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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.
Solving MaxCut to Optimality by Intersecting Semidefinite and Polyhedral Relaxations
, 2007
"... We present a method for finding exact solutions of MaxCut, the problem of finding a cut of maximum weight in a weighted graph. We use a BranchandBound setting, that applies a dynamic version of the bundle method as bounding procedure. This approach uses Lagrangian duality to obtain a “nearly opti ..."
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Cited by 21 (3 self)
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We present a method for finding exact solutions of MaxCut, the problem of finding a cut of maximum weight in a weighted graph. We use a BranchandBound setting, that applies a dynamic version of the bundle method as bounding procedure. This approach uses Lagrangian duality to obtain a “nearly optimal ” solution of the basic semidefinite MaxCut relaxation, strengthened by triangle inequalities. The expensive part of our bounding procedure is solving the basic semidefinite relaxation of the MaxCut problem, which has to be done several times during the bounding process. We review other solution approaches and compare the numerical results with our method. We also extend our experiments to instances of unconstrained quadratic 01 optimization and to instances of the graph equipartition problem. The experiments show, that our method nearly always outperforms all other approaches. In particular, for dense graphs, where linear programming based methods fail, our method performs very well. Exact solutions are obtained in a reasonable time for any instance of size up to n = 100, independent of the density. For some problems of special structure we can solve even larger problem classes. We could prove optimality for several problems of the literature where, to the best of our knowledge, no other method is able to do so. 1 The MaxCut Problem The MaxCut problem is one of the basic NPhard combinatorial optimization problems and has attracted scientific interest from both the discrete (see, e.g.,
Strong Duality for a TrustRegion Type Relaxation of the Quadratic Assignment Problem
, 1998
"... Lagrangian duality underlies many efficient algorithms for convex minimization problems. A key ingredient is strong duality. Lagrangian relaxation also provides lower bounds for nonconvex problems, where the quality of the lower bound depends on the duality gap. Quadratically constrained quadratic p ..."
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Cited by 15 (10 self)
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Lagrangian duality underlies many efficient algorithms for convex minimization problems. A key ingredient is strong duality. Lagrangian relaxation also provides lower bounds for nonconvex problems, where the quality of the lower bound depends on the duality gap. Quadratically constrained quadratic programs (QQPs) provide important examples of nonconvex programs. For the simple case of one quadratic constraint (the trust region subproblem) strong duality holds. In addition, necessary and sufficient (strengthened) second order optimality conditions exist. However, these duality results already fail for the two trust region subproblem. Surprisingly, there are classes of more complex, nonconvex QQPs where strong duality holds. One example is the special case of orthogonality constraints, which arise naturally in relaxations for the quadratic assignment problem (QAP). In this paper we show that strong duality also holds for a relaxation of QAP where the orthogonality constraint is replaced ...
Semidefinite programming for discrete optimization and matrix completion problems
 Discrete Appl. Math
, 2002
"... Survey article for the proceedings of Discrete Optimization '99 where some of these results were presented as a plenary address. y ..."
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Cited by 10 (5 self)
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Survey article for the proceedings of Discrete Optimization '99 where some of these results were presented as a plenary address. y
Semidefinite Programs and Association Schemes
, 1999
"... We consider semidenite programs, where all the matrices dening the problem commute. We show that in this case the semidenite program can be solved through an ordinary linear program. As an application, we consider the maxcut problem, where the underlying graph arises from an association scheme. ..."
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Cited by 8 (3 self)
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We consider semidenite programs, where all the matrices dening the problem commute. We show that in this case the semidenite program can be solved through an ordinary linear program. As an application, we consider the maxcut problem, where the underlying graph arises from an association scheme.
The GaussNewton Direction in Semidefinite Programming
, 1998
"... Primaldual interiorpoint methods have proven to be very successful for both linear programming (LP) and, more recently, for semidefinite programming (SDP) problems. Many of the techniques that have been so successful for LP have been extended to SDP. In fact, interior point methods are currently t ..."
