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Very LargeScale Neighborhood Search for the Quadratic Assignment Problem
 DISCRETE APPLIED MATHEMATICS
, 2002
"... The Quadratic Assignment Problem (QAP) consists of assigning n facilities to n locations so as to minimize the total weighted cost of interactions between facilities. The QAP arises in many diverse settings, is known to be NPhard, and can be solved to optimality only for fairly small size instances ..."
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Cited by 113 (12 self)
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The Quadratic Assignment Problem (QAP) consists of assigning n facilities to n locations so as to minimize the total weighted cost of interactions between facilities. The QAP arises in many diverse settings, is known to be NPhard, and can be solved to optimality only for fairly small size instances (typically, n < 25). Neighborhood search algorithms are the most popular heuristic algorithms to solve larger size instances of the QAP. The most extensively used neighborhood structure for the QAP is the 2exchange neighborhood. This neighborhood is obtained by swapping the locations of two facilities and thus has size O(n²). Previous efforts to explore larger size neighborhoods (such as 3exchange or 4exchange neighborhoods) were not very successful, as it took too long to evaluate the larger set of neighbors. In this paper, we propose very largescale neighborhood (VLSN) search algorithms where the size of the neighborhood is very large and we propose a novel search procedure to heuristically enumerate good neighbors. Our search procedure relies on the concept of improvement graph which allows us to evaluate neighbors much faster than the existing methods. We present extensive computational results of our algorithms on standard benchmark instances. These investigations reveal that very largescale neighborhood search algorithms give consistently better solutions compared the popular 2exchange neighborhood algorithms considering both the solution time and solution accuracy.
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 96 (1 self)
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We describe a few applications of semide nite programming in combinatorial optimization.
An Enabling Framework for MasterWorker Applications on the Computational Grid
 Cluster Computing
, 2000
"... We describe MW  a software framework that allows users to quickly and easily parallelize scientific computations using the masterworker paradigm on the computational grid. MW provides both a "top level" interface to application software and a "bottom level" interface to exi ..."
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Cited by 78 (10 self)
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We describe MW  a software framework that allows users to quickly and easily parallelize scientific computations using the masterworker paradigm on the computational grid. MW provides both a "top level" interface to application software and a "bottom level" interface to existing grid computing toolkits. Both interfaces are briefly described. We conclude with a case study, where the necessary Grid services are provided by the Condor highthroughput computing system, and the MWenabled application code is used to solve a combinatorial optimization problem of unprecedented complexity. This work was supported in part by Grants No. CDA9726385 and CDA9623632 from the National Science Foundation. y Department of Electrical and Computer Engineering, Northwestern University, and Mathematics and Computer Science Division, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, Illinois 60439, goux@mcs.anl.gov z Computer Sciences Department, University of Wisconsin  Madison, 1210 West Dayton Street, Madison, WI 53706, fsanjeevk,yodermeg@cs.wisc.edu x Mathematics and Computer Science Division, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, Illinois 60439, linderot@mcs.anl.gov 1 1
Handbook of semidefinite programming
"... Semidefinite programming (or SDP) has been one of the most exciting and active research areas in optimization during the 1990s. It has attracted researchers with very diverse backgrounds, including experts in convex programming, linear algebra, numerical optimization, combinatorial optimization, con ..."
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Cited by 61 (2 self)
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Semidefinite programming (or SDP) has been one of the most exciting and active research areas in optimization during the 1990s. It has attracted researchers with very diverse backgrounds, including experts in convex programming, linear algebra, numerical optimization, combinatorial optimization, control theory, and statistics. This tremendous research activity was spurred by the discovery of important applications in combinatorial optimization and control theory, the development of efficient interiorpoint algorithms for solving SDP problems, and the depth and elegance of the underlying optimization theory. This book includes nineteen chapters on the theory, algorithms, and applications of semidefinite programming. Written by the leading experts on the subject, it offers an advanced and broad overview of the current state of the field. The coverage is somewhat less comprehensive, and the overall level more advanced, than we had planned at the start of the project. In order to finish the book in a timely fashion, we have had to abandon hopes for separate chapters on some important topics (such as a discussion of SDP algorithms in the
Frequency Assignment Problems
 HANDBOOK OF COMBINATORIAL OPTIMIZATION
, 1999
"... The ever growing number of wireless communications systems deployed around the globe have made the optimal assignment of a limited radio frequency spectrum a problem of primary importance. At issue are planning models for permanent spectrum allocation, licensing, regulation, and network design. Furt ..."
