Matrix analysis
Semidefinite programming in combinatorial optimization
Mathematical Programming: Series A and B - Special issue: papers from ismp97, the 16th international symposium on mathematical programming, Lausanne EPFL
O(√log n) approximation algorithms for min UnCut, min 2CNF deletion, and directed cut problems
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Sums of random symmetric matrices and quadratic optimization under orthogonality constraints
Mathematical Programming: Series A and B
On approximating complex quadratic optimization problems via semidefinite programming relaxations
Mathematical Programming: Series A and B
Approximation Bounds for Quadratic Optimization with Homogeneous Quadratic Constraints
SIAM Journal on Optimization
Convex Approximations of Chance Constrained Programs
SIAM Journal on Optimization
Semidefinite Relaxation Bounds for Indefinite Homogeneous Quadratic Optimization
SIAM Journal on Optimization
Low rank matrix-valued chernoff bounds and approximate matrix multiplication
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part I
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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.