Approximation Bounds for Quadratic Optimization with Homogeneous Quadratic Constraints

We consider the NP-hard problem of finding a minimum norm vector in n-dimensional real or complex Euclidean space, subject to m concave homogeneous quadratic constraints. We show that a semidefinite programming (SDP) relaxation for this nonconvex quadratically constrained quadratic program (QP) provides an O(m) approximation in the real case, and an O(m) approximation in the complex case. Moreover, we show that these bounds are tight up to a constant factor. When the Hessian of each constraint function is of rank 1 (namely, outer products of some given so-called steering vectors) and the phase spread of the entries of these steering vectors are bounded away from π/2, we establish a certain “constant factor” approximation (depending on the phase spread but independent of m and n) for both the SDP relaxation and a convex QP restriction of the original NP-hard problem. Finally, we consider a related problem of finding a maximum norm vector subject to m convex homogeneous quadratic constraints. We show that a SDP relaxation for this nonconvex QP provides an O(1/ ln(m)) approximation, which is analogous to a result of Nemirovski, Roos and Terlaky [14] for the real case. ∗The first author is supported in part by the National Science Foundation, Grant No. DMS-0312416, and by the Natural Sciences and Engineering Research Council of Canada, Grant No. OPG0090391. The second author is supported in part by the U.S. ARO under ERO, Contract No. N62558-03-C-0012, and the EU under U-BROAD STREP, Grant No. 506790. The third author is supported by the National Science Foundation, Grant No. DMS-0511283. The fourth author is supported by Hong Kong RGC Earmarked Grant CUHK418505. †Department of Electrical and Computer Engineering, University of Minnesota, 200 Union Street SE, Minneapolis, MN 55455, U.S.A. ([email protected]) ‡Department of Electronic and Computer Engineering, Technical University of Crete, 73100 Chania Crete, Greece. ([email protected]) §Department of Mathematics, University of Washington, Seattle, Washington 98195, U.S.A. ([email protected]) ¶Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, Hong Kong. ([email protected])