improvement and comparison of heuristics for solving the uncapacitated multisource weber problem the multisource weber problem is to locate simultaneouslym facilities in the euclidean plane to minimize the total transportation cost for satisfying the demand ofn fixed users each supplied from its closest facility many heuristics have been proposed for this problem as well as a few exact algorithms heuristics are needed to solve quickly large problems and to provide good initial solutions for exact algorithms we compare various heuristics i e alternative location allocation cooper 1964 projection bongartz et al 1994 tabu search brimberg and mladenovic 1996a p median plus weber hansen et al 1996 genetic search and several versions of variable neighbourhood search based on empirical tests that are reported it is found that most traditional and some recent heuristics give poor results when the number of facilities to locate is large and that variable neighbourhood search gives consistently best results on average in moderate computing time

coincidence conditions in multifacility location problems with positive and negative weights in minisum multifacility location problems one has to find locations for some new facilities such that the weighted sum of distances between the new and a certain number of old facilities with known locations is minimized in this kind of problem the optimal locations of clusters of facilities frequently tend to coincide by testing conditions for coincidence one has the opportunity to collapse some or even all facilities coinciding at an optimal point into one in this way the dimension of the problem and the degree of nondifferentiability is reduced several conditions for coincidence have been published recently in this paper these conditions are extended and improved with respect to new sufficient coincidence conditions for location problems with attracting and repelling facilities an example shows that these new conditions detect more coincidences than the conditions which are known so far even if all facilities involved are attracting ones

technical note a generalized bounding method for multifacility location models this paper develops a new lower bound for multifacility location problems with lp distances we prove that the method produces superior results to other known procedures the new bound is also computationally efficient

special values of multiple polylogarithms historically the polylogarithm has attracted specialists and non specialists alike with its lovely evaluations much the same can be said for euler sums or multiple harmonic sums which within the past decade have arisen in combinatorics knot theory and high energy physics more recently we have been forced to consider multidimensional extensions encompassing the classical polylogarithm euler sums and the riemann zeta function here we provide a general framework within which previously isolated results can now be properly understood applying the theory developed herein we prove several previously conjectured evaluations including an intriguing conjecture of don zagier

the algebra of multiple harmonic series abstract recently there has been much interest in multiple harmonic series i 1 i 2 i k n 1 n 2 n k 1 1 n 1 i 1 n 2 i 2 n k i k which converge when the exponentsijare positive integers andi1 1 also known as multiple zeta values or euler zagier sums starting with the noncommutative polynomial algebraq x y we define a second multiplication which is commutative and associative and call the resulting structure the harmonic algebra h as a graded commutative algebra h turns out to be a free polynomial algebra with the number of generators in degreengiven by the witt formula for the numbern n of basis elements of degreenin the free lie algebra on two generators multiple harmonic series can be thought of as images under a map h 0 rwhich is a homomorphism with respect to the commutative multiplication where h 0is an appropriate subalgebra of h if we calli1 ikthe weight of the series i1 ik then forn 1 there are at mostn n series of weightnthat are irreducible in the sense that they are not sums of rational multiples of products of multiple harmonic series of lower weight in fact our approach gives an explicit set of algebraically irreducible multiple harmonic series we also show that there is a subalgebra h 0 h 1 h related to the shuffle algebra which contains the algebra s of symmetric functions in fact maps the elements of s h 0to algebraic combinations of zeta values i for integeri 2 the map is not injective we show how several results about multiple harmonic series can be recast as statements about the kernel of and propose some conjectures on the structure of the algebra h 0 ker

a method for globally minimizing concave functions over convex sets a method is described for globally minimizing concave functions over convex sets whose defining constraints may be nonlinear the algorithm generates linear programs whose solutions minimize the convex envelope of the original function over successively tighter polytopes enclosing the feasible region the algorithm does not involve cuts of the feasible region requires only simplex pivot operations and univariate search computations to be performed allows the objective function to be lower semicontinuous and nonseparable and is guaranteed to converge to the global solution computational aspects of the algorithm are discussed

