orientational ordering in two dimensional systems with long range interaction a variational method for calculating gaussian fluctuations of two dimensional 2d moment orientations in lattice systems with arbitrary anisotropic interaction is proposed it is asymptotically exact for very low temperatures and gives a good extrapolation for a certain low temperature range general relationships are obtained that enable one to represent from a unified standpoint and in some cases to refine the known results on 1 the existence of short range order in a degenerate 2d system of moments with isotropic short range interaction 2 long range order stabilization by weak local potentials and dipole interactions 3 the existence of long range order in a 2d system with interaction of the type v r r if 2 4 for systems with anisotropic long range dipole interaction of moments on arbitrary 2d lattices except square ones for which a short range phase takes place the existence of long range order is proved lower limits for temperatures of transitions into phases with long range order are evaluated the low values of the transition temperatures in the dipole systems can be explained due to the small values of interchain interaction in comparison with interchain ones the applicability area for the method used is discussed

sets of efficient points in a normed space abstract sets of efficient points in a normed space with respect to the distances to the points of a given compact set are geometrically characterized hull and closure properties are obtained these results are relevant to geometry of normed spaces and are mostly useful in the context of location theory

continuous location theory under majority rule this paper studies from a theoretical point of view a single facility location problem in a normed space it is assumed that the facility has a finite or an infinite number of users and the location is selected under majority rule where between two feasible locations each user prefers the closer one in general normed space three concepts of solutions condorcet points simpson points and plurality points are geometrically described in a two dimensional space practical methods of construction are given as well as the relationships between condorcet points plurality points and medians in the same framework when the norm is polyhedral or the number of users is finite it is proved that each condorcet point is a solution to the associated fermat weber problem

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

restricted center problems under polyhedral gauges abstract given a finite set ex ex 1 ex 2 ex m in the plane and a forbidden region r we want to find a point x r such that max m 1 m w m m x ex m is minimized this is a variant of the well known continuous center problem where we measure the distance with polyhedral gauges the unit ball of a polyhedral gauge may be any convex polyhedron containing the origin in its interior this large class of distance functions allows very general practical settings such as asymmetry to be modeled each ex m is allowed to have its own gauge m and the forbidden region r enables us to include negative information which means defining the infeasible region explicitly instead of the feasible region in the model efficient algorithms and structural properties for this non convex optimization problem based on combinatorial and geometrical methods are presented a classification scheme for location problems is also introduced

on the uniqueness of optimal solutions in continuous location theory abstract this paper deals with the uniqueness of an optimal solution in general continuous single facility minisum and minimax location problems we define the concept of an s norm and obtain general conditions which guarantee the existence of a unique optimal location some consequences for the uniqueness of optimal locations in multi facility location problems are discussed

a geometric characterisation of the quadratic min power centre abstract for a given set of nodes in the plane the min power centre is a point such that the cost of the star centred at this point and spanning all nodes is minimised the cost of the star is defined as the sum of the costs of its nodes where the cost of a node is an increasing function of the length of its longest incident edge the min power centre problem provides a model for optimally locating a cluster head amongst a set of radio transmitters however the problem can also be formulated within a bicriteria location model involving the 1 centre and a generalised fermat weber point making it suitable for a variety of facility location problems we use farthest point voronoi diagrams and delaunay triangulations to provide a complete geometric description of the min power centre of a finite set of nodes in the euclidean plane when cost is a quadratic function this leads to a new linear time algorithm for its construction when the convex hull of the nodes is given we also provide an upper bound for the performance of the centroid as an approximation to the quadratic min power centre finally we briefly describe the relationship between solutions under quadratic cost and solutions under more general cost functions

on efficient solutions to multiple objective mathematical programs this note develops properties of quasi efficient solutions and explores interrelationships to the classical concept of efficiency in particular a point is a quasi efficient solution to a multiple objective mathematical program if and only if it is an optimal solution to a scalar maximum problem for some set of nonnegative weights on the objectives this result is then used to characterize the set of quasi efficient solutions as the union of efficient solutions to a multiple objective problem over all nonempty subsets of the objectives

