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 heuristic procedure for solving the quadratic assignment problem abstract an heuristic approach to the solution of the quadratic assignment problem is presented a simple procedure is used to get a good feasible starting point then the problem is solved as a nonlinear program ignoring the integrality conditions using minos and lastly the near integer solution is converted into an integer feasible solution using an heuristic procedure the results compare favourably with other procedures in the literature a superior solution to the 19 19 hospital layout problem is found

on the variance of fuzzy random variables abstract this paper deals with an expectation and a real valued variance of fuzzy random variables the expectation and the variance of a fuzzy random variable is characterized by frechet s principle in a metric space we study properties of the variance of a fuzzy random variable and compare it with the common variance of real valued random variables using the expectation and the variance of fuzzy random variables we consider a linear regression problem and limit theorems

solving lift and project relaxations of binary integer programs we propose a method for optimizing the lift and project relaxations of binary integer programs introduced by lovasz and schrijver in particular we study both linear and semidefinite relaxations the key idea is a restructuring of the relaxations which isolates the complicating constraints and allows for a lagrangian approach we detail an enhanced subgradient method and discuss its efficient implementation computational results illustrate that our algorithm produces tight bounds more quickly than state of the art linear and semidefinite solvers

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

approximation orders and shape preserving properties of the multiquadric trigonometric b spline quasi interpolant the purpose of the paper is to derive some properties of the multiquadric trigonometric b spline quasi interpolant firstly the paper captures its approximation orders for high order derivatives based on the error estimate one can choose the shape parameter properly such that the quasi interpolant gives optimal approximations to high order derivatives moreover the approximation orders also show that from the theoretical point of view the considered quasi interpolant can be applied when the approximation of high order derivatives is needed i e numerical solution of some pdes construction of lyapunov function etc secondly the paper derives some shape preserving properties of the quasi interpolant these properties suggest that the quasi interpolant may be used in the geometric modeling cad cam for instance that requires shape preservation finally to illustrate the validity of the results some numerical examples are presented both theoretical and numerical results demonstrate that the quasi interpolant cannot only provide excellent approximations to high order derivatives but also preserve the shapes well

computational experience with a bundle approach for semidefinite cutting plane relaxations of max cut and equipartition we propose a dynamic version of the bundle method to get approximate solutions to semidefinite programs with a nearly arbitrary number of linear inequalities our approach is based on lagrangian duality where the inequalities are dualized and only a basic set of constraints is maintained explicitly this leads to function evaluations requiring to solve a relatively simple semidefinite program our approach provides accurate solutions to semidefinite relaxations of the max cut and the equipartition problem which are not achievable by direct approaches based only on interior point methods

higher order models for fuzzy random variables a fuzzy random variable is viewed as the imprecise observation of the outcomes in a random experiment since randomness and vagueness coexist in the same framework it seems reasonable to integrate fuzzy random variables into imprecise probabilities theory nevertheless fuzzy random variables are commonly presented in the literature as classical measurable functions associated to a classical probability measure we present here a higher order possibility model that represents the imprecise information provided by a fuzzy random variable we compare it with previous classical models in the literature first some aspects about the acceptability function associated to a fuzzy random variable are investigated secondly we present three different higher order possibility models all of them arising in a natural way we investigate their similarities and differences and observe that the first one the fuzzy probability envelope is the most informative finally we compare the fuzzy probability envelope with the classical probability measure induced by the fuzzy random variable we conclude that the classical probability measure does not always contain all relevant information provided by a fuzzy random variable

estimating bounds for quadratic assignment problems associated with hamming and manhattan distance matrices based on semidefinite programming quadratic assignment problems qaps with a hamming distance matrix for a hypercube or a manhattan distance matrix for a rectangular grid arise frequently from communications and facility locations and are known to be among the hardest discrete optimization problems in this paper we consider the issue of how to obtain lower bounds for those two classes of qaps based on semidefinite programming sdp by exploiting the data structure of the distance matrix b we first show that for any permutation matrix x the matrix y alpha e xbx t is positive semidefinite where alpha is a properly chosen parameter depending only on the associated graph in the underlying qap and e ee t is a rank 1 matrix whose elements have value 1 this results in a natural way to approximate the original qaps via sdp relaxation based on the matrix splitting technique our new sdp relaxations have a smaller size compared with other sdp relaxations in the literature and can be solved efficiently by most open source sdp solvers experimental results show that for the underlying qaps of size up to n 200 strong bounds can be obtained effectively

