Semidefinite Relaxation Bounds for Indefinite Homogeneous Quadratic Optimization

This paper studies the relationship between the optimal value of a homogeneous quadratic optimization problem and its semidefinite programming (SDP) relaxation. We consider two quadratic optimization models: (1) $\min \{ x^* C x \mid x^* A_k x \ge 1, k=0,1,\ldots,m, x\in\mathbb{F}^n\}$ and (2) $\max \{ x^* C x \mid x^* A_k x \le 1, k=0,1,\ldots,m, x\in\mathbb{F}^n\}$, where $\mathbb{F}$ is either the real field $\mathbb{R}$ or the complex field $\mathbb{C}$, and $A_k,C$ are symmetric matrices. For the minimization model (1), we prove that if the matrix $C$ and all but one of the $A_k$'s are positive semidefinite, then the ratio between the optimal value of (1) and its SDP relaxation is upper bounded by $O(m^2)$ when $\mathbb{F}=\mathbb{R}$, and by $O(m)$ when $\mathbb{F}=\mathbb{C}$. Moreover, when two or more of the $A_k$'s are indefinite, this ratio can be arbitrarily large. For the maximization model (2), we show that if $C$ and at most one of the $A_k$'s are indefinite while other $A_k$'s are positive semidefinite, then the ratio between the optimal value of (2) and its SDP relaxation is bounded from below by $O(1/\log m)$ for both the real and the complex case. This result improves the bound based on the so-called approximate S-Lemma of Ben-Tal, Nemirovski, and Roos [SIAM J. Optim., 13 (2002), pp. 535-560]. When two or more of the $A_k$'s in (2) are indefinite, we derive a general bound in terms of the problem data and the SDP solution. For both optimization models, we present examples to show that the derived approximation bounds are essentially tight.