On representations of the feasible set in convex optimization

TL;DR

This paper demonstrates that in convex optimization, the KKT conditions are necessary and sufficient for optimality regardless of the convexity of the constraint functions, emphasizing the importance of the feasible set's geometry.

Contribution

It shows that under mild conditions, the KKT conditions characterize optimality even when the constraint functions are not convex, highlighting the role of the feasible set's geometry.

Findings

KKT conditions are necessary and sufficient for optimality under mild assumptions.

The representation of the feasible set does not affect optimality conditions in convex problems.

The geometry of the feasible set is the key factor in optimality analysis.

Abstract

We consider the convex optimization problem $\min \{f(x) : g_j(x)\leq 0, j=1,...,m\}$ where $f$ is convex, the feasible set K is convex and Slater's condition holds, but the functions $g_j$ are not necessarily convex. We show that for any representation of K that satisfies a mild nondegeneracy assumption, every minimizer is a Karush-Kuhn-Tucker (KKT) point and conversely every KKT point is a minimizer. That is, the KKT optimality conditions are necessary and sufficient as in convex programming where one assumes that the $g_j$ are convex. So in convex optimization, and as far as one is concerned with KKT points, what really matters is the geometry of K and not so much its representation.