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 NP-hard, 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 2-exchange 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 3-exchange or 4-exchange 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 large-scale neighborhood search algorithms give consistently better solutions compared the popular 2-exchange neighborhood algorithms considering both the solution time and solution accuracy.

. A new hybrid procedure that combines genetic operators to existing heuristics is proposed to solve the Quadratic Assignment Problem (QAP). Genetic operators are found to improve the performance of both local search and tabu search. Some guidelines are also given to design good hybrid schemes. These hybrid algorithms are then used to improve on the best known solutions of many test problems in the literature. 1. Introduction The quadratic assignment problem (QAP) can be stated as: min OE2P (n) n X i=1 n X j=1 a ij b OE(i)OE(j) ; where A = (a ij ) and B = (b kl ) are two n \Theta n matrices and P (n) is the set of all permutations of f1; :::; ng. Matrix A is often referred to as a distance matrix between sites, and B as a flow matrix between objects. In most cases, the matrices A and B are symmetrical with a null diagonal. A permutation may then be interpreted as an assignment of objects to sites with a quadratic cost associated to it. There are many applications that can be fo...

Guided Local Search (GLS) is an intelligent search scheme for combinatorial optimization problems. A main feature of the approach is the iterative use of local search. Information is gathered from various sources and exploited to guide local search to promising parts of the search space. The application of the method to the Travelling Salesman Problem and the Quadratic Assignment Problem is examined. Results reported show that the algorithm outperforms or compares very favorably with well-known and established optimization techniques such as simulated annealing and tabu search. Given the novelty of the approach and the very encouraging results, the method could have an important contribution to the development of intelligent search techniques for combinatorial optimization. 1. Introduction Guided Local Search is the outcome of a research project with main aim to extend the GENET neural network [29,26,5] for constraint satisfaction problems to partial constraint satisfaction [6,26] and...

Abstract—We propose a new methodology for reliably solving the correspondence problem between sparse sets of points of two or more images. This is a key step in most problems of computer vision and, so far, no general method exists to solve it. Our methodology is able to handle most of the commonly used assumptions in a unique formulation, independent of the domain of application and type of features. It performs correspondence and outlier rejection in a single step and achieves global optimality with feasible computation. Feature selection and correspondence are first formulated as an integer optimization problem. This is a blunt formulation, which considers the whole combinatorial space of possible point selections and correspondences. To find its global optimal solution, we build a concave objective function and relax the search domain into its convex-hull. The special structure of this extended problem assures its equivalence to the original one, but it can be optimally solved by efficient algorithms that avoid combinatorial search. This methodology can use any criterion provided it can be translated into cost functions with continuous second derivatives. Index Terms—Correspondence problem, linear and concave programming, sparse stereo. 1

Quadratically constrained quadratic programs (QQP) play an important modeling role for many diverse problems. These problems are in general NP hard, and numerically intractable. Lagrangian relaxations often provide good approximate solutions to these hard problems. Such relaxations are equivalent to semidefinite programming relaxations. For several special cases of QQP, e.g. convex programs and trust region subproblems, the Lagrangian relaxation provides the exact optimal value, i.e. there is a zero duality gap. However this is not true for the general QQP, or even the QQP with two convex constraints, but a nonconvex objective. In this paper we consider a certain QQP where the quadratic constraints correspond to the matrix orthogonality condition XX T = I. For this problem we show that the Lagrangian dual based on relaxing the constraints XX T = I, and the seemingly redundant constraints X T X = I, has a zero duality gap. This result has natural applications to quadratic assignm...

In the last decade many important applications of eigenvalues and eigenvectors of graphs in combinatorial optimization were discovered. The number and importance of these results is so fascinating that it makes sense to present this survey.

We describe a new convex quadratic programming bound for the quadratic assignment problem (QAP). The construction of the bound uses a semidefinite programming representation of a basic eigenvalue bound for QAP. The new bound dominates the well-known projected eigenvalue bound, and appears to be competitive with existing bounds in the tradeoff between bound quality and computational effort. Keywords: Quadratic Assignment Problem, Eigenvalue Bounds, Quadratic Programming, Semidefinite Programming. Dept. of Management Sciences, University of Iowa, Iowa City, IA 52242 y Dept. of Computer Science, University of Iowa, Iowa City, IA 52242 1 Introduction The quadratic assignment problem (QAP) in "Koopmans-Beckmann" form can be written QAP(A;B;C) : min tr(AXB + C)X T s:t: X 2 \Pi; where A, B and C are n \Theta n matrices, tr denotes the trace of a matrix, and \Pi is the set of n \Theta n permutation matrices. Throughout we assume that A and B are symmetric. The QAP is a very well-know...

A memetic algorithm (MA), i.e. an evolutionary algorithm making use of local search, for the quadratic assignment problem is presented. A new recombination operator for realizing the approach is described, and the behavior of the MA is investigated on a set of problem instances containing between 25 and 100 facilities/locations. The results indicate that the proposed MA is able to produce high quality solutions quickly. A comparison of the MA with some of the currently best alternative approaches -- reactive tabu search, robust tabu search and the fast ant colony system -- demonstrates that the MA outperforms its competitors on all studied problem instances of practical interest. 1 Introduction The problem of assigning a set of facilities (with given flows between them) to a set of locations (with given distances between them) in such a way that the sum of the product between flows and distances is minimized is known as the facilities location problem [1] or the quadratic assignment ...

Many different algorithms and objective functions for topographic mappings have been proposed. We show that several of these approaches can be seen as particular cases of a more general objective function. Consideration of a very simple mapping problem reveals large differences in the form of the map that each particular case favors. These differences have important consequences for the practical application of topographic mapping methods.

A new bounding procedure for the Quadratic Assignment Problem (QAP) is described which extends the Hungarian method for the Linear Assignment Problem (LAP) to QAPs, operating on the four dimensional cost array of the QAP objective function. The QAP is iteratively transformed in a series of equivalent QAPs leading to an increasing sequence of lower bounds for the original problem. To this end, two classes of operations which transform the four dimensional cost array are defined. These have the property that the values of the transformed objective function Z' are the corresponding values of the old objective function Z, shifted by some amount C. In the case that all entries of the transformed cost array are non-negative, then C is a lower bound for the initial QAP. If, moreover, there exists a feasible solution U to the QAP, such that its value in the transformed problem is zero, then C is the optimal value of Z and U is an optimal solution for the original QAP. The transformations are iteratively applied until no significant increase in constant C as above is found, resulting in the so called Dual Procedure (DP). Several strategies are listed for appropriately determining C, or equivalently, transforming the cost array. The goal is the modification of the elements in the cost array so as to obtain new equivalent problems which bring the QAP closer to solution. In some cases the QAP is actually solved, though solution is not guaranteed. The close relationship between the DP and the Linear Programming formulation of Adams and Johnson is presented. The DP attempts to solve Adams and Johnsons CLP, a continuous relaxation of a linearization of the QAP. This explains why the DP produces bounds close to the optimum values for CLP calculated by Johnson in her dissertation and by...