solving general continuous single facility location problems by cutting planes abstract a unified cutting plane method is proposed to solve a large class of continuous single facility location problems these include both minisum and minimax problems with mixed norms and convex transportation costs the method provides an easy stopping rule and permits the post optimal determination of an outer approximation to near optimality regions two examples are given which show the power of the method extensive computational results in dimension two and higher are reported

a solution algorithm for non convex mixed integer optimization problems with only few continuous variables abstract geometric branch and bound techniques are well known solution algorithms for non convex continuous global optimization problems with box constraints several approaches can be found in the literature differing mainly in the bounds used the aim of this paper is to extend geometric branch and bound methods to mixed integer optimization problems i e to objective functions with some continuous and some integer variables mixed integer non linear and non convex optimization problems are extremely hard containing several classes of np hard problems as special cases we identify for which type of mixed integer non linear problems our method can be applied efficiently derive several bounding operations and analyze their rates of convergence theoretically moreover we show that the accuracy of any algorithm for solving the problem with fixed integer variables can be transferred to the mixed integer case our results are demonstrated theoretically and experimentally using the truncated weber problem and the p median problem for both problems we succeed in finding exact optimal solutions

a generalization of the rectangular bounding method for continuous location models abstract we develop a generalized bounding method for the weiszfeld iterative procedure used to solve the hyperbolically approximated l p norm single and multifacility minimum location problems we also show that at optimality the solution to the bound problem coincides with the solution to the original location problem we use this result to show that the rectangular bound value converges to the single facility location problem optimal objective function value

an existence theorem for a nonlinear difference equation where p 1 and s p is a real sequence we establish conditions under which 1 1 has a positive nondecreasing solution here a solution of 1 1 is a real sequence y y which satisfies 1 1 since 1 1 is a recurrence relation given real initial values y and y it is clear that we can inductively obtain y y since p is a real number greater than 1 the sequence yk may not be real that is not a solution note however that any constant multiple of a solution is again a solution the existence question originated in l where it was shown that hardy s inequality for a series 2 can be viewed as a necessary condition for the existence of a positive nondecreasing solution of 1 1 our existence theorem reduces to a nonoscillation criterion in the case p 2 since then 1 1 changes to a second order linear difference equation see e g 3 a brief outline of the paper is as follows section 2 introduces a riccati type transformation and then develops various necessary conditions for the existence of a positive nondecreasing solution section 3 discusses some of the functional analytic background which is needed in order to apply schauder s theorem section 4 contains the proof of our existence theorem and section 5 describes a comparison theorem which allows us to drop the assumption that the coefficients sk in 1 1 are nonnegative

when facilities coincide exact optimality conditions in multifacility location abstract in continuous minisum multifacility location problems it has been observed that optimal solutions often involve coincidences of facilities this paper gives a complete analysis by way of subdifferential calculus of the exact optimality conditions at such solutions these conditions are expressed as the existence of a multidimensional conservative flow in a network satisfying a nonlinear constraint on each edge by way of network flow theory a description with a minimal number of variables is obtained this procedure is illustrated by several examples showing how it enables a sensitivity analysis of the optimal solution with respect to the weights of the problem the optimality conditions are shown to be always directly verifiable if the network is a tree necessary conditions for coincidences at an optimal solution which previously appeared in the literature are generalized and it is shown in which cases these are also sufficient

on existence of positive solutions and bounded oscillations for neutral difference equations abstract a criterion for the existence of a positive solution for the first order difference equation xn cxn m pnxn k 0 pn 0 is established results are also obtained for the oscillation and nonoscillation of solutions of a second order difference equation 2 xn cxn m pnxn k where the pn are either of constant sign or are oscillatory some of the results are sharp

open questions concerning weiszfeld s algorithm for the fermat weber location problem the fermat weber location problem is to find a point in n that minimizes the sum of the weighted euclidean distances fromm given points in n a popular iterative solution method for this problem was first introduced by weiszfeld in 1937 in 1973 kuhn claimed that if them given points are not collinear then for all but a denumerable number of starting points the sequence of iterates generated by weiszfeld s scheme converges to the unique optimal solution we demonstrate that kuhn s convergence theorem is not always correct we then conjecture that if this algorithm is initiated at the affine subspace spanned by them given points the convergence is ensured for all but a denumerable number of starting points

