Regularizing the abstract convex program

Abstract

Characterizations of optimality for the abstract convex program μ = inf{p(x) : g(x) ϵ −S, x ϵ Ω} (P) where S is an arbitrary convex cone in a finite dimensional space, Ω is a convex set, and p and g are respectively convex and S-convex (on Ω), were given in [10]. These characterizations hold without any constraint qualification. They use the “minimal cone” S f of (P) and the cone of directions of constancy D g = ( S f ). In the faithfully convex case these cones can be used to regularize (P), i.e., transform (P) into an equivalent program (P r) for which Slater's condition holds. We present an algorithm that finds both S f and D g =( S f ). The main step of the algorithm consists in solving a particular complementarity problem. We also present a characterization of optimality for (P) in terms of the cone of directions of constancy of a convex functional D φg = rather than D g =( S f ).