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Cited by 30 (3 self)
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The ever growing number of wireless communications systems deployed around the globe have made the optimal assignment of a limited radio frequency spectrum a problem of primary importance. At issue are planning models for permanent spectrum allocation, licensing, regulation, and network design. Further at issue are online algorithms for dynamically assigning frequencies to users within an established network. Applications include aeronautical mobile, land mobile, maritime mobile, broadcast, land fixed (pointto point), and satellite systems. This paper surveys research conducted by theoreticians, engineers, and computer scientists regarding the frequency assignment problem (FAP) in all of its guises. The paper begins by defining some of the more common types of FAPs. It continues with a discussion on measures of optimality relating to the use of spectrum, models of interference, and mathematical representations of the many FAPs, both in graph theoretic terms, and as mathematical pro...
Copositive and semidefinite relaxations of the quadratic assignment problem
 DISCRETE OPTIM
, 2009
"... 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 co ..."
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Cited by 19 (3 self)
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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 can equivalently be formulated as a linear program over the cone of completely positive matrices. Since it is hard to optimize over this cone, we also look at tractable approximations and compare with several relaxations from the literature. We show that several of the wellstudied models are in fact equivalent. It is still a challenging task to solve the strongest of these models to reasonable accuracy on instances of moderate size.
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 14 (9 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 ...
Algorithms for the generalized quadratic assignment problem combining Lagrangean . . .
 COMPUTATIONAL OPTIMIZATION AND APPLICATIONS
"... In this paper, we propose two exact algorithms for the GQAP (generalized quadratic assignment problem). In this problem, given M facilities and N locations, the facility space requirements, the location available space, the facility installation costs, the flows between facilities, and the distance ..."
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Cited by 12 (5 self)
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In this paper, we propose two exact algorithms for the GQAP (generalized quadratic assignment problem). In this problem, given M facilities and N locations, the facility space requirements, the location available space, the facility installation costs, the flows between facilities, and the distance costs between locations, one must assign each facility to exactly one location so that each location has sufficient space for all facilities assigned to it and the sum of the products of the facility flows by the corresponding distance costs plus the sum of the installation costs is minimized. This problem generalizes the wellknown quadratic assignment problem (QAP). Both exact algorithms combine a previously proposed branchandbound scheme with a new Lagrangean relaxation procedure over a known RLT (ReformulationLinearization Technique) formulation. We also apply transformational lower bounding techniques to improve the performance of the new procedure. We report detailed experimental results where 19 out of 21 instances with up to 35 facilities are solved in up to a few days of running time. Six of these instances were open.
A hybrid genetic algorithm for the quadratic assignment problem
 in GECCO2000: Proceedings of the Genetic and Evolutionary Computation Conference
, 2000
"... A heuristic technique that combines a genetic algorithm with a Tabu Search algorithm is applied to the Quadratic Assignment Problem (QAP). The hybrid algorithm improves the results obtained through the application of each of these algorithms separately. The QAP is a NPhard problem and instances of ..."
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Cited by 9 (0 self)
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A heuristic technique that combines a genetic algorithm with a Tabu Search algorithm is applied to the Quadratic Assignment Problem (QAP). The hybrid algorithm improves the results obtained through the application of each of these algorithms separately. The QAP is a NPhard problem and instances of size n> 15 are still considered intractable. The results of our experiments suggest that CHC combined with TS (CHC+TS), and a TS with elitist backtracking algorithm are able to obtain good near optimal solutions within 0.75 % of the bestknown solutions. CHC+TS produces the bestknown solution in 12 of the 16 QAPLIB problems tested, where n ranges from 10 to 256. 1