eigenvalue bounds in the gaps of schrodinger operators and jacobi matrices abstract we consider c a b where a is selfadjoint with a gap a b in its spectrum and b is relatively compact we prove a general result allowing b of indefinite sign and apply it to obtain a v d 2 bound for perturbations of suitable periodic schrodinger operators and a not quite lieb thirring bound for perturbations of algebro geometric almost periodic jacobi matrices

minmax regret location allocation problem on a network under uncertainty abstract we consider a robust location allocation problem with uncertainty in demand coefficients specifically for each demand point only an interval estimate of its demand is known and we consider the problem of determining where to locate a new service when a given fraction of these demand points must be served by the utility the optimal solution of this problem is determined by the minimax regret location i e the point that minimizes the worst case loss in the objective function that may occur because a decision is made without knowing which state of nature will take place for the case where the demand points are vertices of a network we show that the robust location allocation problem can be solved in o min p n p n 3 m time where n is the number of demand points p p n is the fixed number of demand points that must be served by the new service and m is the number of edges of the network

a geometrical solution for quadratic bicriteria location models this paper deals with a single facility unweighted quadratic bicriteria location problem defined in the continuous space first we describe an analytical method for constructing the set of pareto optimal locations for this problem based on the farthest point voronoi diagram then we present an analytical method for finding the solution to the scalarized location problems based on the pareto optimal locations the analytical results enable planners to carry out some sensitivity analyses since both methods are based entirely on geometrical analysis they can be understood intuitively

generalized goal programming polynomial methods and applications in this paper we address a general goal programming problem with linear objectives convex constraints and an arbitrary componentwise nondecreasing norm to aggregate deviations with respect to targets in particular classical linear goal programming problems as well as several models in location and regression analysis are modeled within this framework

finite dominating sets for network location problems a research theme involving location on networks since its inception has been the identification of a finite dominating set fds or a finite set of points to which an optimal solution must belong we attempt to unify and generalize results of this sort we survey the literature and then prove some theorems that subsume most previous results and that are at the same time more general than previous results the paper is aimed primarily at investigators who wish to know whether an fds exists for a specific problem

approximate formulas of average distances associated with regions and their applications to location problems this study is concerned with the problem of measuring average distances between two points in two different coplanar regions the objectives are 1 to derive the approximated average distances associated with circular regions and to check their accuracy and 2 to apply these approximated distances to location problems results show that the simple approximate formulas are accurate and useful the approximated average distances can be applied to the analyses of varied kinds of movement phenomena in cities

minmax distance approximation and separation problems geometrical properties a center hyperplane in the d dimensional space minimizes the maximum of its distances from a finite set of points a with respect to possibly different gauges in this note it is shown that a center hyperplane exists which is at equal maximum distance from at least d 1 points of a moreover the projections of the points among these which lie above the center hyperplane cannot be separated by another hyperplane from the projections of those that are below it when all gauges involved are smooth all center hyperplanes satisfy these properties this geometric property allows us to improve and generalize previously existing results which were only known for the case in which all distances are measured using a common norm the results also extend to the constrained case where for some points it is prespecified on which side of the hyperplane above below or on they must lie in this case the number of points lying on the hyperplane plus those at maximum distance is at least d 1 it follows that solving such global optimization problems reduces to inspecting a finite set of candidate solutions extensions of these results to a separation problem are outlined

the general one center location problem the general one center location problem deals with the location of a point in a real normed space x in order to minimize an objective function g which depends on the distances to a finite number of centers and on initial costs the function g is defined by g x c1 w1 x a1 cn wn x an where a1 an are n given points in x w1 wn are positive numbers c1 cn are nonnegative initial costs and is a monotone norm on n a geometrical description of the set of optimal solutions to the problem minx x g x is provided the peculiar role of the minisum problem where is the l1 norm is emphasized and the minimax problem where is the lx norm is used to illustrate the general geometrical description

