Semidefinite Approximation for Mixed Binary Quadratically Constrained Quadratic Programs

Motivated by applications in wireless communications, this paper develops semidefinite programming (SDP) relaxation techniques for some mixed binary quadratically constrained quadratic programs (MBQCQP) and analyzes their approximation performance. We consider both a minimization and a maximization...

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Bibliographic Details
Published in:SIAM journal on optimization Vol. 24; no. 3; pp. 1265 - 1293
Main Authors: Xu, Zi, Hong, Mingyi, Luo, Zhi-Quan
Format: Journal Article
Language:English
Published: Philadelphia Society for Industrial and Applied Mathematics 01.01.2014
Subjects:
Approximation
Computer engineering
Quality of service
Receivers & amplifiers
Wireless communications
ISSN:1052-6234, 1095-7189
Online Access:Get full text
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Summary:Motivated by applications in wireless communications, this paper develops semidefinite programming (SDP) relaxation techniques for some mixed binary quadratically constrained quadratic programs (MBQCQP) and analyzes their approximation performance. We consider both a minimization and a maximization model of this problem. For the minimization model, the objective is to find a minimum norm vector in $N$-dimensional real or complex Euclidean space, such that $M$ concave quadratic constraints and a cardinality constraint are satisfied with both binary and continuous variables. By employing a special randomized rounding procedure, we show that the ratio between the norm of the optimal solution of the minimization model and its SDP relaxation is upper bounded by $\mathcal{O}(Q^2(M-Q+1)+M^2)$ in the real case and by $\mathcal{O}(M(M-Q+1))$ in the complex case. For the maximization model, the goal is to find a maximum norm vector subject to a set of quadratic constraints and a cardinality constraint with both binary and continuous variables. We show that in this case the approximation ratio is bounded from below by $\mathcal{O}(\epsilon/\ln(M))$ for both the real and the complex cases where $\epsilon > 0$ is a model parameter. Moreover, this ratio is tight up to a constant factor. [PUBLICATION ABSTRACT]
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ISSN:1052-6234
1095-7189
DOI:10.1137/130909597