the quadratic three dimensional assignment problem exact and approximate solution methods abstract this paper reports on algorithm development for solving the quadratic three dimensional assignment problem q3ap the q3ap arises for example in the implementation of a hybrid arq automatic repeat request scheme for enriching diversity among multiple packet re transmissions by optimizing the mapping of data bits to modulation symbols typical practical problem sizes would be 8 16 32 and 64 we present an exact solution method based upon a reformulation linearization technique that is one of the best available for solving the quadratic assignment problem qap our current exact algorithm is useful for q3ap instances of size 13 or smaller we also investigate four stochastic local search algorithms that provide optimum or near optimum solutions for large and difficult qap instances and adapt them for solving the q3ap the results of our experiments make it possible to get good solutions to signal mapping problems of size 8 and 16

regularization theory and neural networks architectures we had previously shown that regularization principles lead to approximation schemes that are equivalent to networks with one layer of hidden units called regularization networks in particular s

on the variability of the concept of variance for fuzzy random variables fuzzy random variables possess several interpretations historically they were proposed either as a tool for handling linguistic label information in statistics or to represent uncertainty about classical random variables accordingly there are two different approaches to the definition of the variance of a fuzzy random variable in the first one the variance of the fuzzy random variable is defined as a crisp number which makes it easier to handle in further processing in the second case the variance is defined as a fuzzy interval thus offering a gradual description of our incomplete knowledge about the variance of an underlying imprecisely observed classical random variable in this paper we also discuss another view of fuzzy random variables which comes down to a set of random variables induced by a fuzzy relation describing an ill known conditional probability this view leads to yet another definition of the variance of a fuzzy random variable in the context of the theory of imprecise probabilities the new variance is a real interval which achieves a compromise between both previous definitions in terms of representation simplicity our main objective is to demonstrate with the help of simple examples the practical significance of these definitions of variance induced by various existing views of fuzzy random variables

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

lower bounding procedure for the asymmetric quadratic traveling salesman problem abstract in this paper we consider the asymmetric quadratic traveling salesman problem aqtsp given a directed graph and a function that maps every pair of consecutive arcs to a cost the problem consists in finding a cycle that visits every vertex exactly once and such that the sum of the costs is minimal we propose an extended linear programming formulation that has a variable for each cycle in the graph since the number of cycles is exponential in the graph size we propose a column generation approach moreover we apply a particular reformulation linearization technique on a compact representation of the problem and compute lower bounds based on lagrangian relaxation we compare our new bounds with those obtained by some linearization models proposed in the literature computational results on some set of benchmarks used in the literature show that our lower bounding procedures are very promising

eigenvalue bounds versus semidefinite relaxations for the quadratic assignment problem it was recently demonstrated that a well known eigenvalue bound for the quadratic assignment problem qap actually corresponds to a semidefinite programming sdp relaxation however for this bound to be computationally useful the assignment constraints of the qap must first be eliminated and the bound then applied to a lower dimensional problem the resulting projected eigenvalue bound is one of the best available bounds for the qap especially when considering the quality of bounds relative to the complexity of obtaining them in this paper we show that the projected eigenvalue bound is also related to an sdp relaxation of the original qap this implicit sdp relaxation is similar to sdp relaxations of the qap proposed by lin and saigal on solving large scale semidefinite programming problems a case study of quadratic assignment problem department of industrial engineering and operations research university of michigan 1997 and zhao et al j combin optim 2 1998 pp 71 109

