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...

• Type: Journal
• Year: 2010
• Where: JGO
• Authors: N. Dinh, Jean-Jacques Strodiot, Van Hien Nguyen
• Keywords: Dual Problem, Equilibrium Problems, JGO 2010, Problems