concave minimization via conical partitions and polyhedral outer approximation an algorithm is proposed for globally minimizing a concave function over a compact convex set this algorithm combines typical branch and bound elements like partitioning bounding and deletion with suitably introduced cuts in such a way that the computationally most expensive subroutines of previous methods are avoided in each step essentially only few linear programming problems have to be solved some preliminary computational results are reported

double bundle method for finding clarke stationary points in nonsmooth dc programming the aim of this paper is to introduce a new proximal double bundle method for unconstrained nonsmooth optimization where the objective function is presented as a difference of two convex dc func

on the convergence of miehle s algorithm for the euclidean multifacility location problem for the euclidean single facility location problem e weiszfeld proposed a simple iterative algorithm in 1937 later it was proved by numerous authors that it is a convergent descent algorithm w miehle extended weiszfeld s algorithm to solve the euclidean multifacility location problem then l m ostresh proved that miehle s algorithm is a descent algorithm recently f rado modified miehle s algorithm and provided several sets of sufficient conditions for the modified algorithm to converge he also indicated that the convergence of miehle s algorithm was an open problem in this paper the relationship between miehle s multifacility location algorithm and weiszfeld s single facility location algorithm is analyzed counterexamples show that miehle s algorithm may converge to nonoptimal points for well structured problems

double shuffle relation for associators it is proved that drinfel 0 d s pentagon equation implies the generalized double shue relation as a corollary an embedding from the grothen

global optimization procedures for the capacitated euclidean and lp distance multifacility location allocation problems in this paper we study the capacitated euclidean andlpdistance location allocation problems there exists no global optimization algorithm that has been developed and tested for this class of problems aside from a total enumeration approach beginning with the euclidean distance problem we design a branch and bound algorithm based on a partitioning of the allocation space that finitely converges to a global optimum for this nonconvex problem for deriving lower bounds at node subproblems in this partial enumeration scheme we employ two types of procedures the first approach computes a lower bound via a projected location space subproblem the second approach derives a significantly enhanced lower bound by using a reformulation linearization technique rlt to transform an equivalent representation of the original nonconvex problem into a higher dimensional linear programming relaxation in addition certain cut set inequalities are generated in the allocation space and objective function based cuts are derived in the location space to further tighten the lower bounding relaxation the rlt procedure is then extended to the generallpdistance problem forp 1 computational experience is provided on a set of test problems to investigate both the projected location space and the rlt lower bounding schemes the results indicate that the proposed global optimization approach using the rlt based scheme offers a promising viable solution procedure in fact among the problems solved for the only two test instances available in the literature for the euclidean distance case we report significantly improved solutions

global convergence of a generalized iterative procedure for the minisum location problem with lp distances this paper considers a general form of the single facility minisum location problem also referred to as the fermat weber problem where distances are measured by an lp norm an iterative solution algorithm is given which generalizes the well known weiszfeld procedure for euclidean distances global convergence of the algorithm is proven for any value of the parameter p in the closed interval 1 2 provided an iterate does not coincide with a singular point of the iteration functions however for p 2 the descent property of the algorithm and as a result global convergence are no longer guaranteed these results generalize the work of kuhn for euclidean p 2 distances

algebraic differential formulas for the shuffle stuffle and duality relations of iterated integrals abstract in this paper we prove certain algebraic identities which correspond to differentiation of the shuffle relation the stuffle relation and the relations which arise from mobius transformations of iterated integrals these formulas provide fundamental and useful tools in the study of iterated integrals on a punctured projective line

an overview of representative problems in location research we present a survey of over 50 representative problems in location research our goal is not to review all variants of different location models or to describe solution results but rather to provide a broad overview of major location problems that have been studied indicating briefly how they are formulated and how they relate to one another we review standard problems such as median center and warehouse location problems as well as less traditional location problems which have emerged in recent years our primary focus is on problems for which operations research type models have been developed most of the problems we review have been formulated as optimization problems

