On convex optimization without convex representation

Abstract

We consider the convex optimization problem \({{\rm {\bf P}}:{\rm min}_{\rm {\bf x}} \{f({\rm {\bf x}})\,:\,{\rm {\bf x}}\in{\rm {\bf K}}\}}\) where f is convex continuously differentiable, and \({{\rm {\bf K}}\subset{\mathbb R}^n}\) is a compact convex set with representation \({\{{\rm {\bf x}}\in{\mathbb R}^n\,:\,g_j({\rm {\bf x}})\geq0, j = 1,\ldots,m\}}\) for some continuously differentiable functions (gj). We discuss the case where the gj’s are not all concave (in contrast with convex programming where they all are). In particular, even if the gj are not concave, we consider the log-barrier function \({\phi_\mu}\) with parameter μ, associated with P, usually defined for concave functions (gj). We then show that any limit point of any sequence \({({\rm {\bf x}}_\mu)\subset{\rm {\bf K}}}\) of stationary points of \({\phi_\mu, \mu \to 0}\) , is a Karush–Kuhn–Tucker point of problem P and a global minimizer of f on K.