Recently, a semidefinite programming (SDP) relaxation approach has been proposed to solve the sensor network localization problem. Although it achieves high accuracy in estimating sensor’s locations, the speed of the SDP approach is not satisfactory for practical applications. In this paper we propose methods to further relax the SDP relaxation; more precisely, to decompose the single semidefin...

We consider semidefinite relaxations of a quadratic optimization problem with polynomial constraints. This is an extension of quadratic problems with boolean variables. Such combinatorial problems can in general not be solved in polynomial time. Semidefinite relaxations has been proposed as a promising technique to give provable good bounds on certain boolean quadratic problems in polynomial ti...

In this paper we analyze a known relaxation for the Sparsest Cut Problem based on positive semidefinite constraints, and we present a branch and bound algorithm and heuristics based on this relaxation. The relaxed formulation and the algorithms were tested on small and moderate sized instances. It leads to values very close to the optimum solution values. The exact algorithm could obtain soluti...

An exact semidefinite linear programming (SDP) relaxation of a nonlinear semidefinite programming problem is a highly desirable feature because a semidefinite linear programming problem can efficiently be solved. This paper addresses the basic issue of which nonlinear semidefinite programming problems possess exact SDP relaxations under a constraint qualification. We do this by establishing exa...

In this paper we study a class of quadratic maximization problems and their semide nite program ming SDP relaxation For a special subclass of the problems we show that the SDP relaxation provides an exact optimal solution Another subclass which is NP hard guarantees that the SDP relaxation yields an approximate solution with a worst case performance ratio of This is a generalization of the well...

Variable selection is a fundamental task in statistical data analysis. Sparsity-inducing regularization methods are a popular class of methods that simultaneously perform variable selection and model estimation. The central problem is a quadratic optimization problem with an `0-norm penalty. Exactly enforcing the `0-norm penalty is computationally intractable for larger scale problems, so diffe...

We obtain a tight semidefinite relaxation of the MAX CUT problem which improves several previous SDP relaxation in the literature. Not only is it a strict improvement over the SDP relaxation obtained by adding all the triangle inequalities to the well-known SDP relaxation, but also it satisfy Slater constraint qualification (strict feasibility).

Notes for lecture given by the author on November 7, 2014 as part of the special course: “Randomness, Matrices and High Dimensional Problems”, at IMPA, Rio de Janeiro, Brazil. The results presented in these notes are from [1]. 1. The problem we will focus on Let n be an even positive integer. Given two sets of n2 nodes consider the following random graph G: For each pair (i, j) of nodes, (i, j)...

Consider the semidefinite relaxation (SDR) of the quadratic integer program (QIP): γ := maxx∈{−1,1}n xT Qx ≤ minD−Qo0 trace(D) =: γ̄ where Q is a given symmetric matrix and D is diagonal. In this paper we consider the relaxation gap γ̄ − γ. We establish the uniqueness of the solution of the semidefinite relaxation problem and prove that γ = γ̄ if and only if γr := 1 n maxx∈{−1,1}n xT V V T x = 1 w...

This report reviews some approximation algorithms for combinatorial optimization problems, based on a semidefinite relaxation followed by randomized rounding.