a quadratically convergent method for minimizing a sum of euclidean norms we consider the problem of minimizing a sum of euclidean norms f x sum nolimits i 1 m r i x here the residuals r i x are affine functions fromr n tor 1 n 1 2 m 2 this arises in a number of applications including single and multi facility location problems the functionf is in general not differentiable atx if at least oner i x is zero

the design centering problem as a d c programming problem the following problem is studied given a compact sets inrn and a minkowski functionalp x find the largest positive numberr for which there existsx s such that the set of ally rn satisfyingp y x r is contained ins it is shown that whens is the intersection of a closed convex set and several complementary convex sets sets whose complements are open convex this design centering problem can be reformulated as the minimization of some d c function difference of two convex functions overrn in the case where moreover p x xtax 1 2 witha being a symmetric positive definite matrix a solution method is developed which is based on the reduction of the problem to the global minimization of a concave function over a compact convex set

the convergence of the weiszfeld algorithm abstract we investigate the convergence properties of the weiszfeld procedure when it is applied to the approximated l p norm single facility location problem where p 2 we show that convergence for p 2 can be obtained by introducing a step size factor to the iterative procedure some numerical test results are also given

a two echelon joint continuous discrete location model abstract the problem of locating up to a given number of facilities in continuous euclidean space that can serve as intermediate transshipment points between multiple stakeholders in a supply chain suppliers and customers who are distributed over the same space is considered the first contribution is in considering the multisource weber problem mwp in the presence of both source points and demand points rather than either alone the second contribution is that the selection of intermediate facilities for further discrete analysis is based on a quantitative determination rather than a subjective selection process which is typical of most popular commercial grade mathematical programming lp and ip based location models while the mathematical programming approach benefits from a degree of richness in features and a sense of computational optimization one limitation is that the candidate locations to be evaluated must be specified often without any computational basis for them computational experiments on randomly generated problem instances and real case studies indicate that significant gains can be achieved with relatively little effort by expanding the boundary of analysis to include multiple suppliers and multiple customers in the analysis and design of a supply chain network an alternating location allocation type heuristic method is developed that is easy to implement the third contribution is the development of two different lower bounding procedures that demonstrate the high quality of this obtained heuristic solution

solution of the multisource weber and conditional weberproblems by d c programming d c programming is a recent technique of global optimization that allows the solution of problems whose objective function and constraints can be expressed as differences of convex i e d c functions many such problems arise in continuous location theory the problem first considered is to locate a known number of source facilities to minimize the sum of weighted euclidean distances between a user s fixed location and the source facility closest to the location of each user we also apply d c programming to the solution of the conditional weber problem an extension of the multisource weber problem in which some facilities are assumed to be already established in addition we consider a generalization of weber s problem the facility location problem with limited distances where the effective service distance becomes a constant when the actual distance attains a given value computational results are reported for problems with up to 10 000 users and two new facilities 50 users and three new facilities 1 000 users 20 existing facilities and one new facility or 200 users 10 existing and two new facilities

an optimal method for solving the generalized multi weber problem abstract the single weber problem and the multi weber problem have troubled generations of mathematicians economists and geographers by enumerating all feasible elements of a solution non overlapping convex hulls linear programming can be employed to obtain optimal solutions to the multi weber problem and the generalized multi weber problem in euclidean space first lists of all feasible convex hulls are generated these are the universe of potential elements of an optimal solution second these convex hulls are used to cover the fixed points of the problem a linear programme based on the set covering problem is then used to cover the fixed points minimizing the total weighted distance computational experience is presented this method guarantees the optimality of the solution

new oscillation criteria for second order linear difference equations with positive and negative coefficients the oscillation of second order neutral difference equations with positive and negative coefficients of the form d 2 x n la nx n t p nx n d q nx n s 0 l 1 is investigated we obtain many new results using the comparison between both first order and second order difference equations an example is given to show the strength of the obtained results

