On Equivalence of Semidefinite Relaxations for Quadratic Matrix Programming

  • Authors:

  • Yichuan Ding;Dongdong Ge;Henry Wolkowicz

  • Affiliations:

  • Department of Management Science and Engineering, Stanford University, Stanford, California 94305;Antai College of Economics and Management, Shanghai Jiao Tong University, 200240 Shanghai, People's Republic of China;Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada

  • Venue:
  • Mathematics of Operations Research
  • Year:
  • 2011

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Abstract

We analyze two popular semidefinite programming relaxations for quadratically constrained quadratic programs with matrix variables. These relaxations are based on vector lifting and on matrix lifting; they are of different size and expense. We prove, under mild assumptions, that these two relaxations provide equivalent bounds. Thus, our results provide a theoretical guideline for how to choose a less expensive semidefinite programming relaxation and still obtain a strong bound. The main technique used to show the equivalence and that allows for the simplified constraints is the recognition of a class of nonchordal sparse patterns that admit a smaller representation of the positive semidefinite constraint.