On maximization of quadratic form over intersection of ellipsoids with common center

Abstract

Abstract. We demonstrate that if A 1 , ..., A m are symmetric positive semidefinite n ×n matrices with positive definite sum and A is an arbitrary symmetric n × n matrix, then the relative accuracy, in terms of the optimal value, of the semidefinite relaxationof the optimization programis not worse than 1 − 1 2 ln(2m 2 ) . It is shown that this bound is sharp in order, as far as the dependence on m is concerned, and that a feasible solution x to (P) withcan be found efficiently. This somehow improves one of the results of Nesterov [4] where bound similar to ( * ) is established for the case when all A i are of rank 1. Key words. semidefinite relaxations -quadratic programming IntroductionLet A i , i = 1, ..., m, be positive semidefinite n × n matrices with positive definite sum, and A be a n × n symmetric matrix. Consider the optimization problemThis problem, in general, is NP-hard (take, e.g., m = n and A i = e i e T i , where e i are the standard basic orths in R n ; then (P) becomes the problem of maximizing a homogeneous quadratic form over the unit cube, which is known to be NP-hard even in the case of positive semidefinite A).

A Unified Theorem on SDP Rank Reduction

“…We remark that the gap between v * maxsdp and v * maxqp can be as large as Ω(log m); see Nemirovski et al [12].…”

“…We first make some standard preparatory moves (see, e.g., Barvinok [3], Luo et al [9], Nemirovski et al [12]). Let X 0 be a solution to the system (1).…”

“…, A m are symmetric psd matrices. Both of these problems arise from various applications (see Luo et al [9], Nemirovski et al [12]) and are NP-hard. Their natural SDP relaxations are given by:…”

“…Suppose that X * maxsdp is an optimal solution to (18). It has been shown in Nemirovski et al [12] that one can extract a rank-1 matrix X 0 from X * maxsdp such that (i) X 0 is always feasible for (18) and (ii)…”

Ellipsoidal bounds for uncertain linear equations and dynamical systems

“…The interested reader is also addressed to [16,21,24] for further discussion of semidefinite relaxations of non-convex quadratic problems.…”