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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 91 (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...
The feasibility pump
 Mathematical Programming
, 2005
"... In this paper we consider the NPhard problem of finding a feasible solution (if any exists) for a generic MIP problem of the form min{cT x: Ax ≥ b, xj integer ∀j ∈ I}. Trivially, a feasible solution can be defined as a point x ∗ ∈ P: = {x: Ax ≥ b} that is equal to its rounding �x, where the rounde ..."
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Cited by 34 (9 self)
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In this paper we consider the NPhard problem of finding a feasible solution (if any exists) for a generic MIP problem of the form min{cT x: Ax ≥ b, xj integer ∀j ∈ I}. Trivially, a feasible solution can be defined as a point x ∗ ∈ P: = {x: Ax ≥ b} that is equal to its rounding �x, where the rounded point �x is defined by �xj: = [x ∗ j] if j ∈ I and �xj: = x ∗ j otherwise, and [·] represents scalar rounding to the nearest integer. Replacing “equal ” with “as close as possible ” relative to a suitable distance function ∆(x ∗ , �x), suggests the following Feasibility Pump (FP) heuristic for finding a feasible solution of a given MIP. We start from any x ∗ ∈ P, and define its rounding �x. At each FP iteration we look for a point x ∗ ∈ P that is as close as possible to the current �x by solving the problem min{∆(x, �x) : x ∈ P}. Assuming ∆(x, �x) is chosen appropriately, this is an easily solvable LP problem. If ∆(x ∗ , �x) = 0, then x ∗ is a feasible MIP solution and we are done. Otherwise, we replace �x by the rounding of x ∗ , and repeat. From a geometric point of view, the FP generates two trajectories of points x ∗ and �x that satisfy feasibility in a complementary but partial way—one satisfies the linear constraints, the other the integer requirement. The FP can also be viewed as a strategy for making a heuristic sequence of roundings that yields a feasible MIP point. We report computational results on a set of 83 difficult 01 MIPs, using the commercial software ILOGCplex 8.1 as a benchmark. The outcome is that FP, in spite of its simple foundation, proves competitive with ILOGCplex both in terms of speed and quality of the first solution delivered. Interestingly, ILOGCplex could not find any feasible solution at the root node for 19 problems in our testbed, whereas FP was unsuccessful in just 3 cases. 1 1
An Interior Point Algorithm to Solve Computationally Difficult Set Covering Problems
, 1990
"... ..."
A Feasibility Pump Heuristic for General MixedInteger Problems
 UNIVERSITÀ DI BOLOGNA – D.E.I.S. – OPERATIONS RESEARCH
, 2007
"... Finding a feasible solution of a given MixedInteger Programming (MIP) model is a very important (NPcomplete) problem that can be extremely hard in practice. Very recently, Fischetti, Glover and Lodi proposed a heuristic scheme for finding a feasible solution to general MIPs, called Feasibility Pum ..."
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Cited by 14 (1 self)
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Finding a feasible solution of a given MixedInteger Programming (MIP) model is a very important (NPcomplete) problem that can be extremely hard in practice. Very recently, Fischetti, Glover and Lodi proposed a heuristic scheme for finding a feasible solution to general MIPs, called Feasibility Pump (FP). According to the computational analysis reported by these authors, FP is indeed quite effective in finding feasible solutions of hard 01 MIPs. However, MIPs with generalinteger variables seem much more difficult to solve by using the FP approach. In this paper we elaborate on the FischettiGloverLodi approach and extend it in two main directions, namely (i) handling as effectively as possible MIP problems with both binary and generalinteger variables, and (ii) exploiting the FP information to drive a subsequent enumeration phase. Extensive computational results on large sets of test instances from the literature are reported, showing the effectiveness of our improved FP scheme for finding feasible solutions to hard MIPs with generalinteger variables.
A local branching heuristic for mixedinteger programs with 2level variables, Research report
 Networks
, 2003
"... Effective heuristic solution methods for general MixedInteger Programs (MIPs) are strongly required in many practical applications, and have been the subject of an intensive research effort in the recent years. Fischetti and Lodi [6] recently proposed an exact solution technique based on the use of ..."
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Cited by 7 (2 self)
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Effective heuristic solution methods for general MixedInteger Programs (MIPs) are strongly required in many practical applications, and have been the subject of an intensive research effort in the recent years. Fischetti and Lodi [6] recently proposed an exact solution technique based on the use of branching conditions expressed through (invalid) linear inequalities called local branching cuts. In the concluding remarks of their paper, these authors anticipated the possibility their method be used to design a genuine MIP metaheuristic framework akin to Tabu Search (TS) or Variable Neighborhood Search (VNS), based on an external MIP solver. In the present paper we introduce and analyze computationally a specific implementation of the above idea. In particular, we address MIPs with binary variables, and propose a variant of the classical VNS scheme that we call Diversification, Refining, and Tightrefining (DRT). The new approach is intended to be of high generality, but exploits the specific structure of some
A Path Relinking Approach for the Generalized Assignment Problem
 Proc. International Symposium on Scheduling
, 2002
"... The generalized assignment problem is a classical combinatorial optimization problem known to be NPhard. It can model a variety of real world applications in location, allocation, machine assignment, and supply chains. The problem has been studied since the late 1960s, and computer codes for practi ..."
