On Convex optimization without convex representation

TL;DR

This paper explores convex optimization problems where the constraints are not necessarily concave, demonstrating that stationary points of the log-barrier function converge to global minimizers even without convex constraint functions.

Contribution

It extends the understanding of barrier methods to non-concave constraint functions, showing convergence to global solutions in broader settings.

Findings

Limit points of stationary points of the log-barrier function are KKT points.

Such limit points are also global minimizers of the original problem.

The results hold even when the constraint functions are not concave.

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.