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53
QAPLIB  A Quadratic Assignment Problem Library
, 1996
"... This report, the data and also most of the best feasible solutions are available via World Wide Web. The URLs of the QAPLIB Home Page are http://www.opt.math.tugraz.ac.at/qaplib/ ..."
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Cited by 163 (6 self)
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This report, the data and also most of the best feasible solutions are available via World Wide Web. The URLs of the QAPLIB Home Page are http://www.opt.math.tugraz.ac.at/qaplib/
Semidefinite Programming Relaxations For The Quadratic Assignment Problem
, 1998
"... Semidefinite programming (SDP) relaxations for the quadratic assignment problem (QAP) are derived using the dual of the (homogenized) Lagrangian dual of appropriate equivalent representations of QAP. These relaxations result in the interesting, special, case where only the dual problem of the SDP re ..."
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Cited by 72 (25 self)
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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 equivalent representations of QAP. These relaxations result in the interesting, special, case where only the dual problem of the SDP relaxation has strict interior, i.e. the Slater constraint qualification always fails for the primal problem. Although there is no duality gap in theory, this indicates that the relaxation cannot be solved in a numerically stable way. By exploring the geometrical structure of the relaxation, we are able to find projected SDP relaxations. These new relaxations, and their duals, satisfy the Slater constraint qualification, and so can be solved numerically using primaldual interiorpoint methods. For one of our models, a preconditioned conjugate gradient method is used for solving the large linear systems which arise when finding the Newton direction. The preconditioner is found by exploiting th...
Solving Large Quadratic Assignment Problems on Computational Grids
, 2000
"... The quadratic assignment problem (QAP) is among the hardest combinatorial optimization problems. Some instances of size n = 30 have remained unsolved for decades. The solution of these problems requires both improvements in mathematical programming algorithms and the utilization of powerful computat ..."
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Cited by 68 (6 self)
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The quadratic assignment problem (QAP) is among the hardest combinatorial optimization problems. Some instances of size n = 30 have remained unsolved for decades. The solution of these problems requires both improvements in mathematical programming algorithms and the utilization of powerful computational platforms. In this article we describe a novel approach to solve QAPs using a stateoftheart branchandbound algorithm running on a federation of geographically distributed resources known as a computational grid. Solution of QAPs of unprecedented complexity, including the nug30, kra30b, and tho30 instances, is reported.
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 50 (19 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 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.
Eigenvalues in combinatorial optimization
, 1993
"... In the last decade many important applications of eigenvalues and eigenvectors of graphs in combinatorial optimization were discovered. The number and importance of these results is so fascinating that it makes sense to present this survey. ..."
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Cited by 42 (0 self)
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In the last decade many important applications of eigenvalues and eigenvectors of graphs in combinatorial optimization were discovered. The number and importance of these results is so fascinating that it makes sense to present this survey.
A Projection Technique for Partitioning the Nodes of a Graph
, 1995
"... Let G = (N; E) be a given undirected graph. We present several new techniques for partitioning the node set N into k disjoint subsets of specified sizes. These techniques involve eigenvalue bounds and tools from continuous optimization. Comparisons with examples taken from the literature show these ..."
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Cited by 41 (13 self)
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Let G = (N; E) be a given undirected graph. We present several new techniques for partitioning the node set N into k disjoint subsets of specified sizes. These techniques involve eigenvalue bounds and tools from continuous optimization. Comparisons with examples taken from the literature show these techniques to be very successful. 1 Introduction Let G = (N; E) be a given undirected graph with node set N = f1; : : : ; ng and edge set E. A common problem in circuit board and microchip design, computer program segmentation, floor planning and other layout problems is to partition the node set N into k disjoint subsets S 1 ; : : : ; S k of specified sizes m 1 m 2 : : : m k ; P k j=1 m j = n, so as to minimize the number of edges connecting nodes in distinct subsets of the partition. We refer to an edge, which connects nodes in distinct subsets of the partition, as being cut by the partition. A recent survey on the graph partitioning problem and further related problems is containe...
Computing lower bounds for the quadratic assignment problem with an interior point algorithm for linear programming
 Operations Research
, 1995
"... A typical example of the quadratic assignment problem (QAP) is the facility location problem, in which a set of n facilities are to be assigned, at minimum cost, to an equal number of locations. Between each pair of facilities, there is a given amount of flow, contributing a cost equal to the produc ..."
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Cited by 33 (3 self)
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A typical example of the quadratic assignment problem (QAP) is the facility location problem, in which a set of n facilities are to be assigned, at minimum cost, to an equal number of locations. Between each pair of facilities, there is a given amount of flow, contributing a cost equal to the product of the flow and the distance between locations to which the facilities are assigned. Proving optimality of solutions to quadratic assignment problems has been limited to instances of small dimension (n less than or equal to 20), in part because known lower bounds for the QAP are of poor quality. In this paper, we compute lower bounds for a wide range of quadratic assignment problems using a linear programmingbased lower bound studied by Drezner (1994). On the majority of quadratic assignment problems tested, the computed lower bound is the new best known lower bound. In 87 percent of the instances, we produced the best known lower bound. On several instances, including some of dimension n equal to 20, the lower bound is tight. The linear programs, which can be large even for moderate values of n, are solved with an interior point code that uses a preconditioned conjugate gradient algorithm to compute the directions taken at each iteration by the interior point algorithm. Attempts to
A New Bound for the Quadratic Assignment Problem Based on Convex Quadratic Programming
 MATHEMATICAL PROGRAMMING
, 1999
"... We describe a new convex quadratic programming bound for the quadratic assignment problem (QAP). The construction of the bound uses a semidefinite programming representation of a basic eigenvalue bound for QAP. The new bound dominates the wellknown projected eigenvalue bound, and appears to be comp ..."
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Cited by 33 (4 self)
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We describe a new convex quadratic programming bound for the quadratic assignment problem (QAP). The construction of the bound uses a semidefinite programming representation of a basic eigenvalue bound for QAP. The new bound dominates the wellknown projected eigenvalue bound, and appears to be competitive with existing bounds in the tradeoff between bound quality and computational effort.