  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.