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

a quasi interpolation scheme for periodic data based on multiquadric trigonometric b splines abstract multiquadric mq quasi interpolation has been studied extensively in the literature however the mq quasi interpolant is not well defined for periodic data since its kernel the mq function itself is not periodic note that in many applications the data may arise from a closed curve surface and thus possess some kind of periodicity for example analysis of geodetic and meteorological data constructing active contours estimating the region of attraction of dynamical systems and so on therefore it is meaningful to construct a quasi interpolant whose kernel itself is periodic for periodic data combining the construction techniques of the mq quasi interpolant and the trigonometric b spline quasi interpolant the paper constructs such a quasi interpolant the quasi interpolant couples together the periodicity of the trigonometric b spline quasi interpolant and the smoothness of the mq quasi interpolant moreover both the quasi interpolant and its derivatives are periodic the quasi interpolant covers the trigonometric b spline quasi interpolant as a special case in addition the error estimate shows that a proper shape parameter can be chosen such that the quasi interpolant provides the same approximation order as a trigonometric b spline quasi interpolant for a periodic function furthermore the quasi interpolant gives better approximations to high order derivatives than the trigonometric b spline quasi interpolant as illustrated by numerical examples presented at the end of the paper

implementation of a variance reduction based lower bound in a branch and bound algorithm for the quadratic assignment problem 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 gilmore lawler 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 problems requiring the examination of much fewer nodes of the branch and bound tree than the same algorithm using the gilmore lawler lower bound

dynamic bundle methods lagrangian relaxation is a popular technique to solve difficult optimization problems however the applicability of this technique depends on having a relatively low number of hard constraints to dualize when there are many hard constraints it may be preferable to relax them dynamically according to some rule depending on which multipliers are active from the dual point of view this approach yields multipliers with varying dimensions and a dual objective function that changes along iterations we discuss how to apply a bundle methodology to solve this kind of dual problems our framework covers many separation procedures to generate inequalities that can be found in the literature including but not limited to the most violated inequality we analyze the resulting dynamic bundle method giving a positive answer for its primal dual convergence properties and under suitable conditions show finite termination for polyhedral problems

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

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

a spectral bundle method for semidefinite programming a central drawback of primal dual interior point methods for semidefinite programs is their lack of ability to exploit problem structure in cost and coefficient matrices this restricts applicability to problems of small dimension typically semidefinite relaxations arising in combinatorial applications have sparse and well structured cost and coefficient matrices of huge order we present a method that allows us to compute acceptable approximations to the optimal solution of large problems within reasonable time semidefinite programming problems with constant trace on the primal feasible set are equivalent to eigenvalue optimization problems these are convex nonsmooth programming problems and can be solved by bundle methods we propose replacing the traditional polyhedral cutting plane model constructed from subgradient information by a semidefinite model that is tailored to eigenvalue problems convergence follows from the traditional approach but a proof is included for completeness we present numerical examples demonstrating the efficiency of the approach on combinatorial examples

approximations of upper and lower probabilities by measurable selections a random set can be regarded as the result of the imprecise observation of a random variable following this interpretation we study to which extent the upper and lower probabilities induced by the random set keep all the information about the values of the probability distribution of the random variable we link this problem to the existence of selectors of a multi valued mapping and with the inner approximations of the upper probability and prove that under fairly general conditions although not in all cases the upper and lower probabilities are an adequate tool for modelling the available information in doing this we generalise a number of results from the literature finally we study the particular case of consonant random sets and we also derive a relationship between aumann and choquet integrals

a new bound for the quadratic assignment problem based on convex quadratic programming 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 trade off between bound quality and computational effort

lower bounds for the quadratic assignment problem via triangle decompositions we consider transformations of the metric quadratic assignment problem qap that exploit the metric structure of a given instance we show in particular how the structural properties of rectangular grids can be used to improve a given lower bound our work is motivated by previous research of palubetskes 1988 and it extends a bounding approach proposed by chakrapani and skorin kapov 1993 our computational results indicate that the present approach is practical it has been applied to problems of dimension up ton 150 moreover the new approach yields by far the best lower bounds on most of the instances of metric qaps that we considered

