Improved approximation bound for quadratic optimization problems with orthogonality constraints

Anthony Man-Cho So

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 ε Rm x 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.