A geometric perspective of the Weiszfeld algorithm for solving the Fermat−Weber problem

The Fermat−Weber problem is a classical location problem that has the Weiszfeld algorithm as its main iterative solution method. This article presents a geometric interpretation of its local convergence for the particular case of three points, with the solution constrained to be an interior point, which is fundamental to the present geometric interpretation. This constraint, on the other hand, implies that the weights associated to each point must obey triangle inequalities. The eigenvalues analysis is developed considering that all weights have the same value, which simplifies calculation and explanation, but the generalization of this analysis is straightforward, as commented in the text. Step-size scaling is also considered for accelerating the convergence rate. The accompanying eigenvalues analysis determines step-size multiplier ranges that ensure convergence. Moreover, the eigenvalues depend on a parameter that is computed based on the sample points configuration

Feedback algorithm for switch location : analysis of complexity and application to network design

An accelerated feedback algorithm to solve the single-facility minisum problem is studied with application to designing networks with the star topology. The algorithm, in which the acceleration with respect to the Weiszfeld procedure is achieved by multiplying the current Weiszfeld iterate by an accelerating feedback factor, is shown to converge faster than the accelerating procedures available in the literature. Singularities encountered in the algorithm are discussed in detail. A simple practical exception handling subroutine is developed. Several applications of the algorithm to designing computer networks with the star topology are demonstrated. Applications of the algorithm as a subroutine for multi-switch location problems are considered. Various engineering aspects involved in acquiring and processing coordinates for geographic locations are discussed. A complete algorithm in pseudocode along with the source code listing in Mathematica 4.1 is presented

A D.C. Algorithm via Convex Analysis Approach for Solving a Location Problem Involving Sets

We study a location problem that involves a weighted sum of distances to closed convex sets. As several of the weights might be negative, traditional solution methods of convex optimization are not applicable. After obtaining some existence theorems, we introduce a simple, but effective, algorithm for solving the problem. Our method is based on the Pham Dinh - Le Thi algorithm for d.c. programming and a generalized version of the Weiszfeld algorithm, which works well for convex location problems

Revisiting several problems and algorithms in continuous location with lp norms

This paper addresses the general continuous single facility location problems in finite dimension spaces under possibly different ℓp norms in the demand points. We analyze the difficulty of this family of problems and revisit convergence properties of some well-known algorithms. The ultimate goal is to provide a common approach to solve the family of continuous ℓp ordered median location problems in dimension d (including of course the ℓp minisum or Fermat-Weber location problem for any p ≥ 1). We prove that this approach has a polynomial worse case complexity for monotone lambda weights and can be also applied to constrained and even non-convex problems.Junta de AndalucíaFondo Europeo de Desarrollo RegionalMinisterio de Ciencia e Innovació

A branch-and-price approach for the continuous multifacility monotone ordered median problem

Acknowledgements The authors of this research acknowledge financial support by the Spanish Ministerio de Ciencia y Tecnología, Agencia Estatal de Investigación and Fondos Europeos de Desarrollo Regional (FEDER) via project PID2020-114594GB-C21. The authors also acknowledge partial support from project B-FQM-322-UGR20. The first, third and fourth authors also acknowledge partial support from projects FEDER-US-1256951, Junta de Andaluca P18-FR-1422, CEI-3-FQM331, FQM-331, and NetmeetData: Ayudas Fundacin BBVA a equipos de investigacin científica 2019. The first and second authors were par- tially supported by research group SEJ-584 (Junta de Andalucía). The first author was also partially supported by the IMAG-Maria de Maeztu grant CEX2020-001105-M/AEI/10.13039/50110 0 011033. The second author was supported by Spanish Ministry of Education and Science grant number PEJ2018-002962-A and the Doctoral Program in Mathematics at the Universidad of Granada. The third author also acknowledges the grant Contratación de Personal Investigador Doctor (Convocatoria 2019) 43 Contratos Capital Humano Línea 2 Paidi 2020, supported by the European Social Fund and Junta de Andalucía.In this paper, we address the Continuous Multifacility Monotone Ordered Median Problem. The goal of this problem is to locate facilities in minimizing a monotone ordered weighted median function of the distances between given demand points and its closest facility. We propose a new branch-and-price procedure for this problem, and three families of matheuristics based on: solving heuristically the pricer problem, aggregating the demand points, and discretizing the decision space. We give detailed discussions of the validity of the exact formulations and also specify the implementation details of all the solution procedures. Besides, we assess their performances in an extensive computational experience that shows the superiority of the branch-and-price approach over the compact formulation in medium-sized instances. To handle larger instances it is advisable to resort to the matheuristics that also report rather good results.Spanish Ministerio de Ciencia y Tecnología, Agencia Estatal de Investigación and Fondos Europeos de Desarrollo Regional (FEDER) via project PID2020-114594GB-C21Partial support from project B-FQM-322-UGR20Partial support from projects FEDER-US-1256951, Junta de Andaluca P18-FR-1422, CEI-3-FQM331, FQM-331, and NetmeetData: Ayudas Fundación BBVA a equipos de investigacin científica 2019Research group SEJ-584 (Junta de Andalucía)Partially supported by the IMAG-Maria de Maeztu grant CEX2020-001105-M/AEI/10.13039/50110 0 011033Spanish Ministry of Education and Science grant number PEJ2018-002962-AEuropean Social Fund and Junta de Andalucí

Improvements and comparison of heuristics for solving the uncapacitated multisource Weber problem

Copyright @ 2000 INFORMSThe multisource Weber problem is to locate simultaneously m facilities in the Euclidean plane to minimize the total transportation cost for satisfying the demand of n 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 ct 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.This study was supported by the Department of National Defence (Canada) Academic Research; Office of Naval Research Grant N00014-92-J-1194, Natural Sciences and Engineering Research Council of Canada Grant GPO 105574 and Fonds pour la Formation des Chercheurs et l’Aide a la Recherche Grant 32EQ 1048; and by an International Postdoctoral Fellowship of the Natural Sciences and Engineering Research Council of Canada, Grant OGPOO 39682

Bearing-Based Network Localization Under Gossip Protocol

This paper proposes a bearing-based network localization algorithm with a randomized gossip protocol. Each sensor node is assumed to be able to obtain the bearing vectors and communicate its position estimates with several neighboring agents. Each update involves two agents, and the update sequence follows a stochastic process. Under the assumption that the network is infinitesimally bearing rigid and contains at least two beacon nodes, we show that the proposed algorithm could successfully estimate the actual positions of the network in probability. The randomized update protocol provides a simple, distributed, and reduces the communication cost of the network. The theoretical result is then supported by a simulation of a 1089-node sensor network.Comment: preprint, 7 pages, 2 figure