a new lift and project operator abstract in this paper we analyze the strength of split cuts in a lift and project framework we first observe that the lovasz schrijver and sherali adams lift and project operator hierarchies can be viewed as applying specific 0 1 split cuts to an appropriate extended formulation and demonstrate how to strengthen these hierarchies using additional split cuts more precisely we define a new operator that adds all 0 1 split cuts to the extended formulation for 0 1 mixed integer sets with k binary variables this new operator is guaranteed to obtain the integer hull in k 2 steps compared to k steps for the lovasz schrijver or the sherali adams operator we also present computational results on the stable set problem with our new operator

a version of the bundle idea for minimizing a nonsmooth function conceptual idea convergence analysis numerical results during recent years various proposals for the minimization of a nonsmooth functional have been made amongst these the bundle concept turned out to be an especially fruitful idea based on this concept a number of authors have developed codes that can successfully deal with nonsmooth problems the aim of the paper is to show that by adding some features of the trust region philosophy to the bundle concept the end result is a distinguished member of the bundle family with a more stable behaviour than some other bundle versions the reliability and efficiency of this code is demonstrated on the standard academic test examples and on some real life problems

semidefinite programming and eigenvalue bounds for the graph partition problem the graph partition problem is the problem of partitioning the vertex set of a graph into a fixed number of sets of given sizes such that the sum of weights of edges joining different sets is optimized in this paper we simplify a known matrix lifting semidefinite programming relaxation of the graph partition problem for several classes of graphs and also show how to aggregate additional triangle and independent set constraints for graphs with symmetry we present an eigenvalue bound for the graph partition problem of a strongly regular graph extending a similar result for the equipartition problem we also derive a linear programming bound of the graph partition problem for certain johnson and kneser graphs using what we call the laplacian algebra of a graph we derive an eigenvalue bound for the graph partition problem that is the first known closed form bound that is applicable to any graph thereby extending a well known result in spectral graph theory finally we strengthen a known semidefinite programming relaxation of a specific quadratic assignment problem and the above mentioned matrix lifting semidefinite programming relaxation by adding two constraints that correspond to assigning two vertices of the graph to different parts of the partition this strengthening performs well on highly symmetric graphs when other relaxations provide weak or trivial bounds

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

a level 2 reformulation linearization technique bound for the quadratic assignment problem abstract this paper studies polyhedral methods for the quadratic assignment problem bounds on the objective value are obtained using mixed 0 1 linear representations that result from a reformulation linearization technique rlt the rlt provides different levels of representations that give increasing strength prior studies have shown that even the weakest level 1 form yields very tight bounds which in turn lead to improved solution methodologies this paper focuses on implementing level 2 we compare level 2 with level 1 and other bounding mechanisms in terms of both overall strength and ease of computation in so doing we extend earlier work on level 1 by implementing a lagrangian relaxation that exploits block diagonal structure present in the constraints the bounds are embedded within an enumerative algorithm to devise an exact solution strategy our computer results are notable exhibiting a dramatic reduction in nodes examined in the enumerative phase and allowing for the exact solution of large instances

iterative solution of systems originating from integral equations and surface interpolation a simple procedure for generating diagonal dominance in certain linear systems is presented and the iterative solution of the transformed systems is discussed the method is presented for systems encountered in the numerical solution of fredholm equations of the first kind with singular kernels and for systems arising in the interpolation of surfaces by certain sets of radial functions in the second case the method provides new sets of basis functions generalizing in some sense the univariate b splines

local minima and convergence in low rank semidefinite programming the low rank semidefinite programming problem lrsdpr is a restriction of the semidefinite programming problem sdp in which a bound r is imposed on the rank of x and it is well known that lrsdpr is equivalent to sdp if r is not too small in this paper we classify the local minima of lrsdpr and prove the optimal convergence of a slight variant of the successful yet experimental algorithm of burer and monteiro 5 which handles lrsdpr via the nonconvex change of variables x rrt in addition for particular problem classes we describe a practical technique for obtaining lower bounds on the optimal solution value during the execution of the algorithm computational results are presented on a set of combinatorial optimization relaxations including some of the largest quadratic assignment sdps solved to date

