qaplib a quadratic assignment problem library abstract after a short introduction into quadratic assignment problems we give a library of problem instances for the quadratic assignment problem the examples are listed y authors in alphabetical order

a parallel depth first search branch and bound algorithm for the quadratic assignment problem abstract we propose a new parallel branch and bound algorithm for the quadratic assignment problem which is a combinatorial optimization problem known to be very hard to solve exactly an original method to distribute work to processors using the notion of feeding tree is presented when adequately used it allows to reduce memory contention and load unbalance therefore a linear speed up in the number of processors is reached on a shared memory multiprocessor the cray 2 and the optimality of solutions for famous problems of size less than 20 nugent 16 elshafei 19 scriabin vergin 20 is proved by this program the implementation analysis shows that these results are more than an improvement due to hardware evolution and confirms the usefulness of our parallel branch and bound algorithm for larger quadratic assignment problems

bounds for the quadratic assignment problem using the bundle method semidefinite programming sdp has recently turned out to be a very powerful tool for approximating some np hard problems the nature of the quadratic assignment problem qap suggests sdp as a way to derive tractable relaxations we recall some sdp relaxations of qap and solve them approximately using a dynamic version of the bundle method the computational results demonstrate the efficiency of the approach our bounds are currently among the strongest ones available for qap we investigate their potential for branch and bound settings by looking also at the bounds in the first levels of the branching tree

the quadratic shortest path problem complexity approximability and solution methods we consider the problem of finding a shortest path in a directed graph with a quadratic objective function the qspp we show that the qspp cannot be approximated unless p np for the case of a convex objective function an n approximation algorithm is presented where n is the number of nodes in the graph and apx hardness is shown furthermore we prove that even if only adjacent arcs play a part in the quadratic objective function the problem still cannot be approximated unless p np in order to solve the problem we first propose a mixed integer programming formulation and then devise an efficient exact branch and bound algorithm for the general qspp where lower bounds are computed by considering a reformulation scheme that is solvable through a number of minimum cost flow problems in our computational experiments we solve to optimality different classes of instances with up to 1000 nodes

some concave convex representations of the quadratic assignment problem abstract the quadratic assignment problem suffers from the non convex non concave form of its objective function in this paper five alternative representations of the problem are given whose convexity concavity properties can in principle be exploited to some advantage

an exact algorithm for the quadratic assignment problem on a tree the tree qap is a special case of the quadratic assignment problem qap where the nonzero flows form a tree no condition is required for the distance matrix this problem is np complete and is also a generalization of the traveling salesman problem in this paper we present a branch and bound algorithm for the exact solution of the tree qap based on an integer programming formulation of the problem the bounds are computed using a lagrangian relaxation of this formulation to solve the relaxed problem we present a dynamic programming algorithm which is polynomially bounded the obtained lower bound is very sharp and equals the optimum in many cases this fact allows us to employ a reduction method to decrease the number of variables and leads to search trees with a small number of nodes compared to those usually encountered in problems of this type computational results are given for problems with size up to 25

convex graph invariant relaxations for graph edit distance the edit distance between two graphs is a widely used measure of similarity that evaluates the smallest number of vertex and edge deletions insertions required to transform one graph to another it is np hard to compute in general and a large number of heuristics have been proposed for approximating this quantity with few exceptions these methods generally provide upper bounds on the edit distance between two graphs in this paper we propose a new family of computationally tractable convex relaxations for obtaining lower bounds on graph edit distance these relaxations can be tailored to the structural properties of the particular graphs via convex graph invariants specific examples that we highlight in this paper include constraints on the graph spectrum as well as tractable approximations of the stability number and the maximum cut values of graphs we prove under suitable conditions that our relaxations are tight i e exactly compute the graph edit distance when one of the graphs consists of few eigenvalues we also validate the utility of our framework on synthetic problems as well as real applications involving molecular structure comparison problems in chemistry

operator log convex functions and operator means we study operator log convex functions on 0 and prove that a continuous nonnegative function on 0 is operator log convex if and only if it is operator monotone decreasing several equivalent conditions related to operator means are given for such functions operator log concave functions are also discussed

