Semidefnite Relaxation Bounds for Indefinite Homogeneous

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{xCx | xAkx ≥ 1, x ∈ F n , k = 0,1, ..., m}; and (2) max{xCx | xAkx ≤ 1, x ∈ F n , 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 2 ) 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/log m) 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.