Regularizing the abstract convex program

Characterizations of optimality for the abstract convex program $\text{μ =}\text{inf}\text{{p(x) : g(x) ? ?S, x ? Ω}}$ (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).