Duality and optimality conditions for generalized equilibrium problems involving DC functions

We consider a generalized equilibrium problem involving DC functions which is called (GEP). For this problem we establish two new dual formulations based on Toland-Fenchel-Lagrange duality for DC programming problems. The first one allows us to obtain a unified dual analysis for many interesting problems. So, this dual coincides with the dual problem proposed by Martinez-Legaz and Sosa in [23] for equilibrium problems in the sense of Blum and Oettli. Furthermore it is equivalent to Mosco’s dual problem [25] when applied to a variational inequality problem. The second dual problem generalizes to our problem another dual scheme that has been recently introduced by Jacinto and Scheimberg in [18] for convex equilibrium problems. Through these schemes, as by products, we obtain new optimality conditions for (GEP) and also, gap functions for (GEP), which cover the ones in [1, 2] for variational inequalities and standard convex equilibrium problems. These results, in turn, when applied to DC and convex optimization problems with convex constraints (considered as special cases of (GEP)) lead to Toland-Fenchel-Lagrange duality for DC problems in [13, 14], Fenchel-Lagrange and Lagrange dualities for convex problems as in [2, 7, 20]. Besides, as consequences of the main results, we obtain some new optimality conditions for DC and convex problems.