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 [5], 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. © Springer-Verlag 2004.