Improved Approximation Bound for Quadratic Optimization Problems with Orthogonality Constraints

Abstract

In this paper we consider the problem of approximating a class of quadratic optimization problems that contain orthogonality constraints, i.e. constraints of the form XTX = I, where X ∊ ℝm×n is the optimization variable. This class of problems, which we denote by (Qp–Oc), is quite general and captures several well–studied problems in the literature as special cases. In a recent work, Nemirovski [17] gave the first non–trivial approximation algorithm for (Qp–Oc). His algorithm is based on semidefinite programming and has an approximation guarantee of O ((m + n)1/3). We improve upon this result by providing the first logarithmic approximation guarantee for (Qp–Oc). Specifically, we show that (Qp–Oc) can be approximated to within a factor of O(ln(max{m, n})). The main technical tool used in the analysis is the so–called non–commutative Khintchine inequality, which allows us to prove a concentration inequality for the spectral norm of a Rademacher sum of matrices. As a by–product, we resolve in the affirmative a conjecture of Nemirovski concerning the typical spectral norm of a sum of certain random matrices. The aforementioned concentration inequality also has ramifications in the design of so–called safe tractable approximations of chance constrained optimization problems. In particular, we use it to simplify and improve a recent result of Ben–Tal and Nemirovski [4] concerning certain chance constrained linear matrix inequality systems.