Evenly Convex Functions

Abstract

In this chapter we introduce evenly convex functions as those whose epigraphs are evenly convex sets, and develop a duality theory for nonlinear programming problems involving evenly convex functions, that is, evenly convex optimization problems. In Sect. 4.1 we present the main properties of this class of convex functions that contains the important subclass of lower semicontinuous convex functions, whose relevance in convex analysis comes from the fact that the Fenchel conjugacy is an involution on most of them. More precisely, any proper lower semicontinuous convex function coincides with its biconjugate. In Sects. 4.2 and 4.3 we introduce the evenly convex hull of a function and appropriate conjugation schemes for evenly convex functions, respectively. Finally, in Sect. 4.4, we use the perturbational approach for developing the so-called c-conjugate duality theory, providing closedness-type regularity conditions. These conditions will be expressed in terms of the even convexity of the involved functions, for both strong and stable strong duality for convex optimization problems.