on the variance of random sets on considere la notion de variance d un ensemble aleatoire on donne un algorithme de determination de la variance des ensembles aleatoires prenant leurs valeurs dans des intervalles compacts non vides sur r

upper and lower probabilities induced by a fuzzy random variable we review two existing interpretations of fuzzy random variables in the first one the fuzzy random variable is viewed as a linguistic random variable in the second case it represents some incomplete knowledge about an otherwise standard random variable for each interpretation the information provided by the frv is described by a specific model namely a standard probability model and a second order imprecise model respectively in this paper we deal with an alternative interpretation guided by simple examples we will observe the usefulness of each interpretation when applied to particular situations then we will demonstrate that the new interpretation leads in a natural way to the pair of order capacities furthermore we show that they are formally related to the former models these results can be applied in future works to make inferences from fuzzy sample data the use of upper lower models instead of second order models will enable us in the future to reach crisp decisions in some specific statistical problems without adding any arbitrary information and taking into account the imprecision in data

the l a mean squared dispersion associated with a fuzzy random variable in this paper we introduce a parameterized real valued measure of the mean dispersion of a fuzzy random variable with respect to an arbitrary fuzzy number this measure extends the second moment of a classical random variable and is based on a parameterized distance between fuzzy numbers properties of the measure presented are analyzed and the extension of the variance of a classical random variable which particularizes the mean squared dispersion is also examined some examples illustrating the computation and possible applications of the measure are included finally a brief discussion about the interest of using a parameterized distance and about some future directions of this study is developed

cutting planes for rlt relaxations of mixed 0 1 polynomial programs the reformulation linearization technique due to sherali and adams can be used to construct hierarchies of linear programming relaxations of mixed 0 1 polynomial programs as one moves up the hierarchy the relaxations grow stronger but the number of variables increases exponentially we present a procedure that generates cutting planes at any given level of the hierarchy by optimally weakening linear inequalities that are valid at any given higher level computational experiments conducted on instances of the quadratic knapsack problem indicate that the cutting planes can close a significant proportion of the integrality gap

on the sqap polytope the sqap polytope was associated to quadratic assignment problems with a certain symmetric objective function structure by rijal 1995 and padberg and rijal 1996 we derive a technique for investigating the sqap polytope that is based on projecting the low dimensional polytope into a lower dimensional vector space where the vertices have a more convenient coordinate structure we exploit this technique in order to prove conjectures by padberg and rijal on the dimension of the sqap polytope as well as on its trivial facets

lower bounds for nonlinear assignment problems using many body interactions abstract this paper concerns lower bounding techniques for the general adic assignment problem the nonlinear objective function is linearized by the introduction of additional variables and constraints thus yielding a mixed integer linear programming formulation of the problem the concept of many body interactions is introduced to strengthen this formulation and incorporated in a modified formulation obtained by lifting the original representation to a higher dimensional space this process involves two steps i addition of new variables and constraints and ii incorporation of the new variables in the objective function if this lifting process is repeated times on an adic assignment problem along with the incorporation of higher order interactions it results in the mixed integer formulation of an equivalent adic assignment problem the incorporation of many body interactions in the higher dimensional formulation improves its degeneracy properties and is also critical to the derivation of decomposition methods for the solution of these large scale mathematical programs in the higher dimensional space it is shown that a lower bound to the optimal solution of the corresponding linear programming relaxation can be obtained by dualizing a subset of constraints in this formulation and solving o n 2 1 linear assignment problems whose coefficients depend on the dual values moreover it is proved that the optimal solution to the lp relaxation is obtained if we use the optimal duals for the solution of the linear assignment problems this concept of many body interactions could be applied in designing algorithms for the solution of formulations obtained by lifting general milp s we illustrate all these concepts on the quadratic assignment problems with these decomposition bounds we have found the provably optimal solutions of two unsolved qap s of size 32 and have also improved upon existing lower bounds for other qap s

