Description:
In this article, we use abstract convexity results to study augmented dual problems for (nonconvex) constrained optimization problems. We consider a nonincreasing function f that is lower semicontinuous at 0 and establish its abstract convexity at 0 with respect to a set of elementary functions defined by nonconvex augmenting functions. We consider three different classes of augmenting functions: nonnegative augmenting functions, bounded-below augmenting functions, and unbounded augmenting functions. We use the abstract convexity results to study augmented optimization duality without imposing boundedness assumptions.
Description:
A subset B of R-+(n) is B-convex if for all x, y is an element of B and all t is an element of [0, 1] one has max (tx, y) is an element of B. These sets were first investigated in [1, 2]. In this paper, we examine radiant B-convex sets and also introduce and study B-convex functions.