variation of discrete spectra a formula see 1 below estimating collectively the variation of eigenvalues of a symmetric matrix under a perturbation is extended to the case of discrete eigenvalues of a selfadjoint operator in hilbert space under the assumption that the perturbation is compact for this purpose the notion of an extended enumeration of discrete eigenvalues is introduced

some properties of location problems with block and round norms the point objective problem and the weber problem are two well known formulations for locating a new facility with respect to a set of fixed facilities when locations are represented as points on a plane the point objective problem is a multiple objective formulation of minimizing the distance from a variable point to each of the fixed points similarly the weber problem is a single objective formulation of minimizing the sum of transportation costs between the variable point and the fixed points where transportation cost is a function of distance generalizing solution properties for these problems from distance measures given by the euclidean rectilinear ip and one infinity norms this paper develops solution properties under the broad classes of distance measures given by block and round norms for the point objective problem we show that i the efficient set for all round norms is the convex hull of the set of fixed points and ii the efficient set under a block norm tends to the convex hull for a sequence of block norms approaching a round norm for the weber problem we prove that i an optimal location for any block norm may be found in a finite set of intersection points belonging to the convex hull and ii this set tends to the convex hull for a sequence of block norms approaching a round norm finally we use these results to propose a synthesis of some of the main properties in continuous and network location theory

the weber problem with regional demand abstract this paper is devoted to the study of the regional weber problem an extension of the weber problem which allows the demand not be concentrated onto a finite set of points the most serious drawback of this formulation from a resolution viewpoint is the high computational cost involved in the evaluation of the objective function a new approach is proposed which requires a low amount of computation and where it is possible to control the error on the approximation this approximation suggests a new methodology to solve the problem this methodology is compared with the existing ones showing its relevance from a practical point of view

simpson points in planar problems with locational constraints the polyhedral gauge case in this paper we address the problem of finding simpson points in planar models with locational constraints when distances are measured by polyhedral gauges making use of the results we have stated in a previous paper we show here the existence of a finite set of points in the plane independent of the weights associated with the users that contains at least a simpson point connection between simpson points as a result of a voting process and weber points as the outcome of a planning process are explored it is shown by means of an example that the existing relations for the unconstrained case are no longer true when locational constraints are imposed in order to reconcile both voting and planning processes a biobjective problem is described for which we construct a finite dominating set

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

on farthest voronoi cells abstract given an arbitrary set t in the euclidean space r n whose elements are called sites and a particular site s the farthest voronoi cell of s denoted by f t s consists of all points which are farther from s than from any other site in this paper we study farthest voronoi cells and diagrams corresponding to arbitrary possibly infinite sets more in particular we characterize for a given arbitrary set t those s t such that f t s is nonempty and study the geometrical properties of f t s in that case we also characterize those sets t whose farthest voronoi diagrams are tesselations of the euclidean space and those sets that can be written as f t s for some t r n and some s t

finite size scaling for the mean spherical model with inverse power law interaction a new analytical technique based on integral transformations with mittag leffler type kernels is used to derive the finite size scaling function for the free energy per particle of the mean spherical model with inverse power law asymptotics of the interaction potential the asymptotic formation of the singularities in the specific heat and magnetic susceptibility at the bulk critical point is studied

group nearest neighbor queries in the l 1 plane let p be a set of n points in the plane the k nearest neighbor abbreviated as k nn query problem is to preprocess p into a data structure that quickly reports k closest points in p for a query point q this paper addresses a generalization of the k nn query problem to a query set q of points namely the group k nearest neighbor query problem in the l 1 plane more precisely a query is assigned with a set q of at most m points and a positive integer k with k n and the distance between a point p of p and a query set q is defined as the sum of l 1 distances from p to all q q the maximum number m of query points q is assumed to be known in advance and to be at most n in this paper we propose two algorithms one based on the range tree and the other based on a data structure for segment dragging queries and obtain the following complexity bounds 1 a group k nn query can be handled in o t min log n k m 2 log log n log m time after preprocessing p using o m 2 n log 2 n space where t min min k m m 2 or 2 a group k nn query can be handled in o k m log 2 n m 2 log n log m time after preprocessing p using o m 2 n space where 0 is an arbitrarily small constant we also show that our approach can be applied to the weighted group k nearest neighbor query problem and the group k farthest neighbor query problem

