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On Equivalence of Semidefinite Relaxations for Quadratic Matrix Programming

Yichuan Ding(y7ding***at***stanford.edu)
Dongdong Ge(ddge***at***sjtu.edu.cn)
Henry Wolkowicz(hwolkowciz***at***uwaterloo.ca)

Abstract: In this paper, we analyze two popular semidefinite programming \SDPb relaxations for quadratically constrained quadratic programs \QCQPb with matrix variables. These are based on \emph{vector-lifting} and on \emph{matrix lifting} and 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 an inexpensive \SDP relaxation and still obtain a strong bound. Our results also shed important insights on how to simplify large-scale \SDP constraints by exploiting the particular sparsity pattern.

Keywords: Semidefinite Programming Relaxations, Quadratic Matrix Programming, Quadratic Constrained Quadratic Programming, Hard Combinatorial Problems.

Category 1: Nonlinear Optimization (Constrained Nonlinear Optimization )

Category 2: Convex and Nonsmooth Optimization (Convex Optimization )

Citation: CORR 2010-02, University of Waterloo, Canada, april/2010.

Download: [PDF]

Entry Submitted: 04/28/2010
Entry Accepted: 04/28/2010
Entry Last Modified: 04/28/2010

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