Approximation Bounds for Quadratic Optimization with Homogeneous Quadratic Constraints

• Authors: Zhi-Quan Luo;Nicholas D. Sidiropoulos;Paul Tseng;Shuzhong Zhang
• Venue: SIAM Journal on Optimization
• Year: 2007

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 $O(m^2)$ 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 $\pi/2$, 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 an SDP relaxation for this nonconvex QP provides an $O(1/\ln(m))$ approximation, which is analogous to a result of Nemirovski &etal; [Math. Program., 86 (1999), pp. 463-473] for the real case.