- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

Abstract and Applied Analysis

Volume 2008 (2008), Article ID 531361, 7 pages

http://dx.doi.org/10.1155/2008/531361

## 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.

#### Abstract

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.

#### 1. Introduction

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 [5], 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 [10] 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
to
with . This is just a regularized gap function [5] 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
only if
which is equivalent
to
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 [11]
This means that is a solution
to QVIP.

The next theorem gives a sufficient condition for the continuity of the function (2.1).

Theorem 2.4. *Let be the function
defined by (2.1) or (2.5). If the set-valued mapping
is continuous
with respect to
in the sense of
set-valued mapping [12], then is also
continuous in . *

*Proof. * In a similar way to [13], 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].

*Remark 2.6. *Fukushima
[10, Theorem 3] has
also proven the directional differentiability of for the
function (2.3)
with polyhedral
convex. This situation is a special case of this theorem.

The next proposition gives a sufficient condition for a stationary point of the equivalent optimization problem (2.11) to be a solution of QVIP (1.1) with the set given by (2.15).

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 [13] 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 [6]. 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:
*

*Proof. *The first half follows directly from Proposition 3.1. The last half also
follows from Lemma 2.2 and Proposition 3.1.

The continuity and the directional differentiability of the D-gap function (3.1) directly follow from those of the regularized gap function (2.1).

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 . *

*Proof. *These
results directly follow from the definition of the D-gap function and Theorems 2.4 and 2.5.

#### Acknowledgment

The author would like to thank an anonymous referee for introducing the paper [10].

#### References

- F. Facchinei and J.-S. Pang,
*Finite-Dimensional Variational Inequalities and Complementarity Problems. Vol. I*, Springer Series in Operations Research, Springer, New York, NY, USA, 2003. - F. Facchinei and J.-S. Pang,
*Finite-Dimensional Variational Inequalities and Complementarity Problems. Vol. II*, Springer Series in Operations Research, Springer, New York, NY, USA, 2003. - M. Fukushima, “Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems,”
*Mathematical Programming*, vol. 53, no. 1, pp. 99–110, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Fukushima, “A class of gap functions for quasi-variational inequality problems,”
*Journal of Industrial and Management Optimization*, vol. 3, no. 2, pp. 165–171, 2007. View at Google Scholar - P. T. Harker, “Generalized Nash games and quasi-variational inequalities,”
*European Journal of Operational Research*, vol. 54, no. 1, pp. 81–94, 1991. View at Publisher · View at Google Scholar - W. Hogan, “Directional derivatives for extremal-value functions with applications to the completely convex case,”
*Operations Research*, vol. 21, pp. 188–209, 1973. View at Google Scholar - W. Hogan, “Point-to-set maps in mathematical programming,”
*SIAM Review*, vol. 15, pp. 591–603, 1973. View at Publisher · View at Google Scholar - J.-S. Pang and M. Fukushima, “Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games,”
*Computational Management Science*, vol. 2, no. 1, pp. 21–56, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J.-M. Peng, “Equivalence of variational inequality problems to unconstrained minimization,”
*Mathematical Programming*, vol. 78, no. 3, pp. 347–355, 1997. View at Publisher · View at Google Scholar - J.-M. Peng and M. Fukushima, “A hybrid Newton method for solving the variational inequality problem via the D-gap function,”
*Mathematical Programming*, vol. 86, no. 2, pp. 367–386, 1999. View at Publisher · View at Google Scholar · View at MathSciNet - K. Taji and M. Fukushima, “A new merit function and a successive quadratic programming algorithm for variational inequality problems,”
*SIAM Journal on Optimization*, vol. 6, no. 3, pp. 704–713, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K. Taji, M. Fukushima, and T. Ibaraki, “A globally convergent Newton method for solving strong monotone variational inequalities,”
*Mathematical Programming*, vol. 58, no. 3, pp. 369–383, 1993. View at Publisher · View at Google Scholar - N. Yamashita, K. Taji, and M. Fukushima, “Unconstrained optimization reformulations of variational inequality problems,”
*Journal of Optimization Theory and Applications*, vol. 92, no. 3, pp. 439–456, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. H. Wu, M. Florian, and P. Marcotte, “A general descent framework for the monotone variational inequality problem,”
*Mathematical Programming*, vol. 61, no. 3, pp. 281–300, 1993. View at Publisher · View at Google Scholar