l_p norm regularization algorithms for optimization over permutation matrices optimization problems over permutation matrices appear widely in facility layout chip design scheduling pattern recognition computer vision graph matching etc since this problem is np hard due to the combinatorial nature of permutation matrices we relax the variable to be the more tractable doubly stochastic matrices and add an l p norm 0 p 1 regularization term to the objective function the optimal solutions of the l p regularized problem are the same as the original problem if the regularization parameter is sufficiently large a lower bound estimation of the nonzero entries of the stationary points and some connections between the local minimizers and the permutation matrices are further established then we propose an l p regularization algorithm with local refinements the algorithm approximately solves a sequence of l p regularization subproblems by the projected gradient method using a nonmonotone line search with the barzilai borwein step sizes its performance can be fu

bounds on the performance of vector quantizers under channel errors vector quantization vq is an effective and widely known method for low bit rate communication of speech and image signals a common assumption in the design of vq based communication systems is that the compressed digital information is transmitted through a perfect channel under this assumption quantization distortion is the only factor in output signal fidelity moreover the assignment of channel symbols to the vq reconstruction vectors is of no importance however under physical channels errors may be present causing degradation in overall system performance in such a case the effect of channel errors on the coding system performance depends on the index assignment of the reconstruction vectors the index assignment problem is a special case of the quadratic assignment problem qap and is known to be np complete for a vq with n reconstruction vectors there are n possible assignments meaning that an exhaustive search over all possible assignments is practically impossible to help the vq designer we present in this correspondence lower and upper bounds on the performance of vq systems under channel errors over all possible assignments the bounds coincide with a general bound for the qap nevertheless the proposed derivation allows us to compare the bounds with published results on vq index assignment a related expression for the average performance is also given and discussed special cases and numerical examples are given in which the bounds and average performance are compared with index assignments obtained by known algorithms

positive definite completions of partial hermitian matrices abstract the question of which partial hermitian matrices some entries specified some free may be completed to positive definite matrices is addressed it is shown that if the diagonal entries are specified and principal minors composed of specified entries are positive then if the undirected graph of the specified entries is chordal a positive definite completion necessarily exists furthermore if this graph is not chordal then examples exist without positive definite completions in case a positive definite completion exists there is a unique matrix in the class of all positive definite completions whose determinant is maximal and this matrix is the unique one whose inverse has zeros in those positions corresponding to unspecified entries in the original partial hermitian matrix additional observations regarding positive definite completions are made

bounds for the spectrum of normal matrices abstract let a be a normal matrix with eigenvalues 1 2 n and let denote the smallest disc containing these eigenvalues we give some inequalities relating the center and radius of to the entries in a when applied to hermitian matrices our results give lower bounds on the spread maxij i j of a when applied to positive definite hermitian matrices they give lower bounds on the kantorovich ratio maxij i j i j

linearization strategies for a class of zero one mixed integer programming problems this paper is concerned with a new linearization strategy for a class of zero one mixed integer programming problems that contains quadratic cross product terms between continuous and binary variables and between the binary variables themselves this linearization scheme provides an equivalent mixed integer linear programming problem which yields a tighter continuous relaxation than that obtainable via the alternative linearization techniques available in the literature moreover the proposed technique provides a unifying framework in the sense that all the alternate methods lead to formulations that are accessible through appropriate surrogates of the constraints of the new linearized formulation extensions to various other types of mixed integer nonlinear programming problems are also discussed

nonlinear 0 1 programming i linearization techniques any real valued nonlinear function in 0 1 variables can be rewritten as a multilinear function we discuss classes of lower and upper bounding linear expressions for multilinear functions in 0 1 variables for any multilinear inequality in 0 1 variables we define an equivalent family of linear inequalities this family contains the well known system of generalized covering inequalities as well as other linear equivalents of the multilinear inequality that are more compact i e of smaller cardinality in a companion paper 7 we discuss dominance relations between various linear equivalents of a multilinear inequality and describe a class of algorithms for multilinear 0 1 programming based on these results

