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38
Improved Approximation Algorithms for Maximum Cut and Satisfiability Problems Using Semidefinite Programming
 Journal of the ACM
, 1995
"... We present randomized approximation algorithms for the maximum cut (MAX CUT) and maximum 2satisfiability (MAX 2SAT) problems that always deliver solutions of expected value at least .87856 times the optimal value. These algorithms use a simple and elegant technique that randomly rounds the solution ..."
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Cited by 958 (14 self)
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We present randomized approximation algorithms for the maximum cut (MAX CUT) and maximum 2satisfiability (MAX 2SAT) problems that always deliver solutions of expected value at least .87856 times the optimal value. These algorithms use a simple and elegant technique that randomly rounds the solution to a nonlinear programming relaxation. This relaxation can be interpreted both as a semidefinite program and as an eigenvalue minimization problem. The best previously known approximation algorithms for these problems had performance guarantees of ...
A Nonlinear Programming Algorithm for Solving Semidefinite Programs via Lowrank Factorization
 Mathematical Programming (series B
, 2001
"... In this paper, we present a nonlinear programming algorithm for solving semidefinite programs (SDPs) in standard form. The algorithm's distinguishing feature is a change of variables that replaces the symmetric, positive semidefinite variable X of the SDP with a rectangular variable R according to t ..."
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Cited by 104 (9 self)
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In this paper, we present a nonlinear programming algorithm for solving semidefinite programs (SDPs) in standard form. The algorithm's distinguishing feature is a change of variables that replaces the symmetric, positive semidefinite variable X of the SDP with a rectangular variable R according to the factorization X = RR T . The rank of the factorization, i.e., the number of columns of R, is chosen minimally so as to enhance computational speed while maintaining equivalence with the SDP. Fundamental results concerning the convergence of the algorithm are derived, and encouraging computational results on some largescale test problems are also presented. Keywords: semidefinite programming, lowrank factorization, nonlinear programming, augmented Lagrangian, limited memory BFGS. 1 Introduction In the past few years, the topic of semidefinite programming, or SDP, has received considerable attention in the optimization community, where interest in SDP has included the investigation of...
Semidefinite Programming and Combinatorial Optimization
 DOC. MATH. J. DMV
, 1998
"... We describe a few applications of semide nite programming in combinatorial optimization. ..."
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Cited by 99 (1 self)
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We describe a few applications of semide nite programming in combinatorial optimization.
Theory of semidefinite programming for sensor network localization
 IN SODA05
, 2005
"... We analyze the semidefinite programming (SDP) based model and method for the position estimation problem in sensor network localization and other Euclidean distance geometry applications. We use SDP duality and interior–point algorithm theories to prove that the SDP localizes any network or graph th ..."
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Cited by 83 (5 self)
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We analyze the semidefinite programming (SDP) based model and method for the position estimation problem in sensor network localization and other Euclidean distance geometry applications. We use SDP duality and interior–point algorithm theories to prove that the SDP localizes any network or graph that has unique sensor positions to fit given distance measures. Therefore, we show, for the first time, that these networks can be localized in polynomial time. We also give a simple and efficient criterion for checking whether a given instance of the localization problem has a unique realization in R 2 using graph rigidity theory. Finally, we introduce a notion called strong localizability and show that the SDP model will identify all strongly localizable sub–networks in the input network.
Semidefinite Programming and Integer Programming
"... We survey how semidefinite programming can be used for finding good approximative solutions to hard combinatorial optimization problems. ..."
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Cited by 48 (7 self)
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We survey how semidefinite programming can be used for finding good approximative solutions to hard combinatorial optimization problems.
Binary Partitioning, Perceptual Grouping, and Restoration with Semidefinite Programming
 IEEE Transactions on Pattern Analysis and Machine Intelligence
, 2003
"... We introduce a novel optimization method based on semidefinite programming relaxations to the field of computer vision and apply it to the combinatorial problem of minimizing quadratic functionals in binary decision variables subject to linear constraints. ..."
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Cited by 30 (6 self)
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We introduce a novel optimization method based on semidefinite programming relaxations to the field of computer vision and apply it to the combinatorial problem of minimizing quadratic functionals in binary decision variables subject to linear constraints.
Further relaxations of the SDP approach to sensor network localization
, 2006
"... 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 prop ..."