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

constrained 0 1 quadratic programming basic approaches and extensions abstract we describe the simplest technique to tackle 0 1 quadratic programs with linear constraints among those that turn out to be successful in practice this method is due to and familiar to the quadratic assignment experts even if it took some time to realize that most approaches to the problem could be interpreted in these terms whereas it does not appear to be widely known outside this community since the technique is completely general and is by far the most successful one in several other cases such as quadratic knapsack we illustrate it in its full generality pointing out its relationship to lagrangian and linear programming relaxations and discussing further extensions we believe that this method should be in the background of every practitioner in combinatorial optimization

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

symmetry in rlt type relaxations for the quadratic assignment and standard quadratic optimization problems the reformulation linearization technique rlt introduced in sherali h d adams w p 1990 a hierarchy of relaxations between the continuous and convex hull representations for zero one programming problems siam journal on discrete mathematics 3 3 411 430 provides a way to compute a hierarchy of linear programming bounds on the optimal values of np hard combinatorial optimization problems in this paper we show that in the presence of suitable algebraic symmetry in the original problem data it is sometimes possible to compute level two rlt bounds with additional linear matrix inequality constraints as an illustration of our methodology we compute the best known bounds for certain graph partitioning problems on strongly regular graphs

a family of multivariate multiquadric quasi interpolation operators with higher degree polynomial reproduction in this paper by using multivariate divided difference rabut 2001 to approximate the partial derivative and the idea of the superposition waldron 2009 we modify a multiquadric quasi interpolation operator ling 2004 based on a dimension splitting technique with the property of linear reproducing to gridded data on multi dimensional spaces such that a family of proposed multivariate multiquadric quasi interpolation operators f r 1 has the property of r 1 email protected z r 0 degree polynomial reproducing and converges up to a rate of r 2 in addition the proposed quasi interpolation operator only demands information of location points rather than the derivatives of the function approximated moreover we give the approximation error of our quasi interpolation operator finally some numerical experiments are shown to confirm the approximation capacity of our quasi interpolation operator

multi symplectic quasi interpolation method for hamiltonian partial differential equations abstract in this paper we propose a multi symplectic quasi interpolation method for solving multi symplectic hamiltonian partial differential equations based on the method of lines we first discretize the multi symplectic pdes using quasi interpolation method and then employ appropriate time integrators to obtain the full discrete system the local conservation properties including multi symplectic conservation laws energy conservation laws and momentum conservation laws are discussed in detail for illustration we provide two concrete examples the nonlinear wave equation and the nonlinear schrodinger equation the salient feature of our multi symplectic quasi interpolation method is that it is valid both on uniform grids and nonuniform grids the numerical results show the good accuracy and excellent conservation properties of the proposed method

advocating the use of imprecisely observed data in genetic fuzzy systems in our opinion and in accordance with current literature the precise contribution of genetic fuzzy systems to the corpus of the machine learning theory has not been clearly stated yet in particular we question the existence of a set of problems for which the use of fuzzy rules in combination with genetic algorithms produces more robust models or classifiers that are inherently better than those arising from the bayesian point of view we will show that this set of problems actually exists and comprises interval and fuzzy valued datasets but it is not being exploited current genetic fuzzy classifiers deal with crisp classification problems where the role of fuzzy sets is reduced to give a parametric definition of a set of discriminant functions with a convenient linguistic interpretation provided that the customary use of fuzzy sets in statistics is vague data we propose to test genetic fuzzy classifiers over imprecisely measured data and design experiments well suited to these problems the same can be said about genetic fuzzy models the use of a scalar fitness function assumes crisp data where fuzzy models a priori do not have advantages over statistical regression

intelligent local search strategies for solving facility layout problems with the quadratic assignment problem formulation abstract in this paper we implement a tabu search heuristic a probabilistic tabu search heuristic a simulated annealing heuristic and a hybrid tabu search heuristic to solve facility layout problems with the quadratic assignment problem formulation classical problems are used as a benchmark for performance comparison extensive computational experiments were conducted and the results were statistically analyzed to observe the behavior of each heuristic and to discriminate the performance of each heuristic the computational results strongly support the competitiveness of intelligent heuristics for the design of facilities the hybrid tabu search heuristic combining the advantages from both tabu search and simulated annealing can generally outperform their parent algorithms