assignment problems a golden anniversary survey abstract having reached the 50th golden anniversary of the publication of kuhn s seminal article on the solution of the classic assignment problem it seems useful to take a look at the variety of models to which it has given birth this paper is a limited survey of what appear to be the most useful of the variations of the assignment problem that have appeared in the literature over the past 50 years the intention here is not to identify every such paper of which there have been hundreds nor to identify the best solution procedure for each variation rather the intention is to identify what these variations are and what they are called so as to make it easier for a researcher trying to develop some variation of the assignment problem for a particular application to find the relevant literature

experimental analysis of simulated annealing based algorithms for the layout problem abstract in this paper results from an experimental analysis involving the 2 way and 3 way exchange algorithms a modified penalty algorithm the simulated annealing algorithm and a hybrid simulated annealing algorithm are presented the hybrid simulated annealing algorithm uses the modified penalty algorithm to generate an initial solution and then improves it using simulated annealing it is tested on single row layout problems with facilities of unequal area and multi row layout problems with facilities of equal area the algorithm produced solutions better than those previously reported in the literature for two of the larger single row problems and five multi row layout problems for all the other single row problems and eight of ten multi row layout test problems it produced the best known solutions the hybrid simulated annealing algorithm appears to produce better quality solutions than the simulated annealing and tabu navigation algorithms but requires more computation time than simulated annealing

a new lower bound for the quadratic assignment problem we introduce a new lower bound for the quadratic assignment problem based on a sequence of equivalent formulations of the problem we present a procedure for obtaining tight bounds by sequentially applying our approach in conjunction with the a assad and w xu bound and the n christofides and m gerrard bound

interpolating the arithmetic geometric mean inequality and its operator version two families of means called heinz means and heron means that interpolate between the geometric and the arithmetic mean are considered comparison inequalities between them are established operator versions of these inequalities are obtained failure of such extensions in some cases is illustrated by a simple example

lower bounds for the quadratic assignment problem based upon a dual formulation 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 nonnegative 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 to obtain new equivalent problems that 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 johnson s 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 resende et al in their interior point algorithm for linear programming the benefit of using dp within a branch and bound algorithm is described then two versions of dp are tested on the nugent test instances from size 5 to size 30 as well as several other test instances from qaplib these compare favorably with earlier bounding methods

an optimal tree search method for the manufacturing systems cell formation problem abstract a solution methodology producting exact solutions to the manufacturing systems cell formation problem is presented a distance matrix representing the closeness between pairs of machines with regard to the parts they process is taken into account the proposed approach is an optimal tree search method which empoys two different bounds a classical bound for the quadratic assignment problem and a recently proposed one for the quadratic transportation problem as well as a new heuristic for the cell formation problem a special tree search was designed in order to reduce the size of the tree minimize the computational effort required in time consuming calculations and exploit symmetries of the problem computational results indicate that the proposed algorithm is very efficient in generating optimal solutions at low computational cost

nonlinear 0 1 programming ii dominance relations and algorithms a nonlinear 0 1 program can be restated as a multilinear 0 1 program which in turn is known to be equivalent to a linear 0 1 program with generalized covering g c inequalities in a companion paper 6 we have defined a family of linear inequalities that contains more compact smaller cardinality linearizations of a multilinear 0 1 program than the one based on the g c inequalities in this paper we analyze the dominance relations between inequalities of the above family in particular we give a criterion that can be checked in linear time for deciding whether a g c inequality can be strengthened by extending the cover from which it was derived we then describe a class of algorithms based on these results and discuss our computational experience we conclude that the g c inequalities can be strengthened most of the time to an extent that increases with problem density in particular the algorithm using the strengthening procedure outperforms the one using only g c inequalities whenever the number of nonlinear terms per constraint exceeds about 12 15 and the difference in their performance grows with the number of such terms

a minimax assignment problem in treelike communication networks abstract a given system of communication centres c 1 c 2 c n has to be embedded into a given undirected network n the centres exchange messages at given rates per time unit through a selected routing pattern if there is no direct connection between c i and c j the messages sent from c i to c j pass through several intermediate centres the messages exchanged between c i and c j may be sent along a single path or they may be split into several parts each part being sent along its own path the goal is to find an embedding of the centres into the network and a routing pattern which minimizes the maximum intermediate traffic over all centres the paper deals mainly with the single path model the complexity of the problem is fully discussed drawing a sharp borderline between np complete and polynomially solvable cases in case that n is a tree a branch and bound algorithm is described and numerical results for random test data are reported and discussed

