A Branch and Bound Algorithm for the Quadratic Assignment Problem using a Lower Bound Based on Linear Programming

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A Branch and Bound Algorithm for the Quadratic Assignment Problem using a Lower Bound Based on Linear Programming (1995)

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by K. G. Ramakrishnan , M. G. C. Resende , P.M. Pardalos

Venue:	In C. Floudas and P.M. Pardalos, editors, State of the Art in Global Optimization: Computational Methods and Applications
Citations:	10 - 2 self

Citations

527	AMPL: A Modeling Language for Mathematical Programming - Fourer, Gay, et al. - 1993 (Show Context) Citation Context ...he system is composed of four components: a branch and bound control module; a greedy randomized adaptive search procedure (GRASP) module to produce the initial upper bound; an AMPL modeling language =-=[10]-=- module to manage the linear programming models; and the linear programming solver ADP to produce the lower bounds for the branch and bound algorithm. The branch and bound control module controls the ...
313	Assignment problems and the location of economic activities - Koopmans, Beckman - 1957 (Show Context) Citation Context ...bound. The LP-based algorithm examines a small portion of the nodes explored by the GLB-based algorithm. 1 Introduction The quadratic assignment problem (QAP), first proposed by Koopmans and Beckmann =-=[16]-=-, can be stated as min p2\Pi n X i=1 n X j=1 a ij b p(i)p(j) ; To appear in Proceedings of State of the Art in Global Optimization: Computational Methods and Applications, Princeton University, April ...
200	QAPLIB --- A Quadratic Assignment Problem Library - Burkard, Karisch, et al. - 1991 (Show Context) Citation Context ...able 1 shows the dimension of theses linear programs for several values of n. Resende, Ramakrishnan, and Drezner [31] tested the LP-based lower bound on the 63 instances of dimension ns30 from QAPLIB =-=[4]-=-, a standard library 4 of QAP test problems. On the majority of the instances tested, the LP-based lower bound was the new best known lower bound. In 87 percent of the cases, the best lower bound was ...
127	The quadratic assignment problem. - Lawler - 1963 (Show Context) Citation Context ...ervation is that the lower bounds are good, resulting in the search of very few branch and bound search tree nodes compared to the same branching scheme using the classical Gilmore-Lawler lower bound =-=[12, 18, 28]-=-. The number of nodes scanned grew less than a cubic function of n, the dimension of the QAP. The instance with the largest number of nodes (nug15 of dimension n = 15) required the solution of 1195 li...
123	A Greedy Randomized Adaptive Search Procedure for the Quadratic Assignment Problem, in: Quadratic assignment and related problems - Li, Pardalos, et al. - 1994 (Show Context) Citation Context ...lle, FL 32611 USA. e-mail: pardalos@ufl.edu 2 where \Pi is the set of all permutations of f1; 2; : : : ; ng, A = (a ij ) 2 R n\Thetan , B = (b ij ) 2 R n\Thetan . Applications of the QAP are abundant =-=[1, 11, 13, 17, 20, 24, 29]-=-. Many classical combinatorial optimization problems, such as the traveling salesman problem, graph partitioning, and maximum clique are special cases of the QAP. A wide range of heuristics have been ...
106	Genetic Hybrids for the Quadratic Assignment Problem - Fleurent, Ferland - 1994 (Show Context) Citation Context ...ems, such as the traveling salesman problem, graph partitioning, and maximum clique are special cases of the QAP. A wide range of heuristics have been applied to find approximate solutions to the QAP =-=[5, 9, 20, 23, 26, 33, 34, 35]-=-. Exact solution approaches have been limited to instances of dimension ns15. General QAPs of dimension n ? 15 are considered difficult large-scale problems. Branch and bound algorithms have been the ...
100	Tabu search applied to the quadratic assignment problem. - Skorin-Kapov - 1990 (Show Context) Citation Context ...ems, such as the traveling salesman problem, graph partitioning, and maximum clique are special cases of the QAP. A wide range of heuristics have been applied to find approximate solutions to the QAP =-=[5, 9, 20, 23, 26, 33, 34, 35]-=-. Exact solution approaches have been limited to instances of dimension ns15. General QAPs of dimension n ? 15 are considered difficult large-scale problems. Branch and bound algorithms have been the ...
98	Optimal and suboptimal algorithms for the quadratic assignment problem,” - Gilmore - 1962 (Show Context) Citation Context ...ervation is that the lower bounds are good, resulting in the search of very few branch and bound search tree nodes compared to the same branching scheme using the classical Gilmore-Lawler lower bound =-=[12, 18, 28]-=-. The number of nodes scanned grew less than a cubic function of n, the dimension of the QAP. The instance with the largest number of nodes (nug15 of dimension n = 15) required the solution of 1195 li...