a globally convergent algorithm for facility location on a sphere abstract in this paper we study the problem of facility location on a sphere this is a generalization of the planar euclidean facility location problem this problem was first studied by katz and cooper and by drezner and wesolowsky where weiszfeld like algorithms were proposed however convergence has never been proved in this paper we first prove a hull property of the problem i e every global minimizer is in the spherical convex hull of the existing facilities we then study the relationship between the spherical facility location problem and a planar euclidean facility location problem corresponding to each approximate solution to the spherical facility location problem optimality conditions for the spherical facility location problem are established in terms of optimality conditions for the corresponding planar euclidean facility location problem and a gradient algorithm is proposed to solve the spherical facility location problem we prove that our algorithm always converges to a global minimizer of the spherical facility location problem computational results are also given

the complexity of vertex enumeration methods the computational complexity of problems relating to the enumeration of all the vertices of a convex polyhedron defined by linear inequalities is examined several published approaches are evaluated in this light a new algorithm is described and shown to have a better complexity estimate than existing methods empirical evidence supporting the theoretical superiority is presented finally vertex enumeration is discussed when the space containing the polyhedra is of fixed dimension and only the size of the inequality system is permitted to vary

on the convergence of a hyperboloid approximation procedure for the perturbed euclidean multifacility location problem for the euclidean single facility location problem e weiszfeld proposed a simple closed form iterative algorithm in 1937 later numerous authors proved that it is a convergent descent algorithm in 1973 j eyster j white and w wierwille extended weiszfeld s idea and proposed a hyperboloid approximation procedure hap for solving the euclidean multifacility location problem they believed based on considerable computational experience that the hap always converges in 1977 ostresh proved that the hap is a descent algorithm under certain conditions in 1981 morris proved that a variant of the hap always converges however no convergence proof for the original hap has ever been given in this paper we prove that the hap is a descent algorithm and that it always converges to the minimizer of the objective function from any initial point

convergence of the weiszfeld algorithm for weber problems using a generalized distance function this paper considers a generalization of the single and multisource weber problem for the case when the distance function is some power k of the usual ip distance properties of the generalized problem are established and an appropriate generalization of the weiszfeld iterative approach is given a convergence proof is supplied for an approximation to the original problem under certain restrictions on p and k

the euclidean multifacility location problem to solve the euclidean multifacility location problem w miehle extended the weiszfeld algorithm from one to several facilities l m ostresh established that the objective function for the iterates of miehle s algorithm decreases in value but his results do not give a convergence proof in this paper we give a new extension of the weiszfeld algorithm and by generalizing the kuhn ostresh convergence theorem prove separate results for convergence and for optimality both under mild hypotheses

methods for global concave minimization a bibliographic survey the global concave minimization problem is to find the constrained global minimum of a concave function since such a function may have many local minima finding the global minimum is a computationally difficult problem in this bibliographic survey which includes most of the recent papers on constrained global concave minimization we have attempted to briefly summarize the main ideas in each paper these recent papers include those concerned with large scale global concave minimization and bilinear programming

existence for nonoscillatory solutions of higher order nonlinear neutral difference equations abstract in this paper we consider the following forced higher order nonlinear neutral difference equation m x n c n x n k s 1 u p s n f s x n r s q n n n 0 where m u 1 k 0 and rs 0 are integers cn p s n s 1 2 u and qn are sequence of real numbers and f s c r r s 1 2 u by using krasnoselskii s fixed point theorem and some new techniques we obtain sufficient conditions for the existence of nonoscillatory solutions for general p s n s 1 2 u and qn which means that we allow oscillatory p s n s 1 2 u and qn

the minisum and minimax location problems revisited the minisum minimax problem consists of locating a single facility in the plane with the aim of minimizing the sum of the weighted distances the maximum weighted distance to m given points we present two solution methods for generalized versions of these problems in which i location is restricted to the union of a finite number of convex polygons ii distances are approximated by norms that may differ with the given points and iii transportation costs are increasing and continuous functions of distance computational experience is described

