In this paper we deal with strong Fenchel duality for infinite dimensional optimization problems where both feasible set and objective function are evenly convex. To this aim, via perturbation approach, a conjugation scheme for evenly convex functions, based on generalized convex conjugation, is used. The key is to extend some well-known results from convex analysis, involving the sum of the epigraphs of two conjugate functions, the infimal convolution and the sum formula of ε-subdifferentials for lower semicontinuous convex functions, to this more general framework.Mathematical subject classification: 52A20, 26B25.

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$EVWUDFW The evenly convex hull of a given set is the intersection of all the open halfspaces which contain such set (hence the convex hull is contained in the evenly convex hull). This paper deals with ¿nite dimensional linear systems containing strict inequalities and (possibly) weak inequalities as well as equalities. The number of inequalities and equalities in these systems is arbitrary (possibly in¿nite). For such kind of systems a consistency theorem is provided and those strict inequalities (weak inequalities, equalities) which are satis¿ed for every solution of a given system are characterized. Such results are formulated in terms of the evenly convex hull of certain sets which depend on the coef¿cients of the system. $06 FODVVL¿FDWLRQ15A39 90C34 52A40..H\ZRUGV convex programming linear programming evenly convex hull existence theorems, consequence relations.

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In this paper we study how Lagrange duality is connected to optimization problems whose objective function is the difference of two convex functions, briefly called DC problems. We present two Lagrange dual problems, each of them obtained via a different approach. While one of the duals corresponds to the standard formulation of the Lagrange dual problem, the other is written in terms of conjugate functions. When one of the involved functions in the objective is evenly convex, both problems are equivalent, but this relation is no longer true in the general setting. For this reason, we study conditions ensuring not only weak, but also zero duality and strong duality between the primal and one of the dual problems written using conjugate functions. For the other dual, and due to the fact that weak duality holds by construction, we just develop conditions for zero duality gap and strong duality between the primal DC problem and its (standard) Lagrange dual problem. Finally, we characterize weak and strong duality together with zero duality gap between the primal problem and its Fenchel-Lagrange dual following techniques used throughout the manuscript.