81	A Thermodynamically Motivated Simulation Procedure for Combinatorial Optimization Problems - Burkard, Rendl - 1983 (Show Context) Citation Context ...ems, such as the traveling salesman problem, graph partitioning, and maximum clique are special cases of the QAP. A wide range of heuristics have been applied to find approximate solutions to the QAP =-=[5, 9, 20, 23, 26, 33, 34, 35]-=-. Exact solution approaches have been limited to instances of dimension ns15. General QAPs of dimension n ? 15 are considered difficult large-scale problems. Branch and bound algorithms have been the ...
71	Robust tabu search for the quadratic assignment problem. - Taillard - 1991 (Show Context) Citation Context ...ems, such as the traveling salesman problem, graph partitioning, and maximum clique are special cases of the QAP. A wide range of heuristics have been applied to find approximate solutions to the QAP =-=[5, 9, 20, 23, 26, 33, 34, 35]-=-. Exact solution approaches have been limited to instances of dimension ns15. General QAPs of dimension n ? 15 are considered difficult large-scale problems. Branch and bound algorithms have been the ...
61	Assignment Methods in Combinatorial Data Analysis. - Hubert - 1987 (Show Context) Citation Context ...lle, FL 32611 USA. e-mail: pardalos@ufl.edu 2 where \Pi is the set of all permutations of f1; 2; : : : ; ng, A = (a ij ) 2 R n\Thetan , B = (b ij ) 2 R n\Thetan . Applications of the QAP are abundant =-=[1, 11, 13, 17, 20, 24, 29]-=-. Many classical combinatorial optimization problems, such as the traveling salesman problem, graph partitioning, and maximum clique are special cases of the QAP. A wide range of heuristics have been ...
51	Assignment Problems in Parallel and Distributed Computing - Bokhari - 1987 (Show Context) Citation Context ...lle, FL 32611 USA. e-mail: pardalos@ufl.edu 2 where \Pi is the set of all permutations of f1; 2; : : : ; ng, A = (a ij ) 2 R n\Thetan , B = (b ij ) 2 R n\Thetan . Applications of the QAP are abundant =-=[1, 11, 13, 17, 20, 24, 29]-=-. Many classical combinatorial optimization problems, such as the traveling salesman problem, graph partitioning, and maximum clique are special cases of the QAP. A wide range of heuristics have been ...
41	Facility Layout and Location - Francis, White - 1974 (Show Context) Citation Context ...lle, FL 32611 USA. e-mail: pardalos@ufl.edu 2 where \Pi is the set of all permutations of f1; 2; : : : ; ng, A = (a ij ) 2 R n\Thetan , B = (b ij ) 2 R n\Thetan . Applications of the QAP are abundant =-=[1, 11, 13, 17, 20, 24, 29]-=-. Many classical combinatorial optimization problems, such as the traveling salesman problem, graph partitioning, and maximum clique are special cases of the QAP. A wide range of heuristics have been ...
40	Computational results of an interior point algorithm for large scale linear programming, - Karmarkar, Ramakrishnan - 1991 (Show Context) Citation Context ...plementation of a dual interior point algorithm, with centering, that uses a preconditioned conjugate gradient algorithm to compute the directions taken at each iteration by the interior point method =-=[15]-=-. Attempts at using the simplex and (direct factorization based) interior point codes of CPLEX 1 were successful only for the smallest instances. Since the interior point algorithm used solves the dua...
38	Assignment and Matching Problems: Solution Methods with Fortran Programs - Burkard, Derigs - 1980 (Show Context) Citation Context ...scale problems. Branch and bound algorithms have been the most successful methods for proving optimality of quadratic assignment problems. Branch and bound algorithms for the QAP have been studied in =-=[8, 3, 32, 27, 22, 21, 6, 28]-=-. Other exact approaches are described in [29]. Lower bounds are key to the computational efficiency of branch and bound algorithms. Recently, Resende, Ramakrishnanm, and Drezner [31] used an efficien...
36	Computing lower bounds for the quadratic assignment problem with an interior point algorithm for linear programming. - Resende, Ramakrishnan, et al. - 1995 (Show Context) Citation Context ...2, 27, 22, 21, 6, 28]. Other exact approaches are described in [29]. Lower bounds are key to the computational efficiency of branch and bound algorithms. Recently, Resende, Ramakrishnanm, and Drezner =-=[31]-=- used an efficient implementation of an interior point algorithm to compute lower bounds for the QAP by solving the linear programming (LP) relaxation of a classical integer programming formulation of...
