and then by applying the existence of a solution of a VVI with a single- valued function. 2. Gap Functions. Let CGRl. + . In this section, we introduce the concept ...
JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS: Vol. 115, No. 2, pp. 407–417, November 2002 ( 2002)
Gap Functions and Existence of Solutions to Set-Valued Vector Variational Inequalities1 X.Q. YANG2 and J.C. YAO3 Communicated by F. Giannessi
Abstract. The variational inequality problem with set-valued mappings is very useful in economics and nonsmooth optimization. In this paper, we study the existence of solutions and the formulation of solution methods for vector variational inequalities (VVI) with set-valued mappings. We introduce gap functions and establish necessary and sufficient conditions for the existence of a solution of the VVI. It is shown that the optimization problem formulated by using gap functions can be transformed into a semi-infinite programming problem. We investigate also the existence of a solution for the generalized VVI with a setvalued mapping by virtue of the existence of a solution of the VVI with a single-valued function and a continuous selection theorem. Key Words. Vector variational inequalities, set-valued mappings, gap functions, existence of a solution, semi-infinite programming.
1. Introduction Let X and Y be topological vector spaces, and let L(X, Y) be the set of all continuous linear operators from X to Y. The value of a linear operator t∈L(X, Y) at x∈X is denoted by 〈t, x〉. Furthermore, let C⊂Y a pointed, closed, and convex cone with apex at the origin and int C ≠ ∅. Let A⊂Y. We will use the following notation: x‚ 兾 A y ⇔ yAx∉A,
∀x, y∈Y.
1
This work was partially supported by Research Grant Council of Hong Kong, Grant B-Q359, and by NCS. 2 Associate Professor, Department of Applied Mathematics, Hong Kong Polytechnic University, Kowloon, Hong Kong. 3 Professor, Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung, Taiwan.
407 0022-3239兾02兾1100-0407兾0 2002 Plenum Publishing Corporation
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Examples of the set A are C, C\{0}, and int C, where int denotes the 兾 intC are partial (topological) interior. The corresponding relations ‚ 兾 C and ‚ orderings of Y. The topological vector space Y with a partial ordering ‚ 兾C is called a partially ordered topological vector space. Let Rl be an l-dimensional vector space, and let Rl+ G{(r1 , . . . , rl)∈Rl兩ri ¤ 0, iG1, 2, . . . , l} be the nonnegative orthant of Rl. Let YGRl, and let K⊂X be a compact l subset. Assume that T: X → → L(X, R ) is a set-valued mapping with a compact set T(x) for each x. The vector variational inequality (VVI) was introduced first in Ref. 1 in finite-dimensional spaces. Consider the VVI with a set-valued mapping T, which consists in finding x¯ ∈K and ¯t ∈T(x¯) such that 〈t¯, yAx¯〉‚ 兾 int C 0,
∀y∈K.
(1)
The VVI (1) can be called weak VVI for the following reason: when T is the gradient mapping of some vector-valued function, then its solutions are related to the solution of a vector optimization problem, which by tradition is called a weakly efficient solution. The VVI (1) is an extension of the VVI with a single-valued mapping introduced in Ref. 2 and is a special case of that considered in Ref. 3. Another VVI with a set-valued mapping is that which consists in finding x¯ ∈K and t¯ ∈T(x¯) such that 〈t¯, yAx¯〉‚ 兾 C \ {0} 0,
∀y∈K.
(2)
Notwithstanding the fact that the set-valued mappings are the same, (1) and (2) are different models which interpret different equilibrium problems, since they are based on different cones, namely int C and C\{0}, respectively. Note also that t¯ of (1) does not depend on the vector y∈K. For the case where the linear operator t¯ depends on the vector y∈K, we have the next definition. Consider a generalized VVI with a set-valued mapping T, which consists in finding x¯ ∈K such that, ∀y∈K, ∃ t¯(y)∈T(x¯) satisfying 〈t¯(y), yAx¯〉‚ 兾 int C 0.
