Deterministic Guarantees for Burer‐Monteiro Factorizations of Smooth Semidefinite Programs

Abstract

AbstractWe consider semidefinite programs (SDPs) with equality constraints. The variable to be optimized is a positive semidefinite matrix X of size n. Following the Burer‐Monteiro approach, we optimize a factor Y of size n × p instead, such that X = YYT. This ensures positive semidefiniteness at no cost and can reduce the dimension of the problem if p is small, but results in a nonconvex optimization problem with a quadratic cost function and quadratic equality constraints in Y. In this paper, we show that if the set of constraints on Y regularly defines a smooth manifold, then, despite nonconvexity, first‐ and second‐order necessary optimality conditions are also sufficient, provided p is large enough. For smaller values of p, we show a similar result holds for almost all (linear) cost functions. Under those conditions, a global optimum Y maps to a global optimum X = YYT of the SDP. We deduce old and new consequences for SDP relaxations of the generalized eigenvector problem, the trust‐region subproblem, and quadratic optimization over several spheres, as well as for the Max‐Cut and Orthogonal‐Cut SDPs, which are common relaxations in stochastic block modeling and synchronization of rotations. © 2019 Wiley Periodicals, Inc.