locating objects in the plane using global optimization techniques we address the problem of locating objects in the plane such as segments arcs of circumferences arbitrary convex sets their complements or their boundaries given a set of points we seek the rotation and translation for such an object optimizing a very general performance measure which includes as a particular case the classical objectives in semi obnoxious facility location in general the above mentioned model yields a global optimization problem whose resolution is dealt with using difference of convex dc techniques such as outer approximation or branch and bound

a note on optimality conditions for the euclidean multifacility location problem thekey problem of the euclidean multifacility location emfl problem is to decide whether a givendead point is optimal if it is not optimal we wish to compute a descent direction this paper extends the optimality conditions of calamai and conn and overton to the case when the rows of the active constraints matrix are linearly dependent we show that linear dependence occurs wheneverg the graph of the coinciding facilities has a cycle in this case the key problem is formulated as a linear least squares problem with bounds on the euclidean norms of certain subvectors

concave minimization via collapsing polytopes we present a procedure for globally minimizing a concave function over a bounded polytope by successively minimizing the function over polytopes containing the feasible region and collapsing to the feasible region the initial containing polytope is a simplex and at the kth iteration the procedure chooses the most promising vertex of the current containing polytope to refine the approximation the method generates a tree whose ultimate terminal nodes coincide with the vertices of the feasible region and accounts for the vertices of the containing polytopes

iterated integrals on p1 0 1 z and a class of relations among multiple zeta values abstract in this paper we consider iterated integrals on p 1 0 1 z and define a class of q linear relations among them which arises from the differential structure of the iterated integrals with respect to z we then define a new class of q linear relations among the multiple zeta values by taking their limits of z 1 which we call confluence relations i e the relations obtained by the confluence of two punctured points one of the significance of the confluence relations is that it gives a rich family and seems to exhaust all the linear relations among the multiple zeta values as a good reason for this we show that confluence relations imply both the regularized double shuffle relations and the duality relations

oscillations of higher order linear difference equations abstract consider the higher order linear difference equation where m 1 is an odd integer and p n is a sequence of nonnegative real numbers we obtain several new sufficient conditions for the oscillation of all solutions of equation examples which dwell upon the importance of our results are also included

on the complexity of some common geometric location problems given n demand points in the plane the p center problem is to find p supply points anywhere in the plane so as to minimize the maximum distance from a demand point to its respective nearest supply point the p median problem is to minimize the sum of distances from demand points to their respective nearest supply points we prove that the p center and the p median problems relative to both the euclidean and the rectilinear metrics are np hard in fact we prove that it is np hard even to approximate the p center problems sufficiently closely the reductions are from 3 satisfiability

motivic multiple zeta values and superstring amplitudes the structure of tree level open and closed superstring amplitudes is analyzed for the open superstring amplitude we find a striking and elegant form which allows one to disentangle its expansion into several contributions accounting for different classes of multiple zeta values this form is bolstered by the decomposition of motivic multiple zeta values i e the latter encapsulate the expansion of the superstring amplitude moreover a morphism induced by the coproduct maps the expansion onto a non commutative hopf algebra this map represents a generalization of the symbol of a transcendental function in terms of elements of this hopf algebra the expansion assumes a very simple and symmetric form which carries all the relevant information equipped with these results we can also cast the closed superstring amplitude into a very elegant form

a smoothing newton method for minimizing a sum of euclidean norms we consider the problem of minimizing a sum of euclidean norms f x sum i 1 m b i a i tx this problem is a nonsmooth problem because f is not differentiable at a point x when one of the norms is zero in this paper we present a smoothing newton method for this problem by applying the smoothing newton method proposed by qi sun and zhou math programming 87 2000 pp 1 35 directly to a system of strongly semismooth equations derived from primal and dual feasibility and a complementarity condition this method is globally and quadratically convergent as applications to this problem smoothing newton methods are presented for the euclidean facilities location problem and the steiner minimal tree problem under a given topology preliminary numerical results indicate that this method is extremely promising

mixed tate motives over z we prove that the category of mixed tate motives over z is spanned by the motivic fundamental group of p 1 minus three points we prove a conjecture by m hoffman which states that every multiple zeta value is a q linear combination of n 1 n r where n i 2 3