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Cited by 7 (2 self)
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The generalized assignment problem is a classical combinatorial optimization problem known to be NPhard. It can model a variety of real world applications in location, allocation, machine assignment, and supply chains. The problem has been studied since the late 1960s, and computer codes for practical applications emerged in the early 1970s. We propose a new algorithm for this problem that proves to be more effective than previously existing methods. The algorithm features a path relinking approach, which is a mechanism for generating new solutions by combining two or more reference solutions. Computational comparisons on benchmark instances show that the method is not only effective in general, but is especially effective for types D and E instances, which are known to be very difficult.
Feasibility pump 2.0
, 2008
"... Finding a feasible solution of a given MixedInteger Programming (MIP) model is a very important N Pcomplete problem that can be extremely hard in practice. Feasibility Pump (FP) is a heuristic scheme for finding a feasible solution to general MIPs that can be viewed as a clever way to round a sequ ..."
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Cited by 5 (0 self)
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Finding a feasible solution of a given MixedInteger Programming (MIP) model is a very important N Pcomplete problem that can be extremely hard in practice. Feasibility Pump (FP) is a heuristic scheme for finding a feasible solution to general MIPs that can be viewed as a clever way to round a sequence of fractional solutions of the LP relaxation, until a feasible one is eventually found. In this paper we study the effect of replacing the original rounding function (which is fast and simple, but somehow blind) with more clever rounding heuristics. In particular, we investigate the use of a divinglike procedure based on rounding and constraint propagation— a basic tool in Constraint Programming. Extensive computational results on binary and general integer MIPs from the literature show that the new approach produces a substantial improvement of the FP success rate, without slowingdown the method and with a significantly better quality of the feasible solutions found.
Repairing mip infeasibility through local branching
 COMPUTERS AND OPERATIONS RESEARCH (2006). DOI 10.1016/J.COR.2006.08.004
, 2005
"... Finding a feasible solution to a generic MixedInteger Program (MIP) is often a very difficult task. Recently, two heuristic approaches called Feasibility Pump and Local Branching have been proposed to address the problem of finding a feasible solution and improving it, respectively. In this paper w ..."
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Cited by 4 (1 self)
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Finding a feasible solution to a generic MixedInteger Program (MIP) is often a very difficult task. Recently, two heuristic approaches called Feasibility Pump and Local Branching have been proposed to address the problem of finding a feasible solution and improving it, respectively. In this paper we introduce and analyze computationally a hybrid algorithm that uses the feasibility pump method to provide, at very low computational cost, an initial (possibly infeasible) solution to the local branching procedure which can indeed work also with infeasible solutions. The overall procedure is reminiscent of Phase I of the two phase simplex algorithm, in which the original LP is augmented with artificial variables that make a known infeasible starting solution feasible and then the augmented model is solved to iteratively reduce that infeasibility by driving the values of the artificial variables to zero. As such, our approach can also to used to find (heuristically) a minimumcardinality set of constraints whose removal converts an infeasible MIP into a feasible one–a very important piece of information in the analysis of infeasible MIP models.
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.
Digital Object Identifier (DOI) 10.1007/s1010700405703
"... Abstract. In this paper we consider the NPhard problem of finding a feasible solution (if any exists) for a generic MIP problem of the form min{cT x: Ax ≥ b, xj integer ∀j ∈ I}. Trivially, a feasible solution can be defined as a point x ∗ ∈ P: = {x: Ax ≥ b} that is equal to its rounding ˜x, where ..."
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Abstract. In this paper we consider the NPhard problem of finding a feasible solution (if any exists) for a generic MIP problem of the form min{cT x: Ax ≥ b, xj integer ∀j ∈ I}. Trivially, a feasible solution can be defined as a point x ∗ ∈ P: = {x: Ax ≥ b} that is equal to its rounding ˜x, where the rounded point ˜x is defined by ˜xj: = [x ∗ j]ifj ∈ I and ˜xj: = x ∗ j otherwise, and [·] represents scalar rounding to the nearest integer. Replacing “equal ” with “as close as possible ” relative to a suitable distance function �(x ∗,˜x), suggests the following Feasibility Pump (FP) heuristic for finding a feasible solution of a given MIP. We start from any x ∗ ∈ P, and define its rounding ˜x. At each FP iteration we look for a point x ∗ ∈ P that is as close as possible to the current ˜x by solving the problem min{�(x,˜x) : x ∈ P}. Assuming �(x,˜x) is chosen appropriately, this is an easily solvable LP problem. If �(x ∗,˜x) = 0, then x ∗ is a feasible MIP solution and we are done. Otherwise, we replace ˜x by the rounding of x ∗ , and repeat. We report computational results on a set of 83 difficult 01 MIPs, using the commercial software ILOGCplex 8.1 as a benchmark. The outcome is that FP, in spite of its simple foundation, proves competitive with ILOGCplex both in terms of speed and quality of the first solution delivered. Interestingly, ILOGCplex could not find any feasible solution at the root node for 19 problems in our testbed, whereas FP was unsuccessful in just 3 cases. 1.