Semidefnite Relaxation Bounds for Indefinite Homogeneous Quadratic Optimization

In this paper we study the relationship between the optimal value of a homogeneous quadratic optimization problem and that of its Semidefinite Programming (SDP) relaxation. We consider two quadratic optimization models: (1) min{x∗Cx | x∗Akx ≥ 1, x ∈ F, k = 0, 1, ...,m}; and (2) max{x∗Cx | x∗Akx ≤ 1, x ∈ F, k = 0, 1, ...,m}. If one of Ak’s is indefinite while others and C are positive semidefinite, we prove that the ratio between the optimal value of (1) and its SDP relaxation is upper bounded by O(m) when F is the real line R, and by O(m) when F is the complex plane C. This result is an extension of the recent work of Luo et al. [8]. For (2), we show that the same ratio is bounded from below by O(1/ logm) for both the real and complex case, whenever all but one of Ak’s are positive semidefinite while C can be indefinite. This result improves the so-called approximate S-Lemma of Ben-Tal et al. [2]. We also consider (2) with multiple indefinite quadratic constraints and derive a general bound in terms of the problem data and the SDP solution. Throughout the paper, we present examples showing that all of our results are essentially tight.