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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 ..."
Abstract

Cited by 33 (4 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 Branch and Bound Algorithm for the Quadratic Assignment Problem using a Lower Bound Based on Linear Programming
 In C. Floudas and P.M. Pardalos, editors, State of the Art in Global Optimization: Computational Methods and Applications
, 1995
"... In this paper, we study a branch and bound algorithm for the quadratic assignment problem (QAP) that uses a lower bound based on the linear programming (LP) relaxation of a classical integer programming formulation of the QAP. Computational experience with the branch and bound algorithm on several Q ..."
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Cited by 10 (2 self)
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In this paper, we study a branch and bound algorithm for the quadratic assignment problem (QAP) that uses a lower bound based on the linear programming (LP) relaxation of a classical integer programming formulation of the QAP. Computational experience with the branch and bound algorithm on several QAP test problems is reported. The linear programming relaxations are solved with an implementation of an interior point algorithm that uses a preconditioned conjugate gradient algorithm to compute directions. The branch and bound algorithm is compared with a similar branch and bound algorithm that uses the GilmoreLawler lower bound (GLB) instead of the LPbased bound. The LPbased algorithm examines a small portion of the nodes explored by the GLBbased 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 Opti...
MixedInteger Nonlinear Optimization in Process Synthesis
, 1998
"... The use of networks allows the representation of a variety of important engineering problems. The treatment of a particular class of network applications, the process synthesis problem, is exposed in this paper. Process Synthesis seeks to develop systematically process flowsheets that convert raw ma ..."
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Cited by 7 (0 self)
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The use of networks allows the representation of a variety of important engineering problems. The treatment of a particular class of network applications, the process synthesis problem, is exposed in this paper. Process Synthesis seeks to develop systematically process flowsheets that convert raw materials into desired products. In recent years, the optimization approach to process synthesis has shown promise in tackling this challenge. It requires the development of a network of interconnected units, the process superstructure, that represents the alternative process flowsheets. The mathematical modeling of the superstructure has a mixed set of binary and continuous variables and results in a mixedinteger optimization model. Due to the nonlinearity of chemical models, these problems are generally classified as MixedInteger Nonlinear Programming (MINLP) problems. A number of local optimization algorithms, developed for the solution of this class of problems, are presented in this pap...
Tight QAP Bounds Via Linear Programming
, 1998
"... . Lower bounds for the quadratic assignment problem (QAP) tend to deteriorate rapidly with the size of the QAP. Recently, Resende, Ramakrishnan, and Drezner (1995) computed a linear programming based lower bound for the QAP using an interior point algorithm for linear programming to solve the linear ..."
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Cited by 4 (0 self)
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. Lower bounds for the quadratic assignment problem (QAP) tend to deteriorate rapidly with the size of the QAP. Recently, Resende, Ramakrishnan, and Drezner (1995) computed a linear programming based lower bound for the QAP using an interior point algorithm for linear programming to solve the linear programming relaxation of a classical integer programming formulation of the QAP. That linear program can be viewed as a two body interaction formulation. Those bounds were found to be the tightest for a large number of instances from QAPLIB, a library of QAP test problems. In this paper, we apply the same interior point approach to compute lower bounds derived from the three body interaction formulation of Ramachandran and Pekny (1996). All instances from QAPLIB, having dimension up to n = 12, were solved. The new approach produces tight lower bounds (lower bounds equal to the optimal solution) for all instances tested. Attempts to solve the linear programming relaxations with CPLEX (prima...
Implementation Of A Variance Reduction Based Lower Bound In A Branch And Bound Algorithm For The Quadratic Assignment Problem
, 1997
"... . The efficient implementation of a branch and bound algorithm for the quadratic assignment problem (QAP), incorporating the lower bound, based on variance reduction, of Li, Pardalos, Ramakrishnan, and Resende (1994), is presented. A new data structure for efficient implementation of branch and boun ..."
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Cited by 4 (1 self)
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. The efficient implementation of a branch and bound algorithm for the quadratic assignment problem (QAP), incorporating the lower bound, based on variance reduction, of Li, Pardalos, Ramakrishnan, and Resende (1994), is presented. A new data structure for efficient implementation of branch and bound algorithms for the QAP is introduced. Computational experiments with the branch and bound algorithm on different classes of QAP test problems are reported. The branch and bound algorithm using the new lower bounds is compared with the same algorithm utilizing the commonly applied GilmoreLawler lower bound. Both implementations use a greedy randomized adaptive search procedure for obtaining initial upper bounds. The algorithms report all optimal permutations. Optimal solutions for previously unsolved instances from the literature, of dimensions n = 16 and n = 20, have been found with the new algorithm. In addition, the new algorithm has been tested on a class of large data variance problem...
Nonlinear and MixedInteger Optimization in Chemical Process Network Systems
, 1998
"... The use of networks allows the representation of a variety of important engineering problems. The treatment of a particular class of network applications, the Process Synthesis problem, is exposed in this paper. Process Synthesis seeks to develop systematically process flowsheets that convert raw ..."
Abstract
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The use of networks allows the representation of a variety of important engineering problems. The treatment of a particular class of network applications, the Process Synthesis problem, is exposed in this paper. Process Synthesis seeks to develop systematically process flowsheets that convert raw materials into desired products. In recent years, the optimization approach to process synthesis has shown promise in tackling this challenge. It requires the development of a network of interconnected units, the process superstructure, that represents the alternative process flowsheets. The mathematical modeling of the superstructure has a mixed set of binary and continuous variables and results in a mixedinteger optimization model. Due to the nonlinearity of chemical models, these problems are generally classified as MixedInteger Nonlinear Programming (MINLP) problems. A number of local optimization algorithms for MINLP problems are outlined in this paper: Generalized Benders Decompositi...
The MINLP approach to structural optimization
"... Abstract: The paper presents the MixedInteger Nonlinear Programming optimization approach (MINLP) to structural optimization. The MINLP is a combined continuous/discrete optimization technique, where a structural topology, discrete materials and standard sizes are optimized simultaneously with th ..."
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Abstract: The paper presents the MixedInteger Nonlinear Programming optimization approach (MINLP) to structural optimization. The MINLP is a combined continuous/discrete optimization technique, where a structural topology, discrete materials and standard sizes are optimized simultaneously with the continuous parameters (e.g. costs, mass). The MINLP optimization is performed through three steps: i.e. the generation of a mechanical superstructure, the modelling of an MINLP model formulation and the solution of the defined MINLP problem. The Modified OuterApproximation/EqualityRelaxation (OA/ER) algorithm and a twophase MINLP strategy are applied. The optimization is performed by a userfriendly version of the MINLP computer package MIPSYN. Two examples are presented at the end of the paper.