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 that arises from the tikhonov regularization method for pseudomonotone equilibrium problems. © copyright 2011 bui van dinh and le dung muu.