Deterministic Guarantees for Burer-Monteiro Factorizations of Smooth Semidefinite Programs

Nicolas Boumal, Vladislav Voroninski, Afonso Bandeira

Research output: Contribution to journalArticle

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 × 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.

Original languageEnglish (US)
JournalCommunications on Pure and Applied Mathematics
DOIs
StatePublished - Jan 1 2019

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Semidefinite Program
Factorization
Cost functions
Equality Constraints
Global Optimum
Eigenvalues and eigenfunctions
Cost Function
Synchronization
Max-cut
Trust Region Subproblem
Second-order Optimality Conditions
Quadratic Constraint
Non-convexity
Quadratic Optimization
Positive Semidefinite Matrix
Nonconvex Optimization
Nonconvex Problems
Necessary Optimality Conditions
Smooth Manifold
Quadratic Function

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

Deterministic Guarantees for Burer-Monteiro Factorizations of Smooth Semidefinite Programs. / Boumal, Nicolas; Voroninski, Vladislav; Bandeira, Afonso.

In: Communications on Pure and Applied Mathematics, 01.01.2019.

Research output: Contribution to journalArticle

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