dynamical analysis of iterative methods for nonlinear systems or how to deal with the dimension abstract this paper deals with the real dynamical analysis of iterative methods for solving nonlinear systems on vectorial quadratic polynomials we use the extended concept of critical point and propose an easy test to determine the stability of fixed points to multivariate rational functions moreover an scaling theorem for different known methods is satisfied we use these tools to analyze the dynamics of the operator associated to known iterative methods on vectorial quadratic polynomials of two variables the dynamical behavior of newton s method is very similar to the obtained in the scalar case but this is not the case for other schemes some procedures of different orders of convergence have been analyzed under this point of view and some dangerous numerical behavior have been found as attracting strange fixed points or periodic orbits

a branch and bound algorithm for the quadratic assignment problem based on the hungarian method this paper presents a new branch and bound algorithm for solving the quadratic assignment problem qap the algorithm is based on a dual procedure dp similar to the hungarian method for solving the linear assignment problem our dp solves the qap in certain cases i e for some small problems n 7 and for numerous larger problems 7 n 16 that arise as sub problems of a larger qap such as the nugent 20 the dp however does not guarantee a solution it is used in our algorithm to calculate lower bounds on solutions to the qap as a result of a number of recently developed improvements the dp produces lower bounds that are as tight as any which might be useful in a branch and bound algorithm these are produced relatively cheaply especially on larger problems experimental results show that the computational complexity of our algorithm is lower than known methods and that its actual runtime is significantly shorter than the best known algorithms for qaplib test instances of size 16 through 22 our method has the potential for being improved and therefore can be expected to aid in solving even larger problems

asymptotic properties of the quadratic assignment problem for the general quadratic assignment problem as well as for a planar version of this problem we extend earlier work by burkard and fincke to prove that the ratio of the maximal to the minimal solution value converges to 1 almost surely in fact any solution value can almost surely be written asymptotically as a simple explicitly given function of the problem size theoretical analysis and computational experiments reveal the convergence to be relatively fast

on the gap between the quadratic integer programming problem and its semidefinite relaxation consider the semidefinite relaxation sdr of the quadratic integer program qip where q is a given symmetric matrix and d is diagonal we consider the sdr gap we establish the uniqueness of the sdr solution and prove that if and only if r n 1max xtvvtx x 1 1 n 1 where v is an orthogonal matrix whose columns span the r dimensional null space of d q and where d is the unique sdr solution we also give a test for establishing whether that involves 2r 1 function evaluations in the case that r 1 we derive an upper bound on which is tighter than thus we show that breaching the sdr gap for the qip problem is as difficult as the solution of a qip with the rank of the cost function matrix equal to the dimension of the null space of d q this reduced rank qip problem has been recently shown to be solvable in polynomial time for fixed r

a distance assignment approach to the facility layout problem the facility layout problem has generally been formulated as a quadratic assignment problem where the interacting facilities are assigned to the various sites in this paper a different approach based on assigning distances between pairs of sites to pairs of facilities is taken a network formulation and analysis of this problem is provided an algorithm composed of three phases is developed the first phase involves preprocessing of the distance matrix the second phase involves solving the generalized assignment network problem the third phase employs a heuristic procedure which transforms the solution of the second phase into a feasible assignment of facilities to sites finally computational experience with the new algorithm is provided

strengthening gilmore s bound for the quadratic assignment problem abstract finding bounds for the quadratic assignment problem is important as part of some procedures for solving this problem in this paper a commonly used bound due to gilmore is extended to a set of possible bounds analytic properties of the functions underlying these bounds are given and computational approaches are suggested which make use of these properties

a novel chaotic search for quadratic assignment problems abstract we propose a novel method for solving the quadratic assignment problems first we realize the conventional tabu search on a neural network and modify it to a chaotic version our novel method includes both effects of chaotic dynamics and tabu search we compare the performance of the novel chaotic search with the conventional tabu search and an exponential tabu search whose memory effect for tabu forbidding previous moves decays exponentially we show that the exponential tabu search has higher performance than the conventional tabu search and further that the novel method with a chaotic neural network exhibits the best performance we also propose a controlling method of the chaotic neural network for realizing easy and robust applications of our method then better performance can be realized without manual parameter setting for various problems