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 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 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

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

imprecise distribution function associated to a random set some different extensions to random sets of the most common parameters of a random variable share a common rationale a random set represents the imprecise observation of a random variable hence the generalized parameter contains the available information about the respective parameter of the imprecisely observed variable following the same principles in this paper it is proposed a new definition of the distribution function of a random set this definition is simpler in its formulation and it can be used in more general cases than previous proposals the properties of the distribution function defined here are discussed some issues about continuity convergence of the images of the distribution function monotonocity and measurability are studied it is also stated that not all the information conveyed by the random set about the original probability measure the probability measure induced by the imprecisely observed random variable is kept by its distribution function

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

solving large quadratic assignment problems on computational grids the quadratic assignment problem qap is among the hardest combinatorial optimization problems some instances of size n 30 have remained unsolved for decades the solution of these problems requires both improvements in mathematical programming algorithms and the utilization of powerful computational platforms in this article we describe a novel approach to solve qaps using a state of the art branch and bound algorithm running on a federation of geographically distributed resources known as a computational grid solution of qaps of unprecedented complexity including the nug30 kra30b and tho30 instances is reported

a path following algorithm for the graph matching problem we propose a convex concave programming approach for the labeled weighted graph matching problem the convex concave programming formulation is obtained by rewriting the weighted graph matching problem as a least square problem on the set of permutation matrices and relaxing it to two different optimization problems a quadratic convex and a quadratic concave optimization problem on the set of doubly stochastic matrices the concave relaxation has the same global minimum as the initial graph matching problem but the search for its global minimum is also a hard combinatorial problem we therefore construct an approximation of the concave problem solution by following a solution path of a convex concave problem obtained by linear interpolation of the convex and concave formulations starting from the convex relaxation this method allows to easily integrate the information on graph label similarities into the optimization problem and therefore perform labeled weighted graph matching the algorithm is compared with some of the best performing graph matching methods on four data sets simulated graphs qaplib retina vessel images and handwritten chinese characters in all cases the results are competitive with the state of the art

a polynomial barrier algorithm for linearly constrained convex programming problems we consider the standard linearly constrained convex program min fx x n ax b x 0 where fx satisfies a certain scaled lipschitz condition using the classical logarithmic barrier method our algorithm obtains an e solution within o n ln e or on ln e total iterations depending on the updating of the barrier parameter

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

on the efficient computation of a generalized jacobian of the projector over the birkhoff polytope we derive an explicit formula as well as an efficient procedure for constructing a generalized jacobian for the projector of a given square matrix onto the birkhoff polytope i e the set of doubly stochastic matrices to guarantee the high efficiency of our procedure a semismooth newton method for solving the dual of the projection problem is proposed and efficiently implemented extensive numerical experiments are presented to demonstrate the merits and effectiveness of our method by comparing its performance against other powerful solvers such as the commercial software gurobi and the academic code pproj hager and zhang in siam j optim 26 1773 1798 2016 in particular our algorithm is able to solve the projection problem with over one billion variables and nonnegative constraints to a very high accuracy in less than 15 min on a modest desktop computer more importantly based on our efficient computation of the projections and their generalized jacobians we can design a highly efficient augmented lagrangian method alm for solving a class of convex quadratic programming qp problems constrained by the birkhoff polytope the resulted alm is demonstrated to be much more efficient than gurobi in solving a collection of qp problems arising from the relaxation of quadratic assignment problems

quasi interpolation for linear functional data quasi interpolation has been studied extensively in the literature however most studies of quasi interpolation are usually only for discrete function values or a finite linear combination of discrete function values note that in practical applications more commonly we can sample the linear functional data the discrete values of the right hand side of some differential equations rather than the discrete function values e g remote sensing seismic data etc therefore it is more meaningful to study quasi interpolation for the linear functional data the main result of this paper is to propose such a quasi interpolation scheme error estimate of the scheme is also given in the paper based on the error estimate one can find a quasi interpolant that provides an optimal approximation order with respect to the smoothness of the right hand side of the differential equation the scheme can be applied in many situations such as the numerical solution of the differential equation construction of the lyapunov function and so on respective examples are presented in the end of this paper

