Breakdown points of Fermat-Weber problems under gauge distances

We compute the robustness of Fermat--Weber points with respect to any finite gauge. We show a breakdown point of $1/(1+\sigma)$ where $\sigma$ is the asymmetry measure of the gauge. We obtain quantitative results indicating how far a corrupted Fermat--Weber point can lie from the true value in terms of the original sample and the size of the corrupted part. If the distance from the true value depends only on the original sample, then we call the gauge 'uniformly robust'. We show that polyhedral gauges are uniformly robust, but locally strictly convex norms are not.Comment: 19 pages, 4 figure

General models in min-max planar location

This paper studies the problem of deciding whether the present iteration point of some algorithm applied to a planar singlefacility min-max location problem, with distances measured by either anl p -norm or a polyhedral gauge, is optimal or not. It turns out that this problem is equivalent to the decision problem of whether 0 belongs to the convex hull of either a finite number of points in the plane or a finite number of differentl q -circles . Although both membership problems are theoretically solvable in polynomial time, the last problem is more difficult to solve in practice than the first one. Moreover, the second problem is solvable only in the weak sense, i.e., up to a predetermined accuracy. Unfortunately, these polynomial-time algorithms are not practical. Although this is a negative result, it is possible to construct an efficient and extremely simple linear-time algorithm to solve the first problem. Moreover, this paper describes an implementable procedure to reduce the second decision problem to the first with any desired precision. Finally, in the last section, some computational results for these algorithms are reported.optimality conditions;continuous location theory;computational geometry;convex hull;Newton-Raphson method

General models in min-max continous location

In this paper, a class of min-max continuous location problems is discussed. After giving a complete characterization of th stationary points, we propose a simple central and deep-cut ellipsoid algorithm to solve these problems for the quasiconvex case. Moreover, an elementary convergence proof of this algorithm and some computational results are presented

Multifacility Location Problems with Tree Structure and Finite Dominating Sets

Multifacility location problems arise in many real world applications. Often, the facilities can only be placed in feasible regions such as development or industrial areas. In this paper we show the existence of a finite dominating set (FDS) for the planar multifacility location problem with polyhedral gauges as distance functions, and polyhedral feasible regions, if the interacting facilities form a tree. As application we show how to solve the planar 2-hub location problem in polynomial time. This approach will yield an ε-approximation for the euclidean norm case polynomial in the input data and 1/ε

An Invitation to Generalized Minkowski Geometry

The present thesis contributes to the theory of generalized Minkowski spaces as a continuation of Minkowski geometry, i.e., the geometry of finite-dimensional normed spaces over the field of real numbers. In a generalized Minkowski space, distance and length measurement is provided by a gauge, whose definition mimics the definition of a norm but lacks the symmetry requirement. This seemingly minor change in the definition is deliberately chosen. On the one hand, many techniques from Minkowski spaces can be adapted to generalized Minkowski spaces because several phenomena in Minkowski geometry simply do not depend on the symmetry of distance measurement. On the other hand, the possible asymmetry of the distance measurement set up by gauges is nonetheless meaningful and interesting for applications, e.g., in location science. In this spirit, the presentation of this thesis is led mainly by minimization problems from convex optimization and location science which are appealing to convex geometers, too. In addition, we study metrically defined objects, which may receive a new interpretation when we measure distances asymmetrically. To this end, we use a combination of methods from convex analysis and convex geometry to relate the properties of these objects to the shape of the unit ball of the generalized Minkowski space under consideration

Proximal and ellipsoid algorithms in convex programming

Issued as Final project report, Project no. G-37-64

An Investigation into Exact Methods for the Continuous p?Centre Problem and its Related Problems

This thesis will analyse, investigate and develop new and interesting ideas to optimally solve a location problem called the continuous p?centre problem. This problem wishes to locate p facilities in a plane or network of n demand points such that the maximum distance or travel time between each demand point and its closest facility is minimised. Several difficulties are identified which are to be overcome to solve the continuous p?centre problem optimally. These relate to producing a finite set of potential facility locations or decreasing the problem size so that less computational time and effort is required. This thesis will examine several schemes that can be applied to this location problem and its related version with the aim to optimally solve large problems that were previously unsolved. This thesis contains eight chapters. The first three chapters provide an introduction into location problems, with a focus on the p?centre problem. Chapter 1 begins with a brief history of location problems, followed by the various classifications and methodologies used to solve them. Chapter 2 provides a review of the methods that have been used to solve the p?centre problem, with a focus on the continuous p?centre problem. An overview of the location models used in this research is given in Chapter 3, alongside an initial investigative work. The next two chapters enhance two well-known optimal algorithms for the continuous p?centre problem. Chapter 4 develops an interesting exact algorithm that was first proposed over 30 years ago. The original algorithm is reexamined and efficient neighbourhood reductions which are mathematically supported are proposed to improve its overall computational performance. The enhanced algorithm shows a substantial reduction of up to 96% of required computational time compared to the original algorithm, and optimal solutions are found for large data sets that were previously unsolved. Chapter 5 develops a relatively new relaxation-based optimal method. Four mathematically supported enhancements are added to the algorithm to improve its efficiency and its overall computational time. The revised reverse relaxation algorithm yields a vast reduction of up to 87% of computational time required, which is then used to solve larger data sets where n ? 1323 optimally. Chapter 6 creates a new relaxation-based matheuristic, called the relaxed p' matheuristic, by combining a well-known heuristic and the optimal method developed in Chapter 5. The unique property of the matheuristic is that it deals with the relaxation of facilities rather than demand points to establish a sub-problem. The matheuristic yields a good, but not necessarily optimal, solution in a reasonable time. To guarantee optimality, the results found from the matheuristic are embedded into the optimal algorithms developed in Chapters 4 and 5. Chapter 7 adapts the optimal algorithm developed in Chapter 5 to solve two related location problems, namely the ??neighbour p?centre problem and the conditional p?centre problem. The ??neighbour p?centre problem is investigated and solved where ? = 2 & 3. A scenario analysis is also conducted for managerial insights by exploring changes in the number of facilities required to cover each demand point. Furthermore, an existing algorithm for the conditional p?centre problem is enhanced by incorporating the optimal algorithm proposed in Chapter 5, and it is used to solve large data sets where the number of preexisting facilities is 20. This chapter therefore demonstrates that an algorithm developed in this research can be adapted or used to enhance existing algorithms to optimally solve more practical and challenging related location problems. Finally, Chapter 8 summarises the findings and main achievements of this research, and outlines any further work that could be worthwhile exploring in the future