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Abstract and Applied Analysis
Volume 2008 (2008), Article ID 531361, 7 pages
On Gap Functions for Quasi-Variational Inequalities
Department of Mechanical Science and Engineering, Graduate School of Engineering, Nagoya University, Furo-cho, Chikusa, Nagoya 464-8603, Japan
Received 14 September 2007; Revised 18 December 2007; Accepted 8 January 2008
Academic Editor: Nobuyuki Kenmochi
Copyright © 2008 Kouichi Taji. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
For variational inequalities, various merit functions, such as the gap function, the regularized gap function, the D-gap function and so on, have been proposed. These functions lead to equivalent optimization formulations and are used to optimization-based methods for solving variational inequalities. In this paper, we extend the regularized gap function and the D-gap functions for a quasi-variational inequality, which is a generalization of the variational inequality and is used to formulate generalized equilibrium problems. These extensions are shown to formulate equivalent optimization problems for quasi-variational inequalities and are shown to be continuous and directionally differentiable.
In this paper, we consider merit functions for quasi-variational inequalities, denoted by QVIP, to find a vector such that where is a mapping, the symbol denotes the inner product in , and is a set-valued mapping of which is a closed convex set in for each . When the mapping is a constant closed convex set for all , QVIP reduces to a well-known variational inequality [1, 2]:
QVIP is used to study and formulate generalized equilibrium problems, such as the generalized Nash equilibrium problem in which a strategy set of each player varies according to the other players' strategies [3, 4]. For variational inequalities, various merit functions, such as the gap function, the regularized gap function , the D-gap functions [6, 7], and so on, have been proposed. These functions are used to make an equivalent optimization formulation for a variational inequality, and based on these formulations, several optimization-based methods are proposed for solving variational inequalities [8, 9].
Fukushima  has proposed gap functions for QVIP, which lead to equivalent optimization formulation for QVIP. In this paper, we extend the results of Fukushima in two directions. One is to show the directional differentiability under more general setting (Theorem 2.5) and to give one sufficient condition for stationary point to be a solution (Proposition 2.7). The other is to propose a so-called D-gap function for QVIP, which leads to an unconstrained minimization optimization formulation for QVIP, and to show its directional differentiability (Section 3).
Notations used in this paper are summarized as follows. The superscript denotes a transpose of vector or matrix, and denotes the Euclidean norm in defined by .
2. Regularized Gap Function for Qvip
In this section, we first generalize the regularized gap function for a variational inequality to a quasi-variational inequality and show its properties.
The (generalized) regularized gap functions for QVIP (1.1) are defined as where an is a positive constant and a function satisfies the following conditions.
(C1) is continuously differentiable on .(C2) is nonnegative on and if and only if .(C3) is strongly convex uniformly in , that is, there is a such that
Remark 2.1. It is easy to
verify that a function for a positive
definite symmetric matrix satisfies the
above conditions (C1)(C3). In this
case, the function (2.1) reduces
with . This is just a regularized gap function  originally proposed for
variational inequalities (1.2).
By the strong convexity of and the closed convexity of , the maximum in (2.1) is uniquely attained and is given by the unique solution of the following mathematical programming problem: and the function (2.1) is written as
Lemma 2.2. A point is a solution to QVIP (1.1) if and only if .
Proof. Since the optimization problem (2.4) is convex with respect to , the point is a solution
to (2.4) if and
which is equivalent
Then by substituting a solution of QVIP
(1.1) to , we have
On the other hand, from the
condition (C3) we have
where the last inequality
follows from the condition (C2). The above two inequalities lead to .
Conversely, suppose that . Then the inequality (2.6) reduces to which shows that is a solution to (1.1).
The next theorem shows that the function (2.1) or (2.5) leads to an equivalent optimization problem for quasi-variational inequalities. The theorem is inherently equivalent to [10, Theorem 2], but for completeness, we provide its proof. We note that our proof is more elementary and simpler than that of [10, Theorem 2].
Theorem 2.3. Let be the function defined by (2.1) or (2.5). Then for all . Furthermore, and if and only if is a solution to QVIP (1.1). Hence, problem (1.1) is equivalent to finding a global optimal solution to the problem:
Proof. The first assertion is obvious from the definition
(2.1) and (C2).
To prove the last assertion, suppose that is a solution
to QVIP. Then, we have
Therefore, from the definition
(2.1), we have .
For the “only if” part, we consider the regularized gap function for fixed , Then, it follows from and that , which implies that is a solution to the variational inequality  This means that is a solution to QVIP.
The next theorem gives a sufficient condition for the continuity of the function (2.1).
Proof. In a similar way to , is shown to be continuous in . Therefore, the function is also continuous.
When the set-valued mapping is expressed as a finite number of convex inequalities, such that where the functions are continuous with respect to and , and are convex for each , then one sufficient condition for the continuity of the set-valued mapping is that Slater's constraint qualification holds, that is, for each , there exists a vector (possibly depending on ) such that In this case, is also continuous and satisfies the KKT condition:
Unfortunately, the function defined by (2.1) or (2.5) is not necessarily differentiable. However, the next theorem gives one sufficient condition of the directional differentiability of the function with the set given by (2.15).
Theorem 2.5. Let the mapping be continuously differentiable. Let also the set-valued mapping be defined as (2.15), where the functions are continuous with respect to and , and is convex for each . If Slater's constraint qualification (2.16) holds, then the function defined by (2.1) or (2.5) is directionally differentiable in any direction , and its directional derivative is given by where is defined by
Proof. This directly follows from [14, Theorem 2].
Proposition 2.7. Suppose that . Suppose also that is positive definite and for all and . If the function used in the regularized gap function (2.1) or (2.5) is , where is differentiable strongly convex function, then a stationary point of the problem (2.11) is a solution to QVIP (1.1).
Proof. It suffices to show that is a solution
to QVIP if the following inequality holds:
It is easy to see that satisfies the conditions (C1)(C3) and that . Then from the definition of directional derivative (2.18) and the KKT condition (2.17) for , we have Since functions are convex with respect to , we have Then we have from the fact and (2.17) that Therefore, it follows from (2.21) and the assumption that which leads to from the positive definiteness of . This shows from Lemma 2.2 that is a solution to QVIP.
Remark 2.8. When the functions are all defined as linearized approximation of convex functions at , that is, , then we have Since the Hesse matrix is positive semidefinite from the convexity of , the assumption of Proposition 2.7 is satisfied. This result has been already obtained by Taji and Fukushima  for this setting, and the above proposition is considered as a generalization in some sense.
3. D-Gap Function for Qvip
For , we consider the function defined by This is a so-called D-gap function and is originally introduced for the variational inequality (1.2) by Peng . D-gap functions are shown to construct a differentiable equivalent unconstrained optimization formulation for VIPs.
We have the next proposition.
Proposition 3.1. For , for the function defined by (3.1),
Proof. From the definition (3.1) and the fact that , we have This shows the left-side hand of the inequality. The right-hand side is shown in a similar way.
This proposition establishes the equivalence between a QVIP and the unconstrained minimization of a D-gap function .
Theorem 3.2. Let the function be defined as (3.1). Then, for , for all . Moreover, if and only if is a solution to QVIP. Hence, the problem (1.1) is equivalent to finding a global optimal solution to the unconstrained minimization problem:
Theorem 3.3. Suppose that the set-valued mapping is continuous in , then the D-gap function defined by (3.1) is continuous in . Moreover, if the set-valued mapping is defined by (2.15) and if Slater's constraint qualification (2.16) holds, then the D-gap function is directionally differentiable in all direction .
The author would like to thank an anonymous referee for introducing the paper .
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