Conditions for Exact Convex Relaxation and No Spurious Local Optima

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.