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A primal-dual active set method for quadratic problems with bound constraints is presented which extends the infeasible active set approach of [K. Kunisch and F. Rendl. An infeasible active set method for convex problems with simple... more
A primal-dual active set method for quadratic problems with bound constraints is presented which extends the infeasible active set approach of [K. Kunisch and F. Rendl. An infeasible active set method for convex problems with simple bounds. SIAM Journal on Optimization, 14(1):35-52, 2003]. Based on a guess of the active set, a primal-dual pair (x,α) is computed that satisfies stationarity and the complementary condition. If x is not feasible, the variables connected to the infeasibilities are added to the active set and a new primal-dual pair (x,α) is computed. This process is iterated until a primal feasible solution is generated. Then a new active set is determined based on the feasibility information of the dual variable α. Strict convexity of the quadratic problem is sufficient for the algorithm to stop after a finite number of steps with an optimal solution. Computational experience indicates that this approach also performs well in practice.
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We investigate an approximation algorithm for the maximum stable set problem based on the Lovasz number ϑ(G) as an initial upper bound. We strengthen this relaxation by adding two classes of cutting planes, odd circuit and triangle... more
We investigate an approximation algorithm for the maximum stable set problem based on the Lovasz number ϑ(G) as an initial upper bound. We strengthen this relaxation by adding two classes of cutting planes, odd circuit and triangle inequalities. We present computational results using this tighter model on several classes of graphs.
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We investigate relaxations for the maximum stable set problem based on the Lovasz number $\vartheta(G)$ as an initial upper bound. We strengthen this relaxation by adding two classes of cutting planes, odd circuit and triangle... more
We investigate relaxations for the maximum stable set problem based on the Lovasz number $\vartheta(G)$ as an initial upper bound. We strengthen this relaxation by adding two classes of cutting planes, odd circuit and triangle inequalities. We present computational results using this tighter model on many classes of graphs.
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We investigate the classical Gilmore-Lawler lower bound for the quadratic assignment problem. We provide evidence of the difficulty of improving the Gilmore-Lawler bound and develop new bounds by means of optimal reduction schemes.... more
We investigate the classical Gilmore-Lawler lower bound for the quadratic assignment problem. We provide evidence of the difficulty of improving the Gilmore-Lawler bound and develop new bounds by means of optimal reduction schemes. Computational results are reported indicating that the new lower bounds have advantages over previous bounds and can be used in a branch-and-bound type algorithm for the quadratic assignment problem.
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Semidefinite relaxations of the quadratic assignment problem (QAP) have recently turned out to provide good approximations to the optimal value of QAP. We take a systematic look at various conic relaxations of QAP. We first show that QAP... more
Semidefinite relaxations of the quadratic assignment problem (QAP) have recently turned out to provide good approximations to the optimal value of QAP. We take a systematic look at various conic relaxations of QAP. We first show that QAP can equivalently be formulated as a linear program over the cone of completely positive matrices. Since it is hard to optimize over
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Two Primal-dual interior point algorithms are presented for theproblem of maximizing the smallest eigenvalue of a symmetric matrixover diagonal perturbations. These algorithms prove to be simple,robust, and efficient. Both algorithms are... more
Two Primal-dual interior point algorithms are presented for theproblem of maximizing the smallest eigenvalue of a symmetric matrixover diagonal perturbations. These algorithms prove to be simple,robust, and efficient. Both algorithms are based on transforming theTechnische Universitat Graz, Institut fur Mathematik, Kopernikusgasse 24, A-8010Graz, Austria. Research support by Christian Doppler Laboratorium fur DiskreteOptimierung.yProgram in Statistics & Operations Research,...
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Abstract The problem of maximizing the smallest eigenvalue of a symmetric matrix subject to modi cations on the main diagonal that sum to zero is important since, for example, it yields the best bounds for graphpartitioning. Current... more
Abstract The problem of maximizing the smallest eigenvalue of a symmetric matrix subject to modi cations on the main diagonal that sum to zero is important since, for example, it yields the best bounds for graphpartitioning. Current algorithms for this problem work well when the multiplicity of the minimum eigenvalue at optimality is one. However, real-world applications have multiplicity at optimality that is greater than one. For such problems, current algorithms break down quickly as the multiplicity increases. We present a primal- ...
