Local Minima and Convergence in Low-Rank Semidefinite Programming

Authors:
Samuel Burer;Renato D. C. Monteiro
Affiliations:
Department of Management Sciences, University of Iowa, 52242-1000, Iowa City, IA, USA;School of Industrial and Systems Engineering, Georgia Institute of Technology, 30332, Atlanta, Georgia, USA
Venue:
Mathematical Programming: Series A and B
Year:
2005

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Abstract

The low-rank semidefinite programming problem LRSDPr 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 LRSDPr is equivalent to SDP if r is not too small. In this paper, we classify the local minima of LRSDPr and prove the optimal convergence of a slight variant of the successful, yet experimental, algorithm of Burer and Monteiro [5], which handles LRSDPr via the nonconvex change of variables X=RRT. 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.