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The Quadratic Assignment Problem: A Survey and Recent Developments
 In Proceedings of the DIMACS Workshop on Quadratic Assignment Problems, volume 16 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science
, 1994
"... . Quadratic Assignment Problems model many applications in diverse areas such as operations research, parallel and distributed computing, and combinatorial data analysis. In this paper we survey some of the most important techniques, applications, and methods regarding the quadratic assignment probl ..."
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Cited by 90 (16 self)
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. Quadratic Assignment Problems model many applications in diverse areas such as operations research, parallel and distributed computing, and combinatorial data analysis. In this paper we survey some of the most important techniques, applications, and methods regarding the quadratic assignment problem. We focus our attention on recent developments. 1. Introduction Given a set N = f1; 2; : : : ; ng and n \Theta n matrices F = (f ij ) and D = (d kl ), the quadratic assignment problem (QAP) can be stated as follows: min p2\Pi N n X i=1 n X j=1 f ij d p(i)p(j) + n X i=1 c ip(i) ; where \Pi N is the set of all permutations of N . One of the major applications of the QAP is in location theory where the matrix F = (f ij ) is the flow matrix, i.e. f ij is the flow of materials from facility i to facility j, and D = (d kl ) is the distance matrix, i.e. d kl represents the distance from location k to location l [62, 67, 137]. The cost of simultaneously assigning facility i to locat...
Quadratic Optimization
, 1995
"... . Quadratic optimization comprises one of the most important areas of nonlinear programming. Numerous problems in real world applications, including problems in planning and scheduling, economies of scale, and engineering design, and control are naturally expressed as quadratic problems. Moreover, t ..."
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Cited by 45 (3 self)
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. Quadratic optimization comprises one of the most important areas of nonlinear programming. Numerous problems in real world applications, including problems in planning and scheduling, economies of scale, and engineering design, and control are naturally expressed as quadratic problems. Moreover, the quadratic problem is known to be NPhard, which makes this one of the most interesting and challenging class of optimization problems. In this chapter, we review various properties of the quadratic problem, and discuss different techniques for solving various classes of quadratic problems. Some of the more successful algorithms for solving the special cases of bound constrained and large scale quadratic problems are considered. Examples of various applications of quadratic programming are presented. A summary of the available computational results for the algorithms to solve the various classes of problems is presented. Key words: Quadratic optimization, bilinear programming, concave pro...
An Interior Point Algorithm to Solve Computationally Difficult Set Covering Problems
, 1990
"... ..."
Reformulations in Mathematical Programming: A Computational Approach
"... Summary. Mathematical programming is a language for describing optimization problems; it is based on parameters, decision variables, objective function(s) subject to various types of constraints. The present treatment is concerned with the case when objective(s) and constraints are algebraic mathema ..."
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Cited by 16 (12 self)
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Summary. Mathematical programming is a language for describing optimization problems; it is based on parameters, decision variables, objective function(s) subject to various types of constraints. The present treatment is concerned with the case when objective(s) and constraints are algebraic mathematical expressions of the parameters and decision variables, and therefore excludes optimization of blackbox functions. A reformulation of a mathematical program P is a mathematical program Q obtained from P via symbolic transformations applied to the sets of variables, objectives and constraints. We present a survey of existing reformulations interpreted along these lines, some example applications, and describe the implementation of a software framework for reformulation and optimization. 1
SmoothandDive Accelerator: A PreMILP Primal Heuristic applied to Scheduling
, 2002
"... This article describes an effective and simple primal heuristic to greedily encourage a reduction in the number of binary or 01 logic variables before an implicit enumerativetype search heuristic is deployed to find integerfeasible solutions to “hard” production scheduling problems. The basis of ..."
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Cited by 3 (1 self)
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This article describes an effective and simple primal heuristic to greedily encourage a reduction in the number of binary or 01 logic variables before an implicit enumerativetype search heuristic is deployed to find integerfeasible solutions to “hard” production scheduling problems. The basis of the technique is to employ wellknown smoothing functions used to solve complementarity problems to the local optimization problem of minimizing the weighted sum over all binary variables the product of themselves multiplied by their complement. The basic algorithm of the “smoothanddive accelerator ” (SDA) is to solve successive linear programming (LP) relaxations with the smoothing functions added to the existing problem’s objective function and to use, if required, a sequence of binary variable fixings known as “diving”. If the smoothing function term is not driven to zero as part of the recursion then a branchandbound or branchandcut search heuristic is called to close the procedure finding at least integerfeasible primal infeasible solutions. The heuristic’s effectiveness is illustrated by its application to an oilrefinery’s crudeoil blendshop scheduling problem, which has commonality to many other production scheduling problems in the continuous and semicontinuous (CSC) process domains.
J.T.: A Continuous Quadratic Programming Formulation of the Vertex Separator Problem
"... Abstract. The Vertex Separator Problem (VSP) for a graph is to find the smallest collection of vertices whose removal breaks the graph into two disconnected subsets of roughly equal size. In a recent paper (Optimality Conditions For Maximizing a Function Over a Polyhedron, Mathematical Programming, ..."
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Abstract. The Vertex Separator Problem (VSP) for a graph is to find the smallest collection of vertices whose removal breaks the graph into two disconnected subsets of roughly equal size. In a recent paper (Optimality Conditions For Maximizing a Function Over a Polyhedron, Mathematical Programming, 2013, doi: 10.1007/s1010701306441), the authors announced a new continuous bilinear quadratic programming formulation of the VSP, and they used this quadratic programming problem to illustrate the new optimality conditions. The current paper develops conditions for the equivalence between this continuous quadratic program and the vertex separator problem, and it examines the relationship between the continuous formulation of the VSP and continuous quadratic programming formulations for both the edge separator problem and maximum clique problem.
Pivot, Cut, and Dive: A Heuristic for 01 Mixed Integer Programming
, 2001
"... We present a heuristic method for general 01 mixed integer programming, intended for eventual incorporation into parallel branchandbound methods for solving such problems exactly. The core of the heuristic is a rounding method based on simplex pivots, employing only gradient information, for a st ..."
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We present a heuristic method for general 01 mixed integer programming, intended for eventual incorporation into parallel branchandbound methods for solving such problems exactly. The core of the heuristic is a rounding method based on simplex pivots, employing only gradient information, for a strictly concave, differentiable merit function measuring integer feasibility. When local minima of this merit function are not integerfeasible, several additional layers of the heuristic come into play. These successive layers include explicit probing of adjacent vertices, modification of the merit function, adjoining of "convexity" cuts to the formulation, and a diving procedure that attempts to fix multiple variables simultaneously. We present "standalone" computational results, running the heuristic by itself without an accompanying branchandbound optimization, on a variety of problems from the MIPLIB collection.
An Exact Penalty Global Optimization Approach for MixedInteger Programming Problems
"... In this work, we propose a global optimization approach for mixedinteger programming problems. To this aim, we preliminarily define an exact penalty algorithm model for globally solving general problems and we show its convergence properties. Then, we describe a particular version of the algorithm ..."
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In this work, we propose a global optimization approach for mixedinteger programming problems. To this aim, we preliminarily define an exact penalty algorithm model for globally solving general problems and we show its convergence properties. Then, we describe a particular version of the algorithm that solves mixed integer problems.