an improved annealing scheme for the qap abstract recently there has been some interest in the use of simulated annealing to obtain good solutions to a number of combinatorial problems this paper reports on the use of this method applied to the quadratic assignment problem i e the assignment of inter communicating objects to locations to minimize the total cost of communication between them the result is a much improved annealing scheme for this problem which performs well on a range of examples finding improved solutions for several of the largest problems available in the literature and requiring only modest amounts of computational effort

cross domain matching with squared loss mutual information the goal of cross domain matching cdm is to find correspondences between two sets of objects in different domains in an unsupervised way cdm has various interesting applications including photo album summarization where photos are automatically aligned into a designed frame expressed in the cartesian coordinate system and temporal alignment which aligns sequences such as videos that are potentially expressed using different features in this paper we propose an information theoretic cdm framework based on squared loss mutual information smi the proposed approach can directly handle non linearly related objects sequences with different dimensions with the ability that hyper parameters can be objectively optimized by cross validation we apply the proposed method to several real world problems including image matching unpaired voice conversion photo album summarization cross feature video and cross domain video to mocap alignment and kinect based action recognition and experimentally demonstrate that the proposed method is a promising alternative to state of the art cdm methods

the asymptotic behaviour of quadratic sum assignment problems a statistical mechanics approach abstract in this paper we apply the statistical mechanics formalism to study quadratic sum assignment problems qsap this formalism allows us to exhibit in the limit of large problems the asymptotic behaviour of the optimal value of the cost function the conclusions of the study are confirmed by the results of metropolis computer simulations

on random quadratic bottleneck assignment problems the relative error between best and worst solution of quadratic bottleneck assignment problems with cost coefficientsdijpq aipbjq is considered whereaip is either arbitrarily given or corresponds to a distance in the plane it is shown that the relative error is bounded by a function m tending to zero with probability tending to one asm provided the data are uniformly distributed this implies that any algorithm for the above mentioned problems yields asymptotically an arbitrarily small relative error with probability tending to one

a classification of formulations for the time dependent traveling salesman problem abstract the time dependent traveling salesman problem tdtsp is a generalization of the classical traveling salesman problem where the cost of any given arc is dependent of its position in the tour the tdtsp can model several real world applications e g one machine sequencing in this paper we present a classification of formulations for the tdtsp this framework includes both new and old formulations the new formulation presented in this paper is derived from a quadratic assignment model for the tdtsp in a first step lawler s transformation procedure is used to derive an equivalent linearized version of the quadratic model in a second step a stronger formulation is obtained by tightening some constraints of the previous formulation it is shown that in terms of linear relaxations the latter formulation is either equivalent or better than other formulations already known from the literature finally we compare these formulations with other well known formulations for the classical traveling salesman problem

on the monotonicity of weighted power means for matrices abstract in this article we provide an alternate proof of the fact that the weighted power means p a b t t a p 1 t b p 1 p 1 p 2 satisfy audenaert s in betweenness property for positive semidefinite matrices we show that the in betweenness property holds with respect to any unitarily invariant norm for p 1 2 and with respect to the euclidean metric for p 1 4 we also show that the only kubo ando symmetric mean that satisfies the in betweenness property with respect to any metric induced by a unitarily invariant norm is the arithmetic mean finally for p 6 we give a counterexample to a conjecture by audenaert regarding the in betweenness property

ranking scalar products to improve bounds for the quadratic assignment problem abstract the eigenvalue bound for the quadratic assignment problem qap is successively improved by considering a set of k best scalar products related to the qap an efficient procedure is proposedto find such a set of k best scalar products a class of qaps is described for which this procedure in general improves existing lower bounds and at the same time generates good suboptimal solutions the method leaves the user with a large flexibility in controlling the quality of the bound however since the method is sensitive to input data it should only be used in combination with other bounding rules

symmetrization of nonsymmetric quadratic assignment problems and the hoffman wielandt inequality abstract a technique is proposed to transform a nonsymmetric quadratic assignment problem qap into an equivalent one consisting of complex hermitian matrices this technique provides several new hoffman wielandt type eigenvalue inequalities for general matrices and extends the eigenvalue bound for symmetric qaps to the general case

