Deterministic Guarantees for Burer‐Monteiro Factorizations of Smooth Semidefinite Programs

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 D Y Y T . 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 D Y Y 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.

Rank Optimality for the Burer--Monteiro Factorization

“…• We show in Theorem 1 that a minor improvement over Inequality 1.1 is possible: Under a stronger geometrical assumption than in [Boumal, Voroninski, and Bandeira, 2018] (but still reasonable), all second-order critical points of Problem (Factorized SDP) are global minimizers, for almost any C, as soon as p(p + 1) 2 + p > m. (1.2) This improves over Inequality (1.1), but only in the low order terms.…”

“…While these works shed light on several important practical situations, they do not provide a general theory on when local optimization algorithms can solve Problem (Factorized SDP). With no restrictive assumptions, essentially the only result is due to Boumal, Voroninski, and Bandeira [2018] 1 : Building on [Burer and Monteiro, 2005] and [Boumal, 2015], these authors show that, under reasonable geometrical hypotheses, all second-order critical points of Problem (Factorized SDP) are global minimizers, for almost any cost matrix C, as soon as…”

SqueezeFit: Label-Aware Dimensionality Reduction by Semidefinite Programming

“…For example, one might consider a reformulation of (1) that optimizes over the O(rd)-dimensional Grassmannian of r-dimensional subspaces of R d instead of the Ω(d 2 )-dimensional cone of positive semidefinite d × d matrices. While such a formulation will be non-convex, there is a growing body of work that provides performance guarantees for such optimization problems [7,21,25,4]. In fact, [32] proposes such a non-convex formulation of LMNN, and it performs well in practice.…”

A Brief Introduction to Manifold Optimization

“…Here, we define the constraint set Since M is non-convex, there may exist many non-global local minima of (4.8). It is claimed in [20] that each local minimum of (4.8) maps to a global minimum of (4.7) if p(p+1) 2 > m. By utilizing the optimality theory of manifold optimization, any second-order stationary point can be mapped to a global minimum of (4.7) under mild assumptions [18]. Note that (4.9) is generally not a manifold.…”