a blaschke type condition for analytic functions on finitely connected domains applications to complex perturbations of a finite band selfadjoint operator abstract this is a sequel of the article by borichev golinskii and kupin 2009 1 where the authors obtain blaschke type conditions for special classes of analytic functions in the unit disk which satisfy certain growth hypotheses these results were applied to get lieb thirring inequalities for complex compact perturbations of a selfadjoint operator with a simply connected resolvent set the first result of the present paper is an appropriate local version of the blaschke type condition from borichev et al 2009 1 we apply it to obtain a similar condition for an analytic function in a finitely connected domain of a special type such condition is by and large the same as a lieb thirring type inequality for complex compact perturbations of a selfadjoint operator with a finite band spectrum a particular case of this result is the lieb thirring inequality for a selfadjoint perturbation of the schatten class of a periodic or finite band jacobi matrix

the effect of axis rotation on distance estimation abstract the objective of this paper is to evaluate the usefulness of the weighted l p norm as a distance measure when axis rotation is included to obtain the best fit we compare the weighted l p and euclidean norms in an empirical study of 18 major country and city road networks a substantial improvement in accuracy is obtained with the weighted l p norm for networks with a significant component of rectangular travel we show that the weighted l p norm and by inference other multi parameter distance estimators are justified in practice provided the reference axes are oriented properly to model the directional bias inherent in the network

weber problems with alternative transportation systems abstract in this paper we address a planar p facility location problem where together with a metric induced by a gauge there exists a series of rapid transit lines which can be used as alternative transportation system to reduce the total transportation cost the location problem is reduced to solving a finite number of multi weber problems from which localization results are obtained in particular it is shown that if the gauge in use is polyhedral then the problem is reduced to finding a p median

perturbations of orthogonal polynomials with periodic recursion coefficients the results of denisov rakhmanov szeg shohat nevai and killip simon are extended from asymptotically constant orthogonal polynomials on the real line oprl and unit circle opuc to asymptotically periodic oprl and opuc the key tool is a characterization of the isospectral torus that is well adapted to the study of perturbations

a heuristic algorithm for constrained multi source weber problem the variational inequality approach abstract for solving the well known multi source weber problem mwp each iteration of the heuristic alternate location allocation algorithm consists of a location phase and an allocation phase the task of the location phase is to solve finitely many single source weber problems swp which are reduced by the heuristic of nearest center reclassification for the customers in the previous allocation phase this paper considers the more general and practical case the mwp with constraints cmwp in particular a variational inequality approach is contributed to solving the involved constrained swp cswp and thus a new heuristic algorithm for cmwp is presented the involved cswp in the location phases are reformulated into some linear variational inequalities whose special structures lead to a new projection contraction pc method global convergence of the pc method is proved under mild assumptions the new heuristic algorithm using the pc method in the location phases approaches to the heuristic solution of cmwp efficiently which is verified by the preliminary numerical results reported in this paper

ground state of an infinite two dimensional system of dipoles on a lattice with arbitrary rhombicity angle abstract a class of possible periodic orientational configurations of a system of dipole moments located at the sites of an infinite flat rhombic lattice with an arbitrary rhombicity angle is studied by using the luttinger and tisza method in the framework of the method it is obtained that for 80 the ground state is ferromagnetic at 60 being continuously degenerate in direction for 80 90 the ground state is antiferromagnetic at 90 being also continuously degenerate with respect to one parameter it is shown that the restriction of the interaction range leads to a change in the type of the ground state at 60 this takes place between the third and the fourth coordination spheres and at 30 at a distance of about 500 lattice constants from comparison with known results of numerical experiments on finite systems it is established that the type of the ground state of a finite and an infinite system may be essentially different