a tight linearization and an algorithm for zero one quadratic programming problems this paper is concerned with the solution of linearly constrained zero one quadratic programming problems problems of this kind arise in numerous economic location decision and strategic planning situations including capital budgeting facility location quadratic assignment media selection and dynamic set covering a new linearization technique is presented for this problem which is demonstrated to yield a tighter continuous or linear programming relaxation than is available through other methods an implicit enumeration algorithm which uses lagrangian relaxation benders cutting planes and local explorations is designed to exploit the strength of this linearization computational experience is provided to demonstrate the usefulness of the proposed linearization and algorithm

an introduction to schoenberg s approximation for a given function b and a non zero real number h schoenberg s approximation defines from some data jh yj j zd the function j zd yj b h j for people not used to this kind of approximation this paper intends to do a summary of the main definitions properties and utilizations of schoenberg s approximation we show that the main tool to handle schoenberg s approximation is the fourier transform of b and even more its modified version the transfer function of b we give conditions for convergence of j zd f jh b h j when h tends to zero and we give various ways to define various b as combinations of translates of some function usually is either some radial function or obtained by a tensor product of some radial function depending on the properties we want for the associated schoenberg s approximation last we show how multi resolution analysis subdivision techniques and wavelets techniques are nicely connected to schoenberg s approximation

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

a new exact discrete linear reformulation of the quadratic assignment problem abstract the quadratic assignment problem qap is a challenging combinatorial problem the problem is np hard and in addition it is considered practically intractable to solve large qap instances to proven optimality within reasonable time limits in this paper we present an attractive mixed integer linear programming milp formulation of the qap we first introduce a useful non linear formulation of the problem and then a method of how to reformulate it to a new exact compact discrete linear model this reformulation is efficient for qap instances with few unique elements in the flow or distance matrices finally we present optimal results obtained with the discrete linear reformulation for some previously unsolved instances with the size n 32 and 64 from the quadratic assignment problem library qaplib

computational results of an interior point algorithm for large scale linear programming this paper gives computational results for an efficient implementation of a variant of dual projective algorithm for linear programming the implementation uses the preconditioned conjugate gradient method for computing projections our computational experience reported in this paper indicates that this algorithm has potential as an alternative for solving very large lps in which the direct methods fail due to memory and cpu time requirements the conjugate gradient algorithm was able to find very accurate directions even when the system was ill conditioned the paper also discusses a new mathematical technique called the reciprocal estimates for estimating the primal variables we have conducted extensive computational experiments on problems representative of large classes of applications of current interest we have also chosen instances of the problems of future potential interest which could not be solved in the past due to the weakness of the prior solution methods but which represent a large class of new applications the hypergraph model is such an example comparison of our implementation with minos 5 1 shows that our implementation is orders of magnitude faster than minos 5 1 for these problems

algorithms for the generalized quadratic assignment problem combining lagrangean decomposition and the reformulation linearization technique in this paper we propose two exact algorithms for the gqap generalized quadratic assignment problem in this problem given m facilities and n locations the facility space requirements the location available space the facility installation costs the flows between facilities and the distance costs between locations one must assign each facility to exactly one location so that each location has sufficient space for all facilities assigned to it and the sum of the products of the facility flows by the corresponding distance costs plus the sum of the installation costs is minimized this problem generalizes the well known quadratic assignment problem qap both exact algorithms combine a previously proposed branch and bound scheme with a new lagrangean relaxation procedure over a known rlt reformulation linearization technique formulation we also apply transformational lower bounding techniques to improve the performance of the new procedure we report detailed experimental results where 19 out of 21 instances with up to 35 facilities are solved in up to a few days of running time six of these instances were open

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