Conditions for Exact Convex Relaxation and No Spurious Local Optima

TL;DR

This paper investigates the conditions under which non-convex optimization problems, like optimal power flow, can be solved exactly through relaxation and have no spurious local optima, explaining their empirical success.

Contribution

It identifies sufficient and necessary conditions for non-convex problems to have exact relaxation and no spurious local optima, enhancing understanding of their solvability.

Findings

Conditions for exact relaxation established

Conditions for absence of spurious local optima identified

Explains empirical success of local algorithms in OPF

Abstract

Non-convex optimization problems can be approximately solved via relaxation or local algorithms. For many practical problems such as optimal power flow (OPF) problems, both approaches tend to succeed in the sense that relaxation is usually exact and local algorithms usually converge to a global optimum. In this paper, we study conditions which are sufficient or necessary for such non-convex problems to simultaneously have exact relaxation and no spurious local optima. Those conditions help us explain the widespread empirical experience that local algorithms for OPF problems often work extremely well.