gbsss the generalized big square small square method for planar single facility location abstract big square small square is a geometrical branch and bound algorithm devised by p hansen et al or 33 1251 1265 for the solution of constrained planar minisum single facility location problems with l p norms and continuous non decreasing costfunctions the method basically works by splitting the studied planar region into squares and either rejecting or further processing these squares by the evaluation of a lower bound we present a modified version of this algorithm aimed at correcting a small failure to converge accelerating the calculations minimising the information to be stored and most importantly determining a region of near optimality furthermore the method is applicable to any planar single facility problem with distances measured by mixed norms and as an objective any continuous function of the distances this includes nearly all the models which have been proposed in the literature

local convexity results in a generalized fermat weber problem abstract a generalized form of the fermat weber problem requires finding a point in r n to minimize a sum of nondecreasing functions of distances to m given points in this paper local convexity properties are investigated for the generalized problem sufficient conditions are derived which guarantee that the hessian matrix of the objective function will be positive definite the analysis also reveals that weiszfeld type iterative algorithms may have sublinear convergence rates since the hessian may only be positive semidefinite at a local minimum

simpson points in planar problems with locational constraints the round norm case the simpson set is one of the most popular solution concepts in location models with voting in this paper we address the problem of finding such a set for planar models with locational constraints when the metric in use is induced by a round norm after formulating the problem as a mathematical program we first propose a general algorithm that enables the evaluation of the corresponding objective function then we show how to solve the problem by proving the existence of a finite dominating set of locations the best of which can be found by inspection using the evaluation algorithm proposed

the multicriteria p facility median location problem on networks abstract in this paper we discuss the multicriteria p facility median location problem on networks with positive and negative weights we assume that the demand is located at the nodes and can be different for each criterion under consideration the goal is to obtain the set of pareto optimal locations in the graph and the corresponding set of non dominated objective values to that end we first characterize the linearity domains of the distance functions on the graph and compute the image of each linearity domain in the objective space the lower envelope of a transformation of all these images then gives us the set of all non dominated points in the objective space and its preimage corresponds to the set of all pareto optimal solutions on the graph for the bicriteria 2 facility case we present a low order polynomial time algorithm also for the general case we propose an efficient algorithm which is polynomial if the number of facilities and criteria is fixed

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

location and shape of a rectangular facility in ℝn convexity properties convexity properties in this paper we address a generalization of the weber problem in which we seek for the center and the shape of a rectangle the facility minimizing the average distance to a given set the demand set which is not assumed to be finite some theoretical properties of the average distance are studied and an expression for its gradient involving solely expected distances to rectangles is obtained this enables the resolution of the problem by standard optimization techniques

efficiency in constrained continuous location abstract we present a geometrical characterization of the efficient weakly efficient and strictly efficient points for multi objective location problems in presence of convex constraints and when distances are measured by an arbitrary norm these results established for a compact set of demand points generalize similar characterizations previously obtained for uncontrained problems they are used to show that in planar problems the set of constrained weakly efficient points always coincides with the closest projection of the set of unconstrained weakly efficient points onto the feasible set this projection property which are known previously only for strictly convex norms allows to easily construct all the weakly efficient points and provides a useful localization property for efficient and strictly efficient points

fibonacci heaps and their uses in improved network optimization algorithms in this paper we develop a new data structure for implementing heaps priority queues our structure fibonacci heaps abbreviated f heaps extends the binomial queues proposed by vuillemin and studied further by brown f heaps support arbitrary deletion from an n item heap in o log n amortized time and all other standard heap operations in o 1 amortized time using f heaps we are able to obtain improved running times for several network optimization algorithms in particular we obtain the following worst case bounds where n is the number of vertices and m the number of edges in the problem graph o n log n m for the single source shortest path problem with nonnegative edge lengths improved from o m log m n 2 n o n 2 log n nm for the all pairs shortest path problem improved from o nm log m n 2 n o n 2 log n nm for the assignment problem weighted bipartite matching improved from o nm log m n 2 n o m m n for the minimum spanning tree problem improved from o m log log m n 2 n where m n min i u log i n m n note that m n log n if m n of these results the improved bound for minimum spanning trees is the most striking although all the results give asymptotic improvements for graphs of appropriate densities

