TY - JOUR

T1 - Deterministic Guarantees for Burer-Monteiro Factorizations of Smooth Semidefinite Programs

AU - Boumal, Nicolas

AU - Voroninski, Vladislav

AU - Bandeira, Afonso S.

N1 - Publisher Copyright:
© 2019 Wiley Periodicals, Inc.

PY - 2020/3/1

Y1 - 2020/3/1

N2 - 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 × 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.

AB - 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 × 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.

UR - http://www.scopus.com/inward/record.url?scp=85066896631&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85066896631&partnerID=8YFLogxK

U2 - 10.1002/cpa.21830

DO - 10.1002/cpa.21830

M3 - Article

AN - SCOPUS:85066896631

VL - 73

SP - 581

EP - 608

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

IS - 3

ER -