D∗ : ‖(u, V ∗)‖ = 1}.Proof. Let H ∈ K be an equilibrium flow. Since the function G is affine, −G is D-convex on X . Then
the sets H and E are convex. From Proposition 6.3, we have (0, 0) �∈ qiE . Since qiE =qriE , we obtain
(0, 0) �∈ qriE , equivalently, cl coneE is not a linear subspace of R × Z . Setting M := E and x0 := 0, it
follows from Lemma 6.2 that there exists a vector (u, V ) ∈ R×Z∗, with (u, V ) �= (0, 0), such that
u� C(H), H −H � + � V , G(H) �� uu+ � V , V �, ∀H ∈ X, ∀ (u, V ) ∈ H. (6.1)
Since H = {(u, V ) ∈ R×Z : u > 0, V �D 0}, letting (u, V ) → (0, 0) in (6.1) yields
u� C(H), H −H � + � V , G(H) �� 0, ∀H ∈ X, (6.2)
proving (i).
We declare that
0 � uu+ � V , V �, ∀ (u, V ) ∈ H. (6.3)
If there is (u0, V0) ∈ H such that uu0+ � V , V0 �< 0, then, as t→ +∞,
u(tu0)+ � V , tV0 �= t(uu0+ � V , V0 �) → −∞,
Li J et al. Sci China Math April 2012 Vol. 55 No. 4 867
which is a contradiction with (6.1) since (tu0, tV0) ∈ H.
It follows from (6.3) that (u, V ) ∈ R+ × D∗. Since (u, V ) �= (0, 0), without loss of generality, we can
suppose that (u, V ) ∈ B.Setting H := H in (6.2), we obtain
� V , G(H) �� 0,
and so
� V , G(H) �= 0,
since G(H) �D 0 and V ∈ D∗. This completes the proof. �
Theorem 6.4. Let H ∈ K. Assume that there is a vector (u, V ) ∈ B such that
u� C(H), H −H � + � V , G(H) �� 0, ∀H ∈ X,
where B is as that in Theorem 6.3. If u > 0, then H is an equilibrium flow.
Proof. The proof is similar to that in Theorem 4.6.
Acknowledgements This work was supported by National Natural Science Foundation of China (Grants Nos.
60804065, 70831005), the Key Project of Chinese Ministry of Education (Grant No. 211163), Sichuan Youth
Science and Technology Foundation and the Research Foundation of China West Normal University (Grant No.
08B075). The first author would like to gratefully thank Prof. Giannessi F and Dr. Mastroeni G for their
hospitality during the period he visited Department of Mathematics, University of Pisa, from March 19, 2011 to
March 27, 2011. The authors are in debt to Dr. Mastroeni G for his valuable suggestions on Sections 5 and 6,
which have essentially upgraded the quality of the paper. Moreover, the authors thank the editors and referees
for their insightful comments.
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