On Penalty and Gap Function Methods for Bilevel Equilibrium Problems

On Penalty and Gap Function Methods for Bilevel Equilibrium Problems

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...

Proximal methods for a class of bilevel monotone equilibrium problems

We consider a bilevel problem involving two monotone equilibrium bifunctions and we show that this problem can be solved by a simple proximal method. Under mild conditions, the weak convergence of the sequences generated by the algorithm is obtained. Using this result we obtain corollaries which improve several corresponding results in this field.

Interior-Point Algorithms, Penalty Methods and Equilibrium Problems

In this paper we consider the question of solving equilibrium problems—formulated as complementarity problems and, more generally, mathematical programs with equilibrium constraints (MPEC’s)—as nonlinear programs, using an interior-point approach. These problems pose theoretical difficulties for nonlinear solvers, including interior-point methods. We examine the use of penalty methods to get ar...

Iterative Methods for Equilibrium Problems, Variational Inequalities and Fixed Points

Gap Functions for Equilibrium Problems

The theory of gap functions, developed in the literature for variational inequalities, is extended to a general equilibrium problem. Descent methods, with exact an inexact line-search rules, are proposed. It is shown that these methods are a generalization of the gap function algorithms for variational inequalities and optimization problems.