a survey for the quadratic assignment problem the quadratic assignment problem qap one of the most difficult problems in the np hard class models many real life problems in several areas such as facilities location parallel and distributed computing and combinatorial data analysis combinatorial optimization problems such as the traveling salesman problem maximal clique and graph partitioning can be formulated as a qap in this paper we present some of the most important qap formulations and classify them according to their mathematical sources we also present a discussion on the theoretical resources used to define lower bounds for exact and heuristic algorithms we then give a detailed discussion of the progress made in both exact and heuristic solution methods including those formulated according to metaheuristic strategies finally we analyze the contributions brought about by the study of different approaches

new characterizations of operator monotone functions abstract in this paper we will show a new characterization of operator monotone functions by a matrix reverse cauchy inequality

a fast convergent numerical method for matrix sign function with application in sdes in this article a new matrix iterative method for computing the sign of a matrix is offered convergence along with asymptotical stability of the presented method is studied an accelerated form of the iteration will further be constructed finally the application of the given approach in numerical solution of stochastic differential equations and in solving algebraic riccati equations is pointed out

a heuristic for quadratic boolean programs with applications to quadratic assignment problems abstract a heuristic for quadratic boolean programs is presented computational tests with quadratic assignment problems qap showed that it finds very good suboptimal solutions in moderate time and behaves computationally stable in the appendix a fortran program for qap is listed which improves an earlier code published by burkard and derigs

a low dimensional semidefinite relaxation for the quadratic assignment problem the quadratic assignment problem qap is arguably one of the hardest np hard discrete optimization problems problems of dimension greater than 25 are still considered to be large scale current successful solution techniques use branch and bound methods which rely on obtaining strong and inexpensive bounds in this paper we introduce a new semidefinite programming sdp relaxation for generating bounds for the qap in the trace formulation we apply majorization to obtain a relaxation of the orthogonal similarity set of the quadratic part of the objective function this exploits the matrix structure of qap and results in a relaxation with much smaller dimension than other current sdp relaxations we compare the resulting bounds with several other computationally inexpensive bounds such as the convex quadratic programming relaxation qpb we find that our method provides stronger bounds on average and is adaptable for branch and bound methods

a new lower bound via projection for the quadratic assignment problem new lower bounds for the quadratic assignment problem qap are presented these bounds are based on the orthogonal relaxation of qap the additional improvement is obtained by making efficient use of a tractable representation of orthogonal matrices having constant row and column sums the new bound is easy to implement and often provides high quality bounds under an acceptable computational effort

a thermodynamically motivated simulation procedure for combinatorial optimization problems abstract exchange algorithms are an important class of heuristics for hard combinatorial optimization problems as e g salesman problems or quadratic assignment problems in kirkpatrick s and cerny s exchange algorithms for the travelling salesman problem and placement problem they propose to perform an exchange not only if the objective function value decreases by this exchange but also in certain cases if the objective function value increases an exchange increasing the objective function value is performed stochastically depending on the size of the increment computational tests with quadratic assignment problems revealed an excellent behaviour in such an approach suboptimal solutions differing 1 2 from the best known solutions are obtained by a simple program in short time by starting this program several times with different starting values all known minimal objective function values were reached thus this approach is well suited also for smaller computers and leads in short time to acceptable solutions

on generalized powers stormer s inequality abstract a generalization of powers stormer s inequality for operator monotone functions on 0 and for positive linear functional on general c algebras will be proved it also will be shown that the generalized powers stormer inequality characterizes the tracial functionals on c algebras

efficient models for the facility layout problem abstract in this paper two new models of the facility layout problem are presented linear continuous with absolute values in the objective function and constraints and linear mixed integer the linear mixed integer models have lesser number of integer variables than any other existing formulation for the facility layout problem while most other linear mixed integer models available in the literature have been obtained through a linearization of the quadratic assignment problem the ones presented in this paper are not the continuous models have an even more compact form an advantage of the formulations presented in this paper is that the location of sites need not be known a priori more importantly two of the formulations model the layout problem with facilities of unequal area solving the models presented with an unconstrained optimization algorithm yields good quality suboptimal solutions in a relatively low computation time the continuous models appear to be more useful for solving the facility layout problem than other models published in the literature

