Local Minima and Convergence in Low-Rank Semidefinite Programming

Abstract

The low-rank semidefinite programming problem (LRSDP r ) is a restriction of the semidefinite programming problem (SDP) in which a bound r is imposed on the rank of X, and it is well known that LRSDP r is equivalent to SDP if r is not too small. In this paper, we classify the local minima of LRSDP r and prove the optimal convergence of a slight variant of the successful, yet experimental, algorithm of Burer and Monteiro [6], which handles LRSDP r via the nonconvex change of variables X = RR T . In addition, for particular problem classes, we describe a practical technique for obtaining lower bounds on the optimal solution value during the execution of the algorithm. Computational results are presented on a set of combinatorial optimization relaxations, including some of the largest quadratic assignment SDPs solved to date.

Conic Optimization Software

“…Website: http://dollar.biz.uiowa.edu/~sburer/ Reference: [41][42][43] SDPlr puts special emphasis on exploiting the special low-rank structure of the coefficient matrices and variables. It is very efficient on some special problems, arising from, e.g., combinatorics.…”

Metric learning with rank and sparsity constraints

“…To illustrate our algorithm, we use a symmetric and non-smooth version of the MLP (1) in the manner of [4,5]. We then use the Nesterov smoothing approach to obtain a smooth cost that has approximation guarantees to the original problem, followed by Burer-Monteiro splitting [10] with quasi-Newton enhancements. Even without the sparsity constraints, our algorithmic approach is novel and leads to improved results over the previous state-of-the-art.…”

An implementable proximal point algorithmic framework for nuclear norm minimization

“…However, the convergence of the projected subgradient method in [40] is not known since problem (2) is a nonsmooth problem. Recht, Fazel and Parrilo [40] also made use of the low rank factorization technique introduced by Burer and Monteiro [8,9] to solve (2) with only linear equality constraints. The potential difficulty of this method is that the low rank factorization formulation is no longer convex and the rank of the optimal matrix is generally unknown a priori.…”

“…Here, (9) means that {λ k } is a nondecreasing sequence of positive parameters that converges to λ ∞ , which is allowed to take the +∞ value. The second algorithm, namely, the dual PPA, is the application of the general proximal point method to the dual problem of (2), which, as a by-product, yields an optimal solution to problem (2).…”

“…Given a sequence {λ k } satisfying (9) and an initial point y 0 ∈ Q * , the sequence {y k } ⊂ Q * generated by the dual PPA is as follows:…”