(3)
The VVI (3) can be called a generalized weak VVI. It is easy to see that any solution of (1) is a solution of (3). Obviously, the converse is not true in general. The VVI with a set-valued mapping is useful in some practical applications. In a traffic equilibrium problem, suppose that the objective is to minimize the travel time and the monetary cost simultaneously. It is known (see Ref.4) that this problem can be formulated as a vector variational inequality with a single-valued mapping when the traffic condition is fixed,
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that is, the travel time between any two given nodes is fixed. However, the traffic conditions may vary. Hence, the travel time between two given nodes is not fixed, but within a time interval. In this case, the corresponding problem may be formulated as a vector variational inequality with a set-valued mapping. There are also several reasons for studying the VVI with a set-valued mapping. The first is that the solution of the VVI with a set-valued mapping is a natural extension of the classic generalized variational inequalities studied, say, in Refs. 5–6. The second is that the concept of a solution of the VVIs (1)–(3) is related to that of nondifferentiable vector optimization problems. See for example Ref. 7, where some equivalence of some particular set-valued VVI and a nondifferentiable and nonconvex vector optimization problem is established. In this paper, we are interested in the gap functions for the VVIs (1) and (3). The motivation is the following: there have been many results regarding gap functions for optimization problems as well as classical variational inequality problems in the literature; but to the best of our knowledge, there are very few results concerning gap functions for the VVIs (see Ref. 8). Therefore, it is our aim in the first part of this paper to construct gap functions for the VVIs with set-valued maps. An interesting finding is that the optimization problem formulated using these gap functions can be transformed into a semi-infinite programming problem. Thus, this provides a solution method for the VVI with set-valued mappings via that of semiinfinite programming problems. We remark also that there are very few results concerning the existence of a solution of the generalized VVI with set-valued mappings. Some results were obtained in Ref. 3. It is our second aim in this paper to derive some existence results for the solution of the generalized VVI with set-valued maps. The technique to be used is a combination of a continuous selection of set-valued mappings and the existence of a solution of the VVI with a single-valued function, which is totally different from technique used in Ref. 3. This is accomplished by establishing a relation between the VVI with a set-valued map and the VVI with a single-valued function and then by applying the existence of a solution of a VVI with a singlevalued function.
2. Gap Functions Let CGRl+ . In this section, we introduce the concept of gap functions for a VVI with set-valued maps.
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Definition 2.1. h: K⊂X → R is said to be a gap function of the VVI (1) or the VVI (3) if (i) h(x)⁄0, ∀x∈K, (ii) 0Gh(x¯ ) if and only if x¯ is a solution of the VVI (1) or the VVI (3). Let x, y∈K and t∈T(x). Denote 〈t, y〉G((〈t, y〉)1 , . . . , (〈t, y〉)l); i.e., (〈t, y〉)i is the ith component of 〈t, y〉, iG1, . . . , l. We define two mappings g1 : KBL(X, Rl) → R and g: K → R as follows: g1 (x, t)Gmin max (〈t, yAx〉)i ,
(4)
g(x)Gmax{g1 (x, t)兩t∈T(x)}.
(5)
y ∈K 1 ⁄ i ⁄ l
Since K is compact, g1 (x, t) is well-defined. If X is a Hausdorff vector space, then g1 (x, t) is a lower semicontinuous function in x; see Corollary 22 in Ref. 9. Since T(x) is a compact set, g(x) is well-defined. For x∈K and t∈T(x), it is easy to see that g1 (x, t)Gmin max (〈t, yAx〉)i ⁄0. y ∈K 1 ⁄ i ⁄ l
Theorem 2.1. g(x) defined by (5) is a gap function of the VVI (1). Proof. It is clear that ∀x∈K, t∈T(x).
g1 (x, t)⁄0, Thus, g(x)⁄0,
∀x∈K.
If 0Gg(x¯ ), then there exists ¯t ∈T(x¯ ) such that g(x¯ , ¯t)G0. Consequently, we have min max (〈t¯, yAx¯ 〉)i G0 y ∈K 1 ⁄ i ⁄ l
if and only if, for any y∈K, max (〈t¯, yAx¯ 〉)i ¤ 0,
1⁄i⁄l
from which it follows that, for any y∈K, 〈t¯, yAx¯ 〉‚ 兾 int C 0 if and only if x¯ is a solution of the VVI (1).
䊐
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It is clear that any solution of the VVI (2) is also a solution of the VVI (1), since CGRl+ . Then, the following result is an easy consequence of Theorem 2.1. Corollary 2.1. If x¯ is a solution of the VVI (2), then 0 Gg(x¯ ). The converse of Corollary 2.1 may not hold. See Ref. 10. By Theorem 2.1, the solution of the VVI (1) is equivalent to finding a global solution x¯ to the following optimization problem: (6)
max g(x), x ∈K
with g(x¯ )G0. Recall that g(x)Gmax{g1 (x, t)兩t∈T(x)}. It is clear that the optimization problem (6) is equivalent to the following generalized semi-infinite programming problem: max
s,
x,s
s.t.
g1 (x, t)⁄ s,
∀ t∈T(x)
x∈K, which is a very important class of optimization problems; see e.g. Ref. 11. Next, let us consider the gap function for the generalized weak VVI (3). To this end, for x∈K, let Sx G{t兩t: K → T(x)}; that is, Sx is the set of all operators t from K to T(x). Let x∈K and t∈Sx . Then, t(y)∈T(x),
∀y∈K.