on weighting two criteria with a parameter in combinatorial optimization problems abstract two criteria in a combinatorial problem are often combined in a weighted sum objective using a weighting parameter between 0 and 1 for special problem types e g when one of the criteria is a bottleneck value efficient algorithms are known that solve for a given value of the weighting parameter we transform the underlying enumeration method into a parametric algorithm solving simultaneously for all values of the weighting parameter efficient implementations are presented for combinatorial problems with criteria as balanced optimization min sum with min max and min sum with balanced optimization considering the spanning tree the linear assignment and the single machine scheduling problem further the new algorithmic scheme can easily incorporate a trade off of the criteria by means of penalty functions again without consequences for the algorithm and its complexity order

finite size scaling for systems with long range interactions abstract the present review is devoted to the fundamental problems of finite size scaling due to the presence of long range interactions the attention is focused on the precise formulation of critical finite size scaling in the case that the bulk correlation length is identically infinite the hypotheses based on the notion of a finite size scaling length are formulated in a way which treats the cases of short and long range interactions on equal grounds as well as the standard and modified above the upper critical dimensionality finite size scaling the general conjectures are tested on the exactly solvable mean spherical model with power law interaction special attention is paid to the adequate mathematical techniques the formulation of finite size scaling at first order phase transitions is also discussed since the finite size scaling length enters into the definition of the relevant scaled external field variable provided the system has d 0 infinite dimensions the review closes with a summary and short discussion of some open problems

a primal dual algorithm for the fermat weber problem involving mixed gauges we give a new algorithm for solving the fermat weber location problem involving mixed gauges this algorithm which is derived from the partial inverse method developed by j e spingarn simultaneously generates two sequences globally converging to a primal and a dual solution respectively in addition the updating formulae are very simple a stopping rule can be defined though the method is not dual feasible and the entire set of optimal locations can be obtained from the dual solution by making use of optimality conditions when polyhedral gauges are used we show that the algorithm terminates in a finite number of steps provided that the set of optimal locations has nonepty interior and a counterexample to finite termination is given in a case where this property is violated finally numerical results are reported and we discuss possible extensions of these results

error bounds for the approximative solution of restricted planar location problems abstract facility location problems in the plane play an important role in mathematical programming when looking for new locations in modeling real world problems we are often confronted with forbidden regions that are not feasible for the placement of new locations furthermore these forbidden regions may have complicated shapes even if we require them to be convex it may be more useful or even necessary to use approximations of such forbidden regions when trying to solve location problems in this paper we develop error bounds for the approximative solution of restricted planar location problems using the so called sandwich algorithm the number of approximation steps required to achieve a specified error bound is analyzed as examples of these approximation schemes we discuss round norms and polyhedral norms computational tests are also included

a variation of the assignment problem abstract in this paper a meaningful variant of the cost minimizing assignment problem with the objective of minimizing the total cost of assignment plus an additional supervisory cost which depends on the total time of completion of the project is formulated an efficient method of finding an optimal solution to such a problem is presented with a numerical example to illustrate the same

on the structure of the solution set for the single facility location problem with average distances this paper analyzes continuous single facility location problems where the demand is randomly defined by a given probability distribution for these types of problems that deal with the minimization of average distances we obtain geometrical characterizations of the entire set of optimal solutions for the important case of total polyhedrality on the plane we derive efficient algorithms with polynomially bounded complexity we also develop a discretization scheme that provides varepsilon approximate solutions of the original problem by solving simpler location problems with points as demand facilities

location and shape of a rectangular facility in r n convexity properties in this paper we address a generalization of the weber problem in which we seek for the center and the shape of a rectangle the facility minimizing the average distance to a given set the demand set which is not assumed to be finite some theoretical properties of the average distance are studied and an expression for its gradient involving solely expected distances to rectangles is obtained this enables the resolution of the problem by standard optimization techniques

