Existence theory for finite-dimensional pseudomonotone equilibrium problems

This article was originally written to be delivered during a short course, but because of its finite-dimensional setting, it can also be addressed to nonspecialists and those only possessing a basic background on real analysis and mathematical programming. Thus, it should be conceived as an introduction to the existence theory for equilibrium ( general optimization) problems including minimization and variational inequality under the assumption of no compactness and possibly having an unbounded solution set. Nevertheless, some of the results that are established here have not appeared elsewhere. Our approach is based on the asymptotic description of the functions and constraint set. In particular, this allows us to give various characterizations of the nonemptiness ( and, in another case, boundedness) of the solution set. Several applications to convex problems in mathematical programming are given, along with applications to vector equilibrium problems. A guide to historical references is also provided.