Local Minima and Convergence in Low-Rank Semidefinite Programming

Samuel Burer (samuel-bureruiowa.edu)
Renato Monteiro (monteiroisye.gatech.edu)

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 \cite{BurMon03-1}, 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.

Keywords:

Category 1: Linear, Cone and Semidefinite Programming (Semi-definite Programming )

Category 2: Combinatorial Optimization

Citation: Technical report, Department of Management Sciences, University of Iowa, September 2003.

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Entry Submitted: 09/29/2003
Entry Accepted: 09/29/2003
Entry Last Modified: 09/29/2003

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