Journal of Applied Mathematics
Volume 2011 (2011), Article ID 646452, 14 pages
http://dx.doi.org/10.1155/2011/646452
Research Article

On Penalty and Gap Function Methods for Bilevel Equilibrium Problems

Bui Van Dinh1 and Le Dung Muu2

1Faculty of Information Technology, Le Quy Don University, Hanoi, Vietnam
2Control and Optimization Department, Institute of Mathematics, VAST, 18 Hoang Quoc Viet, Cau Giay, Hanoi 10307, Vietnam

Received 9 June 2011; Accepted 15 August 2011

Academic Editor: Ya Ping Fang

Copyright © 2011 Bui Van Dinh and Le Dung Muu. 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

We consider bilevel pseudomonotone equilibrium problems. We use a penalty function to convert a bilevel problem into one-level ones. We generalize a pseudo- ∇ -monotonicity concept from ∇ -monotonicity and prove that under pseudo- ∇ -monotonicity property any stationary point of a regularized gap function is a solution of the penalized equilibrium problem. As an application, we discuss a special case that arises from the Tikhonov regularization method for pseudomonotone equilibrium problems.