approximate solutions to the turbine balancing problem abstract the turbine balancing problem tbp is an np hard combinatorial optimization problem arising in the manufacturing and maintenance of turbine engines exact solution methods for solving the tbp are not appropriate since the problem has to be solved in real time and the input data is itself inaccurate in this paper the tbp is formulated as a quadratic assignment problem qap and we propose a heuristic algorithm for solving the resulting problem computational results on a set of instances provided by pratt whitney p w and from the literature indicate that the proposed algorithm outperforms the current methods used for solving the tbp and has the best overall performance with respect to other heuristic algorithms in the literature

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

a new rounding procedure for the assignment problem with applications to dense graph arrangement problems we present a randomized procedure for rounding fractional perfect matchings to integral matchings if the original fractional matching satisfies any linear inequality then with high probability the new matching satisfies that linear inequality in an approximate sense this extends the well known lp rounding procedure of raghavan and thompson which is usually used to round fractional solutions of linear programs we use our rounding procedure to design an additive approximation algorithm to the quadratic assignment problem the approximation error of the algorithm is en 2 and it runs in n o log n e2 time we also describe polynomial time approximation schemes ptass for dense subcases of many well known np hard arrangement problems including minimum linear arrangement minimum cut linear arrangement maximum acyclic subgraph and betweenness

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

a semidefinite optimization based approach for global optimization of multi row facility layout abstract this paper is concerned with the multi row facility layout problem given a set of rectangular departments a fixed number of rows and weights for each pair of departments the problem consists of finding an assignment of departments to rows and the positions of the departments in each row so that the total weighted sum of the center to center distances between all pairs of departments is minimized we show how to extend our recent approach for the space free multi row facility layout problem to general multi row facility layout as well as some special cases thereof to the best of our knowledge this is the first global optimization approach for multi row layout that is applicable beyond the double row case a key aspect of our proposed approach is a model for multi row layout that expresses the problem as a discrete optimization problem and thus makes it possible to exploit the underlying combinatorial structure in particular we can explicitly control the number and size of the spaces between departments we construct a semidefinite relaxation of the discrete optimization formulation and present computational results showing that the proposed approach gives promising results for several variants of multi row layout problems on a variety of benchmark instances

iterative construction of fixed points of asymptotically nonexpansive mappings abstract let t be a completely continuous and asymptotically nonexpansive self mapping of a nonempty closed bounded and convex subset of a hilbert space then under certain conditions the sequence defined by xn 1 ntn xn 1 n x n converges strongly to some fixed point of t a similar result is obtained for asymptotically pseudocontractive mappings whose iterates are lipschitz continuous with respect to some common lipschitz constant

integer programming models for the multidimensional assignment problem with star costs abstract we consider a variant of the multidimensional assignment problem map with decomposable costs in which the resulting optimal assignment is described as a set of disjoint stars this problem arises in the context of multi sensor multi target tracking problems where a set of measurements obtained from a collection of sensors must be associated to a set of different targets to solve this problem we study two different formulations first we introduce a continuous nonlinear program and its linearization along with additional valid inequalities that improve the lower bounds second we state the standard map formulation as a set partitioning problem and solve it via branch and price these approaches were put to test by solving instances ranging from tripartite to 20 partite graphs of 4 to 30 nodes per partition computational results show that our approaches are a viable option to solve this problem a comparative study is presented

invariant functions on lie groups and hamiltonian flows of surface group representations si est le groupe fondamental d une surface orientee fermee s et g est un groupe de lie satisfaisant des conditions tres generales alors l espace hom g g des classes de conjugaison de representation g a une structure symplectique naturelle on etudie la geometrie de cette structure symplectique a l aide d une famille naturelle de fonctions sur hom g g

symplectic structure of the moduli space of flat connection on a riemann surface we consider the canonical symplectic structure on the moduli space of flat g connections on a riemann surface of genus g with a marked points for a being a semisimple lie algebra we obtain an explicit efficient formula for this symplectic form and prove

complexity of a 3 dimensional assignment problem abstract we show that a certain 3 dimensional assignment problem is np complete to do this we show that the following problem is np complete given bipartite graphs g 1 g 2 with the same sets of vertices do there exist perfect matchings m 1 m 2 of g 1 g 2 respectively such that m 1 m 2 o

on poisson actions of compact lie groups on symplectic manifolds let g p be a compact simple poisson lie group equipped with a poisson structure p and m omega be a symplectic manifold assume that m carries a poisson action of g p and there is an equivariant moment map in the sense of lu and weinstein which maps

the quadratic assignment problem with a monotone anti monge and a symmetric toeplitz matrix easy and hard cases this paper investigates a restricted version of the quadratic assignment problem qap where one of the coefficient matrices is an anti monge matrix with non decreasing rows and columns and the other coefficient matrix is a symmetric toeplitz matrix this restricted version is called the anti monge toeplitz qap there are three well known combinatorial problems that can be modeled via the anti monge toeplitz qap pl the turbine problem i e the assignment of given masses to the vertices of a regular polygon such that the distance of the center of gravity of the resulting system to the center of the polygon is minimized p2 the traveling salesman problem on symmetric monge distance matrices p3 the arrangement of data records with given access probabilities in a linear storage medium in order to minimize the average access time we identify conditions on the toeplitz matrixb that lead to a simple solution for the anti monge toeplitz qap the optimal permutation can be given in advance without regarding the numerical values of the data the resulting theorems generalize and unify several known results on problems p1 p2 and p3 we also show that the turbine problem is np hard and consequently that the anti monge toeplitz qap is np hard in general 1998 the mathematical programming society inc published by elsevier science b v