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... algorithm of [5]. To guarantee that there is no duality gap between primal and dual optimalsolutions ... we start the algorithm with (SQK2) as initial relaxation and compute its optimal solution. ... proved that in case of linear cost... more
... algorithm of [5]. To guarantee that there is no duality gap between primal and dual optimalsolutions ... we start the algorithm with (SQK2) as initial relaxation and compute its optimal solution. ... proved that in case of linear cost functions it is superior to the canonical linear relaxation. ...
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... Sherali, Bender," partitioning scheme applied to a new formulation of the quadratic assignmentproblem...,aral Res .o,'41t. Quart. ... [14] N. Christofides and M. Gerrard, Specml cases of the... more
... Sherali, Bender," partitioning scheme applied to a new formulation of the quadratic assignmentproblem...,aral Res .o,'41t. Quart. ... [14] N. Christofides and M. Gerrard, Specml cases of the quadraticassignment problem, Management Sciences Re search Report No. ...
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Semidefinite programming(SDP) relaxations for the quadratic assignment problem (QAP)are derived using the dual of the (homogenized) Lagrangian dual of appropriate equivalentrepresentations of QAP. These relaxations result in the... more
Semidefinite programming(SDP) relaxations for the quadratic assignment problem (QAP)are derived using the dual of the (homogenized) Lagrangian dual of appropriate equivalentrepresentations of QAP. These relaxations result in the interesting, special, case where onlythe dual problem of the SDP relaxation has strict interior, i.e. the Slater constraint qualificationalways fails for the primal problem. Although there is no duality gap in theory,
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... A well known theorem of Birko 21] states that the convex hull of the permutation matrices is the set of doubly stochastic matrices, conv = D: Thus the set of doubly stochastic matrices corresponds to the bipartite perfect matching... more
... A well known theorem of Birko 21] states that the convex hull of the permutation matrices is the set of doubly stochastic matrices, conv = D: Thus the set of doubly stochastic matrices corresponds to the bipartite perfect matching polytope. ...
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ABSTRACT Copositive optimization problems are particular conic programs: optimize linear forms over the copositive cone subject to linear constraints. Every quadratic program with linear constraints can be formulated as a copositive... more
ABSTRACT Copositive optimization problems are particular conic programs: optimize linear forms over the copositive cone subject to linear constraints. Every quadratic program with linear constraints can be formulated as a copositive program, even if some of the variables are binary. So this is an NP-hard problem class. While most methods try to approximate the copositive cone from within, we propose a method which approximates this cone from outside. This is achieved by passing to the dual problem, where the feasible set is an affine subspace intersected with the cone of completely positive matrices, and this cone is approximated from within. We consider feasible descent directions in the completely positive cone, and regularized strictly convex subproblems. In essence, we replace the intractable completely positive cone with a nonnegative cone, at the cost of a series of nonconvex quadratic subproblems. Proper adjustment of the regularization parameter results in short steps for the nonconvex quadratic programs. This suggests to approximate their solution by standard linearization techniques. Preliminary numerical results on three different classes of test problems are quite promising. KeywordsCombinatorial optimization–Copositive programs–Clique number
Max-Cut is one of the most studied combinatorial optimization problems because of its wide range of applications and because of its connections with other fields of discrete mathematics (see, eg, the book by Deza and Laurent [10]). Like... more
Max-Cut is one of the most studied combinatorial optimization problems because of its wide range of applications and because of its connections with other fields of discrete mathematics (see, eg, the book by Deza and Laurent [10]). Like other interesting ...
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ABSTRACT Consider a number of jobs that have to be completed within some fixed period of time. Each job consumes or supplies a certain, job-dependent quantity of a resource thus changing the contents of a stock. Natural examples for this... more
ABSTRACT Consider a number of jobs that have to be completed within some fixed period of time. Each job consumes or supplies a certain, job-dependent quantity of a resource thus changing the contents of a stock. Natural examples for this scenario arise in the case of trucks delivering goods to a trade house and taking other goods away from it, or in the case where processes change their environment. The goal is to find an ordering of the jobs such that all jobs can be completed and such that the maximum stock size needed over the time period becomes minimum. Since this problem can be shown to be NP-hard, we present three polynomial time approximation algorithms which yield job sequences with maximum stock size at most 2, respectively 8/5 times and 3/2 times the optimum stock size. These approximation algorithms are also compared by a numerical simulation. Moreover, we consider a variation of the problem where consecutive processes are allowed to directly exchange resources without using the stock.