34	Computer-aided layout design.” - Krarup, Pruzan - 1978 (Show Context) Citation Context ...lle, FL 32611 USA. e-mail: pardalos@ufl.edu 2 where \Pi is the set of all permutations of f1; 2; : : : ; ng, A = (a ij ) 2 R n\Thetan , B = (b ij ) 2 R n\Thetan . Applications of the QAP are abundant =-=[1, 11, 13, 17, 20, 24, 29]-=-. Many classical combinatorial optimization problems, such as the traveling salesman problem, graph partitioning, and maximum clique are special cases of the QAP. A wide range of heuristics have been ...
34	A parallel algorithm for the quadratic assignment problem, - Pardalos, M, et al. - 1989 (Show Context) Citation Context ...scale problems. Branch and bound algorithms have been the most successful methods for proving optimality of quadratic assignment problems. Branch and bound algorithms for the QAP have been studied in =-=[8, 3, 32, 27, 22, 21, 6, 28]-=-. Other exact approaches are described in [29]. Lower bounds are key to the computational efficiency of branch and bound algorithms. Recently, Resende, Ramakrishnanm, and Drezner [31] used an efficien...
30	editors. Quadratic assignment and related problems. A merican - Pardalos, Wolkowicz - 1994 (Show Context) Citation Context ...lle, FL 32611 USA. e-mail: pardalos@ufl.edu 2 where \Pi is the set of all permutations of f1; 2; : : : ; ng, A = (a ij ) 2 R n\Thetan , B = (b ij ) 2 R n\Thetan . Applications of the QAP are abundant =-=[1, 11, 13, 17, 20, 24, 29]-=-. Many classical combinatorial optimization problems, such as the traveling salesman problem, graph partitioning, and maximum clique are special cases of the QAP. A wide range of heuristics have been ...
24	Theoretical Comparisons of Search Strategies in Branch-and-Bound Algorithms - Ibaraki - 1976 (Show Context) Citation Context ...eta\Theta\Theta\Theta\Theta\Theta\Theta\Theta\Theta\Theta\Theta\Theta\Theta\Theta\Theta\Theta\Theta\Theta optimal QAP bound Figure 1: CPLEX simplex and ADP iterates on nug15 6 The notation of Ibaraki =-=[14]-=- is employed to formally define a branch-andbound algorithm that will be needed in the sequel. Let P 0 denote an optimization problem and f denote the objective function to be minimized. The decomposi...
22	A parallel branch and bound algorithm for the quadratic assignment problem - Roucairol - 1987
21	A.: An improved simulating annealing algorithm for the quadratic assignment problem. Working paper, - Thonemann, Bolte - 1994 (Show Context) Citation Context ...ems, such as the traveling salesman problem, graph partitioning, and maximum clique are special cases of the QAP. A wide range of heuristics have been applied to find approximate solutions to the QAP =-=[5, 9, 20, 23, 26, 33, 34, 35]-=-. Exact solution approaches have been limited to instances of dimension ns15. General QAPs of dimension n ? 15 are considered difficult large-scale problems. Branch and bound algorithms have been the ...
20	Lower bounds for the quadratic assignment problem - Li, Pardalos, et al. - 1994 (Show Context) Citation Context ... Ramakrishnan, Resende, and Li [28] describe a branch and bound algorithm used to study the effectiveness of a variance reduction based lower bound proposed by Li, Pardalos, Ramakrishnan, and Resende =-=[19]-=-. In this paper, we use this branch and bound algorithm in conjunction with the LPbased lower bound described in Section 2. We next describe the branch and bound algorithm and illustrate the algorithm...
15	Solving Nonlinear Integer Programs with Large-scale Optimization Software, - Mawergkang, Murtagh - 1986 (Show Context) Citation Context ...ems, such as the traveling salesman problem, graph partitioning, and maximum clique are special cases of the QAP. A wide range of heuristics have been applied to find approximate solutions to the QAP =-=[5, 9, 20, 23, 26, 33, 34, 35]-=-. Exact solution approaches have been limited to instances of dimension ns15. General QAPs of dimension n ? 15 are considered difficult large-scale problems. Branch and bound algorithms have been the ...
13	Using an interior point method in a branch and bound algorithm for integer programming, - Borchers, Mitchell - 1992 (Show Context) Citation Context ... point algorithm is used to compute the lower bounds. The use of interior point methods in the context of branch and bound methods for integer programming was first described by Mitchell and Borchers =-=[2]-=-. See Mitchell [25] for a survey on interior point algorithms for integer programming. Our implementation successfully solved to optimality all instances of QAPLIB [4] with dimension ns15. The main ob...