Define two mappings g*1 and g* as follows: g*1 (x, t)Gmin max (〈t(y), yAx〉)i , y ∈K 1 ⁄ i ⁄ l
(7)
where (〈t(y), y〉)i is the ith component of 〈t(y), y〉, iG1, . . . , l, and g*(x)Gmax{g*1 (x, t)兩t∈Sx }.
(8)
We have the following result. Theorem 2.2. g*(x) defined by (8) is a gap function of the VVI (3).
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Proof. It is clear that g*1 (x, t)⁄0,
∀x∈K, t∈Sx ,
and hence, ∀x∈K.
g*(x)⁄0,
Assume that x¯ is a solution of the VVI (3). Let y∈K. Since x¯ is a solution of the VVI (3), it follows that, for each y∈K, there is a ¯t(y)∈T(x¯ ) such that 〈t¯(y), yAx¯ 〉‚ 兾 int C 0, from which it follows that max (〈t¯(y), yAx¯ 〉)i ¤ 0.
1⁄i⁄l
Thus, an operator t¯ from K into T(x¯ ) has been defined. Then, t¯ ∈Sx¯ and max (〈t¯(y), yAx¯ 〉)i ¤ 0,
1⁄i⁄l
∀y∈K.
Hence, g*1 (x¯ , t¯)Gmin max (〈t¯(y), yAx¯ 〉)i ¤ 0. y ∈K 1 ⁄ i ⁄ l
So, g*1 (x¯ , t¯)G0. Also, it is clear that, for any t∈Sx¯ , max (〈t(y), x¯ Ax¯ 〉)i G0,
1⁄i⁄l
from which it follows that g*1 (x¯ , t)G0, and consequently, 0Gg*(x¯ ). If 0Gg*(x¯ ), then there exists t¯ ∈Sx¯ such that g*1 (x¯ , t¯)G0. Thus, min max (〈t¯(y), yAx¯ 〉)i G0. y ∈K 1 ⁄ i ⁄ l
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So we have that, for any y∈K, max (〈t¯(y), yAx¯ 〉)i ¤ 0.
1⁄i⁄l
Hence, for any y∈K, 〈t¯(y), yAx¯ 〉‚ 兾 int C 0. Therefore, x¯ is a solution of the VVI (3).
䊐
3. Existence of a Solution of VVI Let T: K → → L(X, Y) and f: K → L(X, Y). Recall that f is called a selection of T on K if f (x)∈T(x),
∀x∈K.
Furthermore, the function f is called a continuous selection of T on K if f is a selection of T on K and it is also continuous on K. Assume that, for each x∈K, C(x) is a closed and convex cone with apex at the origin in Y. In this section, we shall consider the generalized VVI with a set-valued mapping T, which consists in finding x¯ ∈K and ¯t ∈T(x¯ ) such that 〈t¯, yAx¯ 〉‚ 兾 int C (x) 0,
∀y∈K.
(9)
Recall the VVI with a single-valued mapping f (Ref. 3), which consists in finding x¯ ∈ such that 〈 f (x¯ ), yAx¯ 〉‚ 兾 intC (x)0,
∀y∈K.
(10)
The following lemma provides a relation between (9) and (10), which is the key to the establishment of the existence of a solution of the generalized VVI with a set-valued mapping via that of the VVI with a singlevalued mapping. Lemma 3.1. If f is a selection of T on K, then every solution of the VVI (10) is a solution of the VVI (9). Proof. Assume that x¯ ∈K is a solution of the VVI (10). That is, 〈 f (x¯ ), yAx¯ 〉‚ 兾 intC (x)0,
∀y∈K.
Let t¯ Gf (x¯ ). Then, t¯ ∈T(x¯ ) and 〈t¯, yAx¯ 〉‚ 兾 intC (x)0,
∀y∈K.
Therefore, x¯ ∈K is a solution of the VVI (9).