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

symplectic geometry of the moduli space of projective structures in homological coordinates we study the symplectic geometry of the space of linear differential equations with holomorphic coefficients of the form varphi u varphi 0 on riemann surfaces of genus g this space coincides with the moduli space of projective connections which is an affine bundle modelled on the cotangent bundle t mathcal m g we show that for several choices of the origin or base holomorphic projective connection such as bergman wirtinger or schottky the canonical poisson structure on t mathcal m g induces the goldman bracket on the monodromy character variety these different choices give rise to equivalent symplectic structures on the space of projective connections but different symplectic polarizations we find the corresponding generating functions combined with a prior theorem of kawai our results show the symplectic equivalence between the embeddings of t mathcal m g induced by the bers and bergman projective connections into the space of projective structures the main technical tools are variational formulas with respect to homological darboux coordinates on the space of holomorphic quadratic differentials in particular we get a new system of differential equations for the prym matrix of the canonical two sheeted covering of a riemann surface defined by a quadratic differential

invariant approximations and r subweakly commuting maps the existence of invariant approximations for r subweakly commuting mappings is proved our results extend most of the known results to a new class of noncommuting mappings

simulated annealing and tabu search in the long run a comparison on qap tasks simulated annealing sa and tabu search ts are compared on the quadratic assignment problem a recent work on the same benchmark suite argued that sa could achieve a reasonable solution quality with fewer function evaluations than ts the discussion is extended by showing that the conclusions must be changed if the task is hard or a very good approximation of the optimal solution is desired or if cpu time is the relevant parameter in addition a recently proposed version of ts the reactive tabu search solves the problem of finding the proper list size with an automatic memory based reaction mechanism

a grasp for the biquadratic assignment problem the biquadratic assignment problem biqap is a generalization of the quadratic assignment problem qap it is a nonlinear integer programming problem where the objective function is a fourth degree multivariable polynomial and the feasible domain is the assignment polytope biqap problems appear in vlsi synthesis due to the difficulty of this problem only heuristic solution approaches have been proposed in this paper we propose a new heuristic for the biqap a greedy randomized adaptive search procedure grasp computational results on instances described in the literature indicate that this procedure consistently finds better solutions than previously described algorithms

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

on new classes of nonnegative symmetric tensors in this paper we introduce three new classes of nonnegative forms or equivalently symmetric tensors and their extensions the newly identified nonnegative symmetric tensors constitute distinctive convex cones in the space of general symmetric tensors order six or above for the special case of quartic forms they collapse into the set of convex quartic homogeneous polynomial functions we discuss the properties and applications of the new classes of nonnegative symmetric tensors in the context of polynomial and tensor optimization numerical experiments for solving certain polynomial optimization models based on the new classes of nonnegative symmetric tensors are presented

the local structure of poisson manifolds varietes de poisson et applications decomposition structures de poisson lineaires approximation lineaire systemes hamiltoniens le probleme de linearisation groupes de fonction realisations et applications impulsion paires duales et groupes de jauge existence des realisations unicite des realisations les problemes des 3 corps restreints et autres exemples

heuristics for biquadratic assignment problems and their computational comparison abstract the biquadratic assignment problem biqap is a generalization of the quadratic assignment problem qap as for any hard optimization problem also for biqap a reasonable effort to cope with the problem is trying to derive heuristics which solve it suboptimally and which possibly yield a good trade off between the solution quality and the time and memory requirements in this paper we describe several heuristics for biqaps in particular pair exchange algorithms improvement methods and variants of simulated annealing and taboo search we implement these heuristics as c codes and analyze their performances

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

the hyperbolic moduli space of flat connections and the isomorphism of symplectic multiplicity spaces the hyperbolic moduli space of flat connections and the isomorphism of symplectic multiplicity spaces

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

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

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

strong convergence theorems for accretive operators in banach spaces abstract let e be a reflexive banach space with a uniformly g teaux differentiable norm and let a e e be m accretive assume that s cl d a e is bounded strongly accretive and continuous for x e and t 0 let xt be the unique solution of the inclusion x sxt taxt it is proved that if there exists a nonexpansive retraction p of e onto cl d a then the strong limt 0xt exists moreover it is shown that if every weakly compact convex subset of e has the fixed point property for nonexpansive mappings and 0 r a then the strong limt xt exists and belongs to a 10 an interesting application to a convergence result for an implicit iterative scheme is also included

