On Penalty and Gap Function Methods for Bilevel Equilibrium Problems

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-$\nabla$-monotonicity concept from $\nabla$-monotonicity and prove that under pseudo-$\nabla$-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.