network based formulations of the quadratic assignment problem abstract we present two formulations of the quadratic assignment problem qap that result in network flow problems with integer variables and side constraints a linearization of the qap is obtained in both cases by considering each facility to consist of two parts a source for all outgoing flows from that facility and a sink for all incoming flows to the facility preliminary computational experience with both approaches is presented

lower bounds for the spread of a hermitian matrix abstract some simple lower bounds for the spread of a hermitian matrix are derived these bounds are explicit functions of the entries of the matrix one of which is sharper than a recent result due to e r barnes and a j hoffman comparisons are made with several known results

algodesk an experimental comparison of eight evolutionary heuristics applied to the quadratic assignment problem abstract this work compares the effectiveness of eight evolutionary heuristic algorithms applied to the quadratic assignment problem qap reputedly one of the most difficult combinatorial optimization problems qap is merely used as a carrier for the comparison we do not attempt to compare any heuristics with solving algorithms specific for it results are given both with respect to the best result achieved by each algorithm in a limited time span and to its speed of convergence to that result

a generalized steffensen s method for matrix sign function a direct generalized steffensen s method is proposed for solving the quadratic matrix equation f x x2 i 0 in this way when the matrix a is nonsingular we derive a new numerical scheme for finding matrix sign function the local and global convergence of the proposed generalization is brought up via the concept of basins of attraction we also report some numerical results of the proposed method which show its applicability

a constructive method for improving lower bounds for a class of quadratic assignment problems a new approach to evaluate lower bounds for a class of quadratic assignment problems qap is presented an instance of a qap of size n is specified by two n n matrices d and f and is denoted by qapd f this approach is presented for qaps where d is the matrix of rectilinear distances between points on a regular grid however it can easily be generalized to a wider class of problems two matrices fopt and fres are constructed such that f fopt fres and the optimal solution to qapd fopt is known any existing lower bound can then be applied to qapd fres which in sum with the optimal value for qapd fopt provides a lower bound to qapd f thus this method could serve as a reduction process to possibly improve the results from a variety of bounding techniques this approach is tested on two bounds from the literature the gilmore lawler bound glb and an eigenvalue bound pb for various problems of size ranging from 6 49 computational results show a good improvement in bounds for all the test problems an extension of our method to a more general class of qaps is also presented

computing lower bounds for the quadratic assignment problem with an interior point algorithm for linear programming an example of the quadratic assignment problem qap is the facility location problem in which n facilities are assigned at minimum cost to n sites 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 sites to which the facilities are assigned proving optimality of qaps has been limited to instances having fewer than 20 facilities largely because known lower bounds are weak we compute lower bounds for a wide range of qaps using a linear programming based lower bound studied by z drezner 1995 on the majority of qaps tested a new best known lower bound is computed on 87 of the instances we produced the best known lower bound on several instances including some having more the 20 facilities the lower bound is tight the linear programs which can be large even for small qaps are solved with an interior point code that uses a preconditioned conjugate gradient algorithm to compute the interior point directions attempts to solve these instances using the cplex primal simplex algorithm as well as the cplex barrier primal dual interior point method were successful only for the smallest instances

chaotic dynamics of a third order newton type method the dynamics of a classical third order newton type iterative method is studied when it is applied to degrees two and three polynomials the method is free of second derivatives which is the main limitation of the classical third order iterative schemes for systems moreover each iteration consists only in two steps of newton s method having the same derivative with these two properties the scheme becomes a real alternative to the classical newton method affine conjugacy class of the method when is applied to a differentiable function is given chaotic dynamics have been investigated in several examples applying the root finding method to a family of degree three polynomials we have find a bifurcation diagram as those that appear in the bifurcation of the logistic map in the interval

recent advances in the solution of quadratic assignment problems the quadratic assignment problem qap is notoriously difficult for exact solution methods in the past few years a number of long open qaps including those posed by steinberg 1961 nugent et al 1968 and krarup 1972 were solved to optimality for the first time the solution of these problems has utilized both new algorithms and novel computing structures we describe these developments as well as recent work which is likely to result in the solution of even more difficult instances