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

lie poisson structure on some poisson lie groups poisson lie groups appeared in the work of drinfel d see e g drl dr2 as classical objects corresponding to quantum groups going in the other direction we may say that a poisson lie group is a group of symmetries of a phase space that are allowed to twist in a certain sense the symplectic or poisson structure the poisson structure on the group controls this twisting in a precise way quantizing both the phase space and the symmetries one may obtain a quantum group acting on a quantum phase space in recent work lu and ratiu lr used so called standard poisson structures on a compact semisimple lie group k and on its poisson dual k in order to give a new proof of the nonlinear convexity theorem of kostant ko their method is analogous to the famous symplectic proof of the linear convexity theorem given by atiyah a and guillemin and stemnberg gs the nonlinear convexity theorem like the linear one follows from a very general result on convexity of the image of the momentum map a gs however in lr it is applied not to a coadjoint orbit in t but to a symplectic leaf in k the main result of this paper is that the standard poisson structure on the poisson dual k to a compact semisimple poisson lie group k is actually isomorphic to the linear one on t this theorem seems to be related to some facts in the theory of quantum groups namely for generic q the universal enveloping algebra u t and its quantum deformation uq t are isomorphic as algebras though not as coalgebras of course since the quantum version is not cocommutative in particular there is a bijective correspondence between their representations a direct connection between our work and its quantum analogues though is still to be found the present work supplies a positive answer to question 5 1 in lr and strongly depends on that paper to simplify reading we keep the notations of lr wherever possible on one hand but give all necessary definitions on the other our work is also related to du in which the nonlinear convexity theorem is reduced to the linear one by a deformation argument not unlike the one that we use in 5 the paper is organized as follows in 2 we define poisson lie groups discuss

best simultaneous approximations asymptotically nonexpansive mappings and variational inequalities in banach spaces abstract the existence of common fixed points is established for three mappings where t is either generalized f g nonexpansive or asymptotically f g nonexpansive on a nonempty subset of a banach space as applications the invariant best simultaneous approximation results are proved and the existence of solution of variational inequalities is obtained our results unify and substantially improve several recent results existing in the current literature

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

simulated annealing for the qap optimal tradeoff between simulation time and solution quality abstract this paper reports on the optimization of solution quality for a simulated annealing algorithm applied to the quadratic assignment problem the optimization is performed by varying two of the parameters in the algorithm the size of the ensemble i e the number of simulations and the length of each simulation the variation is performed in such a way that the algorithm always uses the same amount of time for a given problem size it is found that the simulation length is indeed optimizable but also that a rather broad spectrum of values produce a near optimal solution quality generally it seems that the optimal simulation length is rather short compared to results found for other optimization problems

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

application of the simulated annealing algorithm to the combinatorial optimisation problem with permutation property an investigation of generation mechanism focusing on the generation mechanism of random permutation solutions this paper investigates the application of the simulated annealing sa algorithm to the combinatorial optimisation problems with permutation property six types of perturbation scheme for generating random permutation solutions are introduced they are proved to satisfy the asymptotical convergence requirements the results of experimental evaluations on traveling salesman problem tsp flow shop scheduling problem fsp and quadratic assignment problem qap reveal that the efficiencies of the perturbation schemes are different in searching a solution space by adopting a proper perturbation scheme the sa algorithm has shown to produce very efficient solutions to different combinatorial optimisation problems with permutation property

box inequalities for quadratic assignment polytopes linear programming based lower bounds have been considered both for the general as well as for the symmetric quadratic assignment problem several times in the recent years their quality has turned out to be quite good in practice investigations of the polytopes underlying the corresponding integer linear programming formulations the non symmetric and the symmetric quadratic assignment polytope have been started during the last decade 34 31 21 22 they have lead to basic knowledge on these polytopes concerning questions like their dimensions affine hulls and trivial facets however no large class of facet defining inequalities that could be used in cutting plane procedures had been found we present in this paper the first such class of inequalities the box inequalities which have an interesting origin in some well known hypermetric inequalities for the cut polytope computational experiments with a cutting plane algorithm based on these inequalities show that they are very useful with respect to the goal of solving quadratic assignment problems to optimality or to compute tight lower bounds the most effective ones among the new inequalities turn out to be indeed facet defining for both the non symmetric as well as for the symmetric quadratic assignment polytope

approximation of fixed points of uniformly r subweakly commuting mappings we introduce a new class of uniformly r subweakly commuting mappings and then using this class study the problem of approximation of common fixed points of asymptotically s nonexpansive mappings in a banach space with uniformly g teaux differentiable norm

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

stratified symplectic spaces and reduction let m w be a hamiltonian g space with proper momentum map j m g it is well known that if zero is a regular value of j and g acts freely on the level set j 0 then the reduced space mo j 0 g is a symplectic manifold we show that if the regularity assumptions are dropped the space mo is a union of symplectic manifolds i e it is a stratified symplectic space arms et al 2 proved that mo possesses a natural poisson bracket using their result we study hamiltonian dynamics on the reduced space in particular we show that hamiltonian flows are strata preserving and give a recipe for lifting a reduced hamiltonian flow to the level set j 0 finally we give a detailed description of the stratification of mo and prove the existence of a connected

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