On Approximating Complex Quadratic Optimization Problems via Semidefinite Programming Relaxations

In this paperwe study semidefinite programming (SDP)models for a class of discrete and continuous quadratic optimization problems in the complex Hermitian form. These problems capture a class of well-known combinatorial optimization problems, as well as problems in control theory. For instance, they include theMAX-3-CUT problem where the Laplacian matrix is positive semidefinite (in particular, some of the edge weights can be negative). We present a generic algorithmand a unified analysis of the SDP relaxationswhich allowus to obtain good approximation guarantees for our models. Specifically, we give an (k sin(πk )) 2/(4π)-approximation algorithm for the discrete problem where the decision variables are k-ary and the objective matrix is positive semidefinite. To the best of our knowledge, this is the first known approximation result for this family of problems. For the continuous problem where the objective matrix is positive semidefinite, we obtain the well-known π/4 result due to Ben-Tal et al. A preliminary version of this paper has appeared in the Proceedings of the 11th Conference on Integer Programming and Combinatorial Optimization (IPCO XI), 2005. This research was supported in part by NSF grant DMS-0306611. A. Man-Cho So (B) Department of Computer Science, Stanford University, Stanford, CA 94305, USA e-mail: [email protected] J. Zhang Department of Information, Operations, and Management Sciences, Stern School of Business, New York University, New York, NY 10012, USA e-mail: [email protected] Y. Ye Department of Management Science and Engineering, and, by courtesy, Electrical Engineering, Stanford University, Stanford, CA 94305, USA e-mail: [email protected] A. Man-Cho So et al. [Math Oper Res 28(3):497–523, 2003], and independently, Zhang and Huang [SIAM J Optim 16(3):871–890, 2006]. However, our techniques simplify their analyses and provide a unified framework for treating those problems. In addition, we show for the first time that the gap between the optimal value of the original problem and that of the SDP relaxation can be arbitrarily close to π/4. We also show that the unified analysis can be used to obtain an (1/ log n)approximation algorithm for the continuous problem in which the objective matrix is not positive semidefinite.