random assignment problems abstract analysis of random instances of optimization problems provides valuable insights into the behavior and properties of problem s solutions feasible region and optimal values especially in large scale cases a class of problems that have been studied extensively in the literature using the methods of probabilistic analysis is represented by the assignment problems and many important problems in operations research and computer science can be formulated as assignment problems this paper presents an overview of the recent results and developments in the area of probabilistic assignment problems including the linear and multidimensional assignment problems quadratic assignment problem etc

a note on lack of strong duality for quadratic problems with orthogonal constraints abstract the general quadratically constrained quadratic program qqp is an important modelling tool for many diverse problems the qqp is 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 and can be solved efficiently for several special cases of qqp the lagrangian relaxation provides the exact optimal value this means that there is a zero duality gap and the problem is tractable it is important to know for which cases this is true since they can then be used as subproblems to improve lagrangian relaxation for intractable qqps in this paper we study the special qqp with orthogonal matrix constraints xx t i if c 0 the zero duality gap result holds if the redundant orthogonal constraints x t x i are added we show that this is not true in the general case however we show how to close the duality gap in the pure linear case by adding variables in addition to constraints

one dimensional machine location problems in a multi product flowline with equidistant locations an assignment of machines to locations along a straight track is required to optimize material flow in many manufacturing systems the assignment of m unique machines to m locations along a linear material handling track with the objective of minimizing the total machine to machine material transportation cost is formulated as a quadratic assignment problem qap the distance matrix in problems involving equally spaced machine locations in one dimension is seen to possess some unique characteristics called amoebic properties since an optimal solution to a problem with large m is computationally intractable the qap is np hard a number of the amoebic properties are exploited to devise heuristics and a lower bound on the optimal solution computational results which demonstrate the performance of the heuristics and the lower bound are presented

minimax nonredundant channel coding the distortion of a message due to channel noise can be alleviated significantly without redundant error control bits by judicious assignment of binary indices to message symbols the nonredundant coding gain relies only on a notion of distance between symbols in this paper we consider the index assignment problem and adopt a minimax design criterion instead of the usual mean squared error mse measure the problem is found to be np hard and an effective heuristic polynomial time algorithm is presented for computing approximate solutions the minimax criterion yields greatly improved worst case performance while maintaining good average performance in addition the familiar mse criterion is shown likewise to yield an np hard index assignment task the mse problem is a special case of the classical quadratic assignment problem for which computationally and theoretically useful results are available from the discrete mathematics literature

balancing hydraulic turbine runners a quadratic assignment problem abstract hydraulic turbine runners are used in electricity generation these consist of a cylinder around which are welded at regular spacings a number of blades whose weights differ slightly it is desired to locate the blades around the cylinder in order to minimize the distance between the center of mass of the blades and the geometric center of the cylinder the problem can be formulated as a quadratic assignment problem it is solved by adapting an interchange algorithm devised by or for the travelling salesman problem computational results are reported

quadratic assignment problems abstract this paper surveys quadratic assignment problems qap at first several applications of this problem class are described and mathematical formulations of qaps are given then some exact solution methods and good heuristics are outlined their computational behaviour is illustrated by numerical results further recent results on the asymptotic probabilistic behaviour of qaps are outlined

applications of parametric programming and eigenvalue maximization to the quadratic assignment problem we investigate new bounding strategies based on different relaxations of the quadratic assignment problem in particular we improve the lower bound found by using an eigenvalue decomposition of the quadratic part and by solving a linear program for the linear part the improvement is accomplished by applying a steepest ascent algorithm to the sum of the two bounds

a class of quasi newton generalized steffensen methods on banach spaces we consider a class of generalized steffensen iterations procedure for solving nonlinear equations on banach spaces without any derivative we establish the convergence under the kantarovich ostrowski s conditions the majorizing sequence will be a newton s type sequence thus the convergence can have better properties finally a numerical comparation with the classical methods is presented

on a steffensen s type method and its behavior for semismooth equations abstract a generalization of steffensen s method is proposed our goal is to obtain similar convergence as newton s method but without to evaluate any derivative and without to have stability problems convergence analysis and numerical results for semismooth equations are presented