Asymptotic dual conditions characterizing optimality for infinite convex programs
Journal of Optimization Theory and Applications
Gap Functions for Equilibrium Problems
Journal of Global Optimization
A new geometric condition for Fenchel's duality in infinite dimensional spaces
Mathematical Programming: Series A and B
Liberating the Subgradient Optimality Conditions from Constraint Qualifications
Journal of Global Optimization
On Nash---Cournot oligopolistic market equilibrium models with concave cost functions
Journal of Global Optimization
Mathematical Programming: Series A and B - Series B - Special Issue: Well-posedness, stability and related topics
SIAM Journal on Optimization
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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 (J Glob Optim 25:311---319, 2006) for equilibrium problems in the sense of Blum and Oettli. Furthermore it is equivalent to Mosco's dual problem (Mosco in J Math Anal Appl 40:202---206, 1972) 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 (Optimization 57:795---805, 2008) 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 Antangerel et al. (J Oper Res 24:353---371, 2007, Pac J Optim 2:667---678, 2006) 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 Dinh et al. (Optimization 1---20, 2008, J Convex Anal 15:235---262, 2008), Fenchel-Lagrange and Lagrange dualities for convex problems as in Antangerel et al. (Pac J Optim 2:667---678, 2006), Bot and Wanka (Nonlinear Anal to appear), Jeyakumar et al. (Applied Mathematics research report AMR04/8, 2004). Besides, as consequences of the main results, we obtain some new optimality conditions for DC and convex problems.