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12
An InteriorPoint Method for Semidefinite Programming
, 2005
"... We propose a new interior point based method to minimize a linear function of a matrix variable subject to linear equality and inequality constraints over the set of positive semidefinite matrices. We show that the approach is very efficient for graph bisection problems, such as maxcut. Other appli ..."
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Cited by 255 (18 self)
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We propose a new interior point based method to minimize a linear function of a matrix variable subject to linear equality and inequality constraints over the set of positive semidefinite matrices. We show that the approach is very efficient for graph bisection problems, such as maxcut. Other applications include maxmin eigenvalue problems and relaxations for the stable set problem.
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 109 (1 self)
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We describe a few applications of semide nite programming in combinatorial optimization.
A semidefinite framework for trust region subproblems with applications to large scale minimization
, 2002
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On Lagrangian relaxation of quadratic matrix constraints
 SIAM J. Matrix Anal. Appl
, 2000
"... Abstract. Quadratically constrained quadratic programs (QQPs) play an important modeling role for many diverse problems. These problems are in general NP hard and numerically intractable. Lagrangian relaxations often provide good approximate solutions to these hard problems. Such relaxations are equ ..."
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Cited by 52 (18 self)
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Abstract. Quadratically constrained quadratic programs (QQPs) play an important modeling role for many diverse problems. These problems are in general NP hard and numerically intractable. Lagrangian relaxations often provide good approximate solutions to these hard problems. Such relaxations are equivalent to semidefinite programming relaxations. For several special cases of QQP, e.g., convex programs and trust region subproblems, the Lagrangian relaxation provides the exact optimal value, i.e., there is a zero duality gap. However, this is not true for the general QQP, or even the QQP with two convex constraints, but a nonconvex objective. In this paper we consider a certain QQP where the quadratic constraints correspond to the matrix orthogonality condition XXT = I. For this problem we show that the Lagrangian dual based on relaxing the constraints XXT = I and the seemingly redundant constraints XT X = I has a zero duality gap. This result has natural applications to quadratic assignment and graph partitioning problems, as well as the problem of minimizing the weighted sum of the largest eigenvalues of a matrix. We also show that the technique of relaxing quadratic matrix constraints can be used to obtain a strengthened semidefinite relaxation for the maxcut problem. Key words. Lagrangian relaxations, quadratically constrained quadratic programs, semidefinite programming, quadratic assignment, graph partitioning, maxcut problems
Semidefinite Programming and Graph Equipartition
 IN TOPICS IN SEMIDEFINITE AND INTERIORPOINT METHODS
, 1998
"... Semidefinite relaxations are used to approximate the problem of partitioning a graph into equally sized components. The relaxations extend previous eigenvalue based models, and combine semidefinite and polyhedral approaches. Computational results on graphs with several hundred vertices are given, an ..."
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Cited by 27 (7 self)
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Semidefinite relaxations are used to approximate the problem of partitioning a graph into equally sized components. The relaxations extend previous eigenvalue based models, and combine semidefinite and polyhedral approaches. Computational results on graphs with several hundred vertices are given, and indicate that semidefinite relaxations approximate the equipartition problem quite well.
Combining Semidefinite and Polyhedral Relaxations for Integer Programs
, 1995
"... We present a general framework for designing semidefinite relaxations for constrained 01 quadratic programming and show how valid inequalities of the cutpolytope can be used to strengthen these relaxations. As examples we improve the #function and give a semidefinite relaxation for the quadrati ..."
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Cited by 19 (11 self)
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We present a general framework for designing semidefinite relaxations for constrained 01 quadratic programming and show how valid inequalities of the cutpolytope can be used to strengthen these relaxations. As examples we improve the #function and give a semidefinite relaxation for the quadratic knapsack problem. The practical value of this approach is supported by numerical experiments which make use of the recent development of efficient interior point codes for semidefinite programming.
Semidefinite Programming and Combinatorial Optimization
 Appl. Numer. Math
, 1998
"... Semidefinite Programs have recently turned out to be a powerful tool for approximating integer problems. To survey the development in this area over the last few years, the following topics are addressed in some detail. First, we investigate ways to derive semidefinite programs from discrete opti ..."
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Cited by 17 (4 self)
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Semidefinite Programs have recently turned out to be a powerful tool for approximating integer problems. To survey the development in this area over the last few years, the following topics are addressed in some detail. First, we investigate ways to derive semidefinite programs from discrete optimization problems. The duality theory for semidefinite programs is the key to understand algorithms to solve them. The relevant duality results are therefore summarized. The second part of the paper deals with the approximation of integer problems both in a theoretical setting, and from a computational point of view. 1 Introduction The interest in Semidefinite Programming (SDP) has been growing rapidly in the last few years. Here are some possible explanations for this sudden rise of interest. The algorithmic development of interiorpoint methods for Linear Programs indicated the potential of this approach to solve general convex problems. Semidefinite Programs are a natural generaliza...
Reductions between Disjoint NPPairs
 Information and Computation
, 2004
"... We prove that all of the following assertions are equivalent: There is a manyone complete disjoint NPpair; there is a strongly manyone complete disjoint NPpair; there is a Turing complete disjoint NPpair such that all reductions are smart reductions; there is a complete disjoint NPpair for one ..."
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Cited by 15 (4 self)
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We prove that all of the following assertions are equivalent: There is a manyone complete disjoint NPpair; there is a strongly manyone complete disjoint NPpair; there is a Turing complete disjoint NPpair such that all reductions are smart reductions; there is a complete disjoint NPpair for onetoone, invertible reductions; the class of all disjoint NPpairs is uniformly enumerable. Let A, B, C, and D be nonempty sets belonging to NP. A smart reduction between the disjoint NPpairs (A, B) and (C, D) is a Turing reduction with the additional property that if D. We prove under the reasonable assumption UP coUP has a Pbiimmune set that there exist disjoint NPpairs (A, B) and (C, D) such that (A, B) is truthtable reducible to (C, D), but there is no smart reduction between them. This paper contains several additional separations of reductions between disjoint NPpairs. We exhibit an oracle relative to which DisjNP has a truthtablecomplete disjoint NPpair, but has no manyonecomplete disjoint NPpair.
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 (2 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.
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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Cited by 5 (0 self)
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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.