Deterministic guarantees for Burer-Monteiro factorizations of smooth semidefinite programs

TL;DR

This paper establishes conditions under which the non-convex Burer-Monteiro factorization approach guarantees global optimality for certain smooth semidefinite programs, connecting local optimality conditions to global solutions.

Contribution

It proves that for smooth constraint sets, first- and second-order optimality conditions are sufficient for global optimality in Burer-Monteiro factorizations when the factor size is large enough, and nearly so for smaller sizes.

Findings

Global optima correspond to solutions of the original SDP under certain conditions.

First- and second-order optimality conditions are sufficient for global optimality in the factorized problem.

Results apply to various SDP relaxations including Max-Cut, orthogonal-cut, and problems in stochastic block modeling.

Abstract

We 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 $n \times p$ instead, such that $X = YY^T$. This ensures positive semidefiniteness at no cost and can reduce the dimension of the problem if $p$ is small, but results in a non-convex 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 non-convexity, 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 $X = YY^T$ 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.