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Cited by 5 (3 self)
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Primaldual interiorpoint methods have proven to be very successful for both linear programming (LP) and, more recently, for semidefinite programming (SDP) problems. Many of the techniques that have been so successful for LP have been extended to SDP. In fact, interior point methods are currently the only successful techniques for SDP. Research supported by Natural Sciences Engineering Research Council Canada. Email sgkruk@acm.org y Department of Mechanical Engineering, Sophia University, 71 Kioicho, Chiyodaku, Tokyo 102 Japan z Technische Universitat Graz, Institut fur Mathematik, Steyrergasse 30, A8010 Graz, Austria x EMS Program Director, ACE42 EQuad, Princeton University, Princeton NJ 08544, Tel: 6092580876, Fax: 6092583796, Email rvdb@princeton.edu, http://www.princeton.edu/~rvdb/  Research supported by Natural Sciences Engineering Research Council Canada. Email hwolkowi@orion.math.uwaterloo.ca, http://orion.math.uwaterloo.ca/~hwolkowi 0 This report is av...
On the Equivalence of Convex Programming Bounds for Boolean Quadratic Programming
, 1998
"... Recent papers have shown the equivalence of several tractable bounds for Boolean quadratic programming. In this note we give simplified proofs for these results, and also show that all of the bounds considered are simultaneously attained by one diagonal perturbation of the quadratic form. Keywords: ..."
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Cited by 3 (0 self)
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Recent papers have shown the equivalence of several tractable bounds for Boolean quadratic programming. In this note we give simplified proofs for these results, and also show that all of the bounds considered are simultaneously attained by one diagonal perturbation of the quadratic form. Keywords: Boolean Quadratic Programming, Semidefinite Programming, Eigenvalue Optimization, TrustRegion Methods. 1 Introduction Consider the Boolean Quadratic Programming problem, BQP : max q(x) = x T Qx \Gamma 2c T x x j 2 f\Gamma1; 1g; j = 1; : : : ; n; where Q is an n \Theta n symmetric matrix. Let z denote the solution objective value in BQP. It is well known that BQP is an NPhard problem; for example the maxcut problem can be written as an instance of BQP. In [7], several convexprogramming relaxations of BQP are considered in the interest of obtaining computable upper bounds B z. These relaxations correspond to parametric convex quadratic programming over box constraints, parametric qu...
Approximation Bounds for Quadratic Maximization with Semidefinite Programming Relaxation
, 2003
"... In this paper, we consider a class of quadratic maximization problems. One important instance in that class is the famous quadratic maximization formulation of the maxcut problem studied by Goemans and Williamson [6]. Since the problem is NPhard in general, following Goemans and Williamson, we app ..."
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Cited by 2 (0 self)
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In this paper, we consider a class of quadratic maximization problems. One important instance in that class is the famous quadratic maximization formulation of the maxcut problem studied by Goemans and Williamson [6]. Since the problem is NPhard in general, following Goemans and Williamson, we apply the approximation method based on the semidefinite programming (SDP) relaxation. For a subclass of the problems, including the ones studied by Helmberg [9] and Zhang [23], we show that the SDP relaxation approach yields an approximation solution with the worstcase performance ratio at least alpha = 0.87856... . This is a generalization...
Presolving for Semidefinite Programs Without Constraint Qualifications
, 1998
"... Presolving for linear programming is an essential ingredient in many commercial packages. This step eliminates redundant constraints and identically zero variables, and it identifies possible infeasibility and unboundedness. In semidefinite programming, identically zero variables corresponds to lack ..."
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Presolving for linear programming is an essential ingredient in many commercial packages. This step eliminates redundant constraints and identically zero variables, and it identifies possible infeasibility and unboundedness. In semidefinite programming, identically zero variables corresponds to lack of a constraint qualification which can result in both theoretical and numerical difficulties. A nonzero duality gap can exist which nullifies the elegant and powerful duality theory. Small perturbations can result in infeasibility and/or large perturbations in solutions. Such problems fall into the class of illposed problems. It is interesting to note that classes of problems where constraint qualifications fail arise from semidefinite programming relaxations of hard combinatorial problems. We look at several such classes and present two approaches to find regularized solutions. Some preliminary numerical results are included. Contents 1 Introduction 2 1.1 Notation . . . . . . . . . . ....