continuous multifacility ordered median location problems abstract in this paper we propose a general methodology for solving a broad class of continuous multifacility location problems in any dimension and with l norms proposing two different methodologies 1 by a new second order cone mixed integer programming formulation and 2 by formulating a sequence of semidefinite programs that converges to the solution of the problem each of these relaxed problems solvable with sdp solvers in polynomial time

on pareto optima the fermat weber problem and polyhedral gauges this paper deals with multiobjective programming in which the objective functions are nonsymmetric distances derived from different gauges to the points of a fixed finite subset of n it emphasizes the case in which the gauges are polyhedral in this framework the following result is known if the gauges are polyhedral then each pareto optimum is the solution to a fermat weber problem with strictly positive coefficients we give a new proof of this result and we show that it is useful in finding the whole set of efficient points of a location problem with polyhedral gauges also we characterize polyhedral gauges in terms of a property of their subdifferential

hull properties in location problems abstract some properties of the solution set for single and multifacility continuous location problems with l p distances are given a set reduction algorithm is developed for problems in k dimensional space having rectangular distances

geometrical properties of the fermat weber problem abstract this paper is devoted to the fermat weber problem with mixed gauges in order to take into account nonsymmetric distances a geometrical description of the set of solutions is obtained then the concept of metric hull is defined general sufficient conditions are given such that the metric hull of existing facilities intersects the set of optimal locations particularly it is shown that an optimal location can be found in the metric hull in dimension two whatever the gauges are and in dimension n with the rectilinear norm the connection between metric hull convex hull and octagonal hull is studied

optimal expected distance separating halfspace one recently proposed criterion to separate two data sets in discriminant analysis is to use a hyperplane which minimizes the sum of distances to it from all the misclassified data points here all distances are supposed to be measured by way of some fixed norm while misclassification means lying in the wrong halfspace in this paper we study the problem of determining such an optimal halfspace when points are distributed according to an arbitrary random vector x in rd in the unconstrained case in dimension d we prove that any optimal separating halfspace always balances the misclassified points moreover under polyhedrality assumptions on the support of x there always exists an optimal separating halfspace passing through d affinely independent points these results extend in a natural way when different norms or a fixed gauge are used to measure distances and we allow constraints modeling that certain points are forced to be correctly classified

a polynomial algorithm for the multicriteria cent dian location problem given a network with several weights per node and several lengths per edge we address the problem of locating a facility on the network such that the convex combinations of the center and median objective functions are minimized since we consider several weights and several lengths various objective functions should be minimized and hence we have to solve a multicriteria cent dian location problem a polynomial algorithm to characterize the efficient location point set on the network is developed furthermore this model can generalize other problems such as the multicriteria center problem and the multicriteria median problem computing time results on random planar networks considering different combinations of weights and lengths are reported which strengthen the polynomial complexity of the procedure

an approach to location models involving sets as existing facilities in this paper we deal with single facility location problems in a general normed space in which the existing facilities are represented by convex sets of points the criterion to be satisfied by the service facility is the minimization of an increasing convex function of the distances from the service facility to the closest point of each demand set we obtain a geometrical characterization of the set of optimal solutions for this problem two remarkable cases the classical weber problem and the minimax problem with demand sets are studied as particular instances of our problem finally for the planar polyhedral case we give an algorithm to find the solution set of the considered problems

solving combinatorial problems with combined min max min sum objective and applications the purpose of this paper is to introduce and study a new class of combinatorial optimization problems in which the objective function is the algebraic sum of a bottleneck cost function min max and a linear cost function min sum general algorithms for solving such problems are described and general complexity results are derived a number of examples of application involving matchings paths and cutsets matroid bases and matroid intersection problems are examined and the general complexity results are specialized to each of them the interest of these various problems comes in particular from their strong relation to other important and difficult combinatorial problems such as weighted edge coloring of a graph optimum weighted covering with matroid bases optimum weighted partitioning with matroid intersections etc another important area of application of the algorithms given in the paper is bicriterion analysis involving a min max criterion and a min sum one

