On Convex optimization without convex representation

Abstract

We consider the convex optimization problem P: min {f(x): x in K} where "f" is convex continuously differentiable, and K is a compact convex set in Rn with representation {x: g_j(x) >=0, j=1,;;,m} for some continuously differentiable functions (g_j). We discuss the case where the g_j's are not all concave (in contrast with convex programming where they all are). In particular, even if the g_j's are not concave, we consider the log-barrier function phi_\mu with parameter \mu, associated with P, usually defined for concave functions (g_j). We then show that any limit point of any sequence (x_\mu) of stationary points of phi_\mu, \mu ->0, is a Karush-Kuhn-Tucker point of problem P and a global minimizer of f on K.