Abstract and Applied Analysis
Volume 2008 (2008), Article ID 531361, 7 pages
Research Article

On Gap Functions for Quasi-Variational Inequalities

Kouichi Taji

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.