Semidefinite Relaxation Bounds for Indefinite Homogeneous Quadratic Optimization

This paper studies 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, k = 0, 1, ...,m, x ∈ Fn} and (2) max{x∗Cx | x∗Akx ≤ 1, k = 0, 1, ..., m, x ∈ Fn}, where F is either the real field R or the complex field C, and Ak, C are symmetric matrices. For the minimization model (1), we prove that, if the matrix C and all but one of Ak’s are positive semidefinite, then the ratio between the optimal value of (1) and its SDP relaxation is upper bounded by O(m) when F = R, and by O(m) when F = C. Moreover, when two or more of Ak’s are indefinite, this ratio can be arbitrarily large. For the maximization model (2), we show that, if C and at most one of Ak’s are indefinite while other Ak’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 complex case. This result improves the bound based on the so-called approximate S-Lemma of Ben-Tal et al. [3]. When two or more of Ak 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.