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Cited by 23 (0 self)
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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 relax the single semidefinite matrix cone into a set of smallsize semidefinite matrix cones, which we call the smaller SDP (SSDP) approach. We present two such relaxations; and they are, although weaker than the original SDP relaxation, retaining the key theoretical property and tested to be both efficient and accurate in computation. The speed of the SSDP is even faster than that of other further weaker approaches. The SSDP approach may also pave a way to efficiently solve general SDP relaxations without sacrificing their solution quality.
A Remark On The Rank Of Positive Semidefinite Matrices Subject To Affine Constraints
 Discrete and Computational Geometry
, 2001
"... . Let Kn be the cone of positive semidefinite n \Theta n matrices and let A be an affine subspace of the space of symmetric matrices such that the intersection Kn " A is nonempty and bounded. Suppose that n 3 and that codim A = \Gamma r+2 2 \Delta for some 1 r n \Gamma 2. Then there is a ..."
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Cited by 21 (0 self)
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. Let Kn be the cone of positive semidefinite n \Theta n matrices and let A be an affine subspace of the space of symmetric matrices such that the intersection Kn " A is nonempty and bounded. Suppose that n 3 and that codim A = \Gamma r+2 2 \Delta for some 1 r n \Gamma 2. Then there is a matrix X 2 Kn "A such that rank X r. We give a short geometric proof of this result, use it to improve a bound on realizability of weighted graphs as graphs of distances between points in Euclidean space, and describe its relation to theorems of Bohnenblust, Friedland and Loewy, and AuYeung and Poon. 1. Introduction Let Sym n be the space of n \Theta n symmetric matrices. Thus Sym n is a real vector space of dimension \Gamma n+1 2 \Delta . Let K n ae Sym n be the convex cone of positive semidefinite matrices. The following result is well known, see for example [Barvinok 1995], Section 31.5 of [Deza and Laurent 1997] and [Pataki 1996]. (1.1) Theorem. Let A ae Sym n be an affine su...
An implementable proximal point algorithmic framework for nuclear norm minimization
, 2010
"... The nuclear norm minimization problem is to find a matrix with the minimum nuclear norm subject to linear and second order cone constraints. Such a problem often arises from the convex relaxation of a rank minimization problem with noisy data, and arises in many fields of engineering and science. In ..."
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Cited by 21 (3 self)
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The nuclear norm minimization problem is to find a matrix with the minimum nuclear norm subject to linear and second order cone constraints. Such a problem often arises from the convex relaxation of a rank minimization problem with noisy data, and arises in many fields of engineering and science. In this paper, we study inexact proximal point algorithms in the primal, dual and primaldual forms for solving the nuclear norm minimization with linear equality and second order cone constraints. We design efficient implementations of these algorithms and present comprehensive convergence results. In particular, we investigate the performance of our proposed algorithms in which the inner subproblems are approximately solved by the gradient projection method or the accelerated proximal gradient method. Our numerical results for solving randomly generated matrix completion problems and real matrix completion problems show that our algorithms perform favorably in comparison to several recently proposed stateoftheart algorithms. Interestingly, our proposed algorithms are connected with other algorithms that have been studied in the literature. Key words. Nuclear norm minimization, proximal point method, rank minimization, gradient projection method, accelerated proximal gradient method.
Quadratic matrix programming
 SIAM J. Optim
"... We introduce and study a special class of nonconvex quadratic problems in which the objective and constraint functions have the form f(X) = Tr(X T AX) + 2Tr(B T X) + c, X ∈ R n×r The latter formulation is termed quadratic matrix programming (QMP) of order r. We construct a specially devised semidef ..."
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Cited by 18 (2 self)
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We introduce and study a special class of nonconvex quadratic problems in which the objective and constraint functions have the form f(X) = Tr(X T AX) + 2Tr(B T X) + c, X ∈ R n×r The latter formulation is termed quadratic matrix programming (QMP) of order r. We construct a specially devised semidefinite relaxation (SDR) and dual for the QMP problem and show that under some mild conditions strong duality holds for QMP problems with at most r constraints. Using a result on the equivalence of two characterizations of the nonnegativity property of quadratic functions of the above form, we are able to compare the constructed SDR and dual problems to other known SDR and dual formulations of the problem. An application to robust least squares problems is discussed. 1