13	FORTRAN subroutines for approximate solution of dense quadratic assignment problems using GRASP - Resende, Pardalos, et al. (Show Context) Citation Context ...ed at this node. The binary tree is organized in maximum order, i.e. the node with the largest lower bound is first. The initial best known upper bound is computed by the GRASP heuristic described in =-=[20, 30]-=-. The initial search tree consists of n nodes with SA = fig and SE = ; for i = 1; : : : ; n, and q(i) = p(i), where p is the permutation obtained by the GRASP and for k 6= i, q(k) = 0 and a key of 0. ...
9	A branch and bound algorithm for the Koopmans-Beckman quadratic assignment problem - Edwards - 1980 (Show Context) Citation Context ...scale problems. Branch and bound algorithms have been the most successful methods for proving optimality of quadratic assignment problems. Branch and bound algorithms for the QAP have been studied in =-=[8, 3, 32, 27, 22, 21, 6, 28]-=-. Other exact approaches are described in [29]. Lower bounds are key to the computational efficiency of branch and bound algorithms. Recently, Resende, Ramakrishnanm, and Drezner [31] used an efficien...
8	Lower bounds based on linear programming for the quadratic assignment problem - Drezner - 1994 (Show Context) Citation Context ...e algorithm and implementation being used. That study also showed that the quality of the lower bounds produced was good, suggesting that they be incorporated in a branch and bound algorithm. Drezner =-=[7]-=- had previously studied this lower bound. In this paper, we show how this lower bound performs when implemented in a branch and bound algorithm for the QAP. We report computational experiments on a se...
6	Difficulties of exact methods for solving the quadratic assignment problem - Mautor, Roucairol - 1994 (Show Context) Citation Context ...scale problems. Branch and bound algorithms have been the most successful methods for proving optimality of quadratic assignment problems. Branch and bound algorithms for the QAP have been studied in =-=[8, 3, 32, 27, 22, 21, 6, 28]-=-. Other exact approaches are described in [29]. Lower bounds are key to the computational efficiency of branch and bound algorithms. Recently, Resende, Ramakrishnanm, and Drezner [31] used an efficien...
6	Interior point algorithms for integer programming - Mitchell - 1996 (Show Context) Citation Context ...s used to compute the lower bounds. The use of interior point methods in the context of branch and bound methods for integer programming was first described by Mitchell and Borchers [2]. See Mitchell =-=[25]-=- for a survey on interior point algorithms for integer programming. Our implementation successfully solved to optimality all instances of QAPLIB [4] with dimension ns15. The main observation is that t...
5	A heuristic procedure for solving the quadratic assignment problem - Murtagh, Jefferson, et al. - 1982 (Show Context) Citation Context ...ems, such as the traveling salesman problem, graph partitioning, and maximum clique are special cases of the QAP. A wide range of heuristics have been applied to find approximate solutions to the QAP =-=[5, 9, 20, 23, 26, 33, 34, 35]-=-. Exact solution approaches have been limited to instances of dimension ns15. General QAPs of dimension n ? 15 are considered difficult large-scale problems. Branch and bound algorithms have been the ...
5	Implementation of a variance reduction based lower bound in a branch and bound algorithm for the quadratic assignment problem - Pardalos, Ramakrishnan, et al. - 1997 (Show Context) Citation Context ...scale problems. Branch and bound algorithms have been the most successful methods for proving optimality of quadratic assignment problems. Branch and bound algorithms for the QAP have been studied in =-=[8, 3, 32, 27, 22, 21, 6, 28]-=-. Other exact approaches are described in [29]. Lower bounds are key to the computational efficiency of branch and bound algorithms. Recently, Resende, Ramakrishnanm, and Drezner [31] used an efficien...
3	Human Factors Engineering - McCormik - 1970 (Show Context) Citation Context ...lle, FL 32611 USA. e-mail: pardalos@ufl.edu 2 where \Pi is the set of all permutations of f1; 2; : : : ; ng, A = (a ij ) 2 R n\Thetan , B = (b ij ) 2 R n\Thetan . Applications of the QAP are abundant =-=[1, 11, 13, 17, 20, 24, 29]-=-. Many classical combinatorial optimization problems, such as the traveling salesman problem, graph partitioning, and maximum clique are special cases of the QAP. A wide range of heuristics have been ...
1	Announcement on Discrete Mathematics and Algorithms Network (DIMANET - Clausen - 1994 (Show Context) Citation Context ...scale problems. Branch and bound algorithms have been the most successful methods for proving optimality of quadratic assignment problems. Branch and bound algorithms for the QAP have been studied in =-=[8, 3, 32, 27, 22, 21, 6, 28]-=-. Other exact approaches are described in [29]. Lower bounds are key to the computational efficiency of branch and bound algorithms. Recently, Resende, Ramakrishnanm, and Drezner [31] used an efficien...