linear time algorithms for two and three variable linear programs o n time algorithms for linear programming problems with two or three variables and n constraints are described the approach uses convexity dominance of linear functions and linear time median finding algorithms the algorithms improve the previously known best bounds of o n log n time for both of these problems

optimal locations and inner products abstract in a normed space x we consider objective functions which depend on the distances between a variable point and the points of certain finite sets a a point where such a function attains its minimum on x is generically called an optimal location in this paper we obtain characterizations of inner product spaces with properties connecting optimal locations and the convex hull of a or barycenters of points of a with well chosen weights we thus generalize several classical results about characterization of inner product spaces

a fractional model for locating semi desirable facilities on networks abstract in this paper we address the problem of locating a series of facilities on a network maximizing the average distance to population centers assumed to be distributed in the plane per unit transportation cost a function of the network distances to users a finite dominating set is constructed allowing the resolution of the problem by standard integer programming techniques we also discuss some extensions of the model including in particular the weber problem with attraction and repulsion in networks for which e dominating sets are derived

a discretizing algorithm for location problems abstract a new and simple methodology is proposed to solve both constrained and unconstrained planar continuous single facility location problems as particular instances the classical location problems with mixed gauges can be solved theoretical convergence is proved and numerical examples are given showing a fast convergence in a small number of steps

continuous location under the effect of refraction in this paper we address the problem of locating a new facility on a d dimensional space when the distance measure ell p lp or polyhedral norms is different at each one of the sides of a given hyperplane mathcal h h we relate this problem with the physical phenomenon of refraction and extend it to any finite dimensional space and different distances at each one of the sides of any hyperplane an application to this problem is the location of a facility within or outside an urban area where different distance measures must be used we provide a new second order cone programming formulation based on the ell p lp norm representation given in blanco et al comput optim appl 58 3 563 595 2014 that allows to solve the problem in any finite dimensional space with second order cone or semidefinite programming tools we also extend the problem to the case where the hyperplane is considered as a rapid transit media a different third norm is also considered over mathcal h h that allows the demand to travel whenever it is convenient through mathcal h h to reach the new facility extensive computational experiments run in gurobi are reported in order to show the effectiveness of the approach some extensions of these models are also presented

optimality conditions in preference based spanning tree problems abstract spanning tree problems defined in a preference based environment are addressed in this approach optimality conditions for the minimum weight spanning tree problem mst are generalized for use with other more general preference orders the main goal of this paper is to determine which properties of the preference relations are sufficient to assure that the set of most preferred trees is the set of spanning trees verifying the optimality conditions finally algorithms for the construction of the set of spanning trees fulfilling the optimality conditions are designed improving the methods in previous papers

classical dipoles on a finite triangular lattice the spherical model approximation abstract in the framework of the spherical model it is shown that a phase transition at finite temperature to a ferromagnetic phase in a two dimensional system with dipole dipole and exchange interactions on a lattice with arbitrary rhombicity angle exists if the ferromagnetic ground state is not continuously degenerate in spin direction when the degeneracy exists the system exhibits essential singularities in the thermodynamic functions at t 0 finite size effects in such systems are investigated in the example of the triangular latice under periodic boundary conditions the existence of only one relevant variable determining the crossover from infinite to finite size behavior is found as t 0 is approached despite that there are two characteristic lengths determining the scale of variation of the correlation function the validity of the general finite size scalling relations in terms of this variable is established

localization in multifacility location theory abstract we present two localization theorems for a multifacility location problem involving linear costs and a single norm in a two dimensional space it is shown that optimal locations for all the new facilities can be found in the metric hull of existing facilities whatever the norm is in the polyhedral norm case we study if only finite set of intersections points which belong to the metric hull needs to be considered the result is obtained in particular for the l 1 norm and a counter example is given for a general polyhedral norm