䊐
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We now apply this lemma to derive the existence of a solution of the VVI (9). Let W: K → → Y be a set-valued mapping. The graph of W on K is defined by
G (W )G{(x, y)兩y∈W(x), x∈K}. Definition 3.1. See Ref. 3. The set-valued mapping T: K → → L(X, Y) is C(x)-pseudomonotone if, for every pair of points x∈K, y∈K and for all t′∈T(x), t′′∈T(y), we have that 〈t′, yAx〉‚ 兾 int C (x) 0 implies〈t′′, yAx〉‚ 兾 intC (x) 0. Definition 3.2. See Ref. 3. f: K → L(X, Y) is C(x)-pseudomonotone if, for every pair of points x∈K, y∈K, we have that 〈 f (x), yAx〉‚ 兾 int C (x) 0 implies 〈 f (y), yAx〉‚ 兾 intC (x) 0. The following lemma provides a connection between the pseudomonotonicity properties of a set-valued mapping and that of its selection. Its proof follows directly from the definitions and is omitted. Lemma 3.2. Let T: K → → L(X, Y) be a set-valued mapping, and let f be a selection of T; i.e., f (x)∈T(x),
∀x∈K.
If T is C(x)-pseudomonotone, then f is also C(x)-pseudomonotone. Now, we can state the main result of this section as follows. Theorem 3.1. Let X and Y be Banach spaces, and let K be a nonempty weakly compact and convex subset of X. Let C: K → → Y be a set-valued mapping such that, for each x∈K, C(x) is a proper, closed, and convex cone with apex at the origin and int C(x) ≠ ∅; and let W: K → → Y defined by W(x)GY \ (−int C(x)) be such that the graph G (W ) of W is weakly closed in XBY. Suppose that T(x) is a nonempty set of L(X, Y), for each x∈K. Assume further that: (P) T: K → → L(X, Y) is C(x)-pseudomonotone; (C) there is a continuous selection f of T on K. Then, there exists a solution to the VVI (9).
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Remark 3.1. A variant of Theorem 3.1 without Assumption (C) was proved in Ref. 3. Here, we obtain a simpler proof by using Lemmas 3.1 and 3.2 and a result in Ref. 3. Proof of Theorem 3.1. By the assumption, there is a continuous function f: K → L(X, Y) such that f (x)∈T(x),
∀x∈K.
It follows from Lemma 3.2 that f is also C(x)-pseudomonotone. Then, all the conditions of Theorem 3.1 in Ref. 3 are satisfied. Thus, there is a solution to the following problem: Find x¯ ∈K, such that 〈 f (x¯ ), yAx〉‚ 兾 intC (x¯ ) 0,
∀y∈K.
So x¯ is a solution of the VVI (10). By Lemma 3.1, x¯ is a solution of the VVI (9). 䊐 Corollary 3.1. Assume that all the conditions in Theorem 3.2 are satisfied, except Condition (C), which is replaced by (C’) T: K → → L(X, Y) is continuous on K. Then, there exists a solution to the VVI (9). Proof. It follows from the selection theorem of Ref. 12 that there is a continuous selection f: K → L(X, Y) such that f (x)∈T(x),
∀x∈K.
By Theorem 3.1, the conclusion holds.
䊐
Recall that the mapping T: K → → L(X, Y) is said to be generalized vcoercive on K (Ref. 3) if there exist a weakly compact subset B of X and y0 ∈B∩K, such that, for every t∈T(x), 〈t, y0Ax〉∈AintC(x),
for all x∈K\B.
We now have the following result for the existence of a solution of the VVI (9) where the set K need not be weakly compact. Theorem 3.2. Let X, Y, C, W, and G (W ) be the same as in Theorem 3.1. Let K be a nonempty, closed, and convex subset of X. Suppose that T(x) is a nonempty set of Y, for each x∈K. Assume further that:
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(P) T: K → L(X, Y) is C(x)-pseudomonotone and generalized û-coercive on K; (C) there is a continuous selection f of T on K. Then, there exists a solution to the VVI (9).
3.1.
Proof. The result follows from Theorem 3.2 of Ref. 3 and Lemma 䊐
4. Conclusions In this paper, we have introduced the concept of gap function for both the VVIs (1) and (3). We have derived also several existence results for the solutions to the VVI (9). We remark that Ref. 3 is the first paper concerning the existence of a solution to the VVI (2). Here, we have used the method of a continuous selection, which is totally different from the scalarization method used in Ref. 3. However, it is possible to derive more existence results of solutions of the VVI with set-valued mappings under weaker continuity assumption if one can to establish the following selection theorem: if T: K → → L(X, Y) is a u-hemicontinuous set-valued function, then there is a u-hemicontinuous function f: K → L(X, Y) such that f (x)∈T(x), ∀x∈K.
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