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

Optimal expected-distance separating halfspace

One recently proposed criterion to separate two datasets in discriminant analysis, is to use a hyperplane which minimises 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 on the wrong side of the hyperplane, or rather 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. It follows that the problem is polynomially solvable in fixed dimension by an algorithm of O(n d+1) when the support of X consists of n points. All these results are strengthened in the one-dimensional case, yielding an algorithm with complexity linear in the cardinality of the support of X. If a different norm is used for each data set in order to measure distances to the hyperplane, or if all distances are measured by a fixed gauge, the balancing property still holds, and we show that, under polyhedrality assumptions on the support of X, there always exists an optimal separating halfspace passing through d − 1 affinely independent data points. These results extend in a natural way when we allow constraints modeling that certain points are forced to be correctly classified.Ministerio de Ciencia y Tecnologí

A dissimilarity-based approach for Classification

The Nearest Neighbor classifier has shown to be a powerful tool for multiclass classification. In this note we explore both theoretical properties and empirical behavior of a variant of such method, in which the Nearest Neighbor rule is applied after selecting a set of so-called prototypes, whose cardinality is fixed in advance, by minimizing the empirical mis-classification cost. With this we alleviate the two serious drawbacks of the Nearest Neighbor method: high storage requirements and time-consuming queries. The problem is shown to be NP-Hard. Mixed Integer Programming (MIP) programs are formulated, theoretically compared and solved by a standard MIP solver for problem instances of small size. Large sized problem instances are solved by a metaheuristic yielding good classification rules in reasonable time.operations research and management science;

Location and design of a competitive facility for profit maximisation

A single facility has to be located in competition with fixed existing facilities of similar type. Demand is supposed to be concentrated at a finite number of points, and consumers patronise the facility to which they are attracted most. Attraction is expressed by some function of the quality of the facility and its distance to demand. For existing facilities quality is fixed, while quality of the new facility may be freely chosen at known costs. The total demand captured by the new facility generates income. The question is to find that location and quality for the new facility which maximises the resulting profits. It is shown that this problem is well posed as soon as consumers are novelty oriented, i.e. attraction ties are resolved in favor of the new facility. Solution of the problem then may be reduced to a bicriterion maxcovering-minquantile problem for which solution methods are known. In the planar case with Euclidean distances and a variety of attraction functions this leads to a finite algorithm polynomial in the number of consumers, whereas, for more general instances, the search of a maximal profit solution is reduced to solving a series of small-scale nonlinear optimisation problems. Alternative tie-resolution rules are finally shown to result in ill-posed problems.Dirección General de Enseñanza Superio

Alternating local search based VNS for linear classification

We consider the linear classification method consisting of separating two sets of points in d-space by a hyperplane. We wish to determine the hyperplane which minimises the sum of distances from all misclassified points to the hyperplane. To this end two local descent methods are developed, one grid-based and one optimisation-theory based, and are embedded in several ways into a VNS metaheuristic scheme. Computational results show these approaches to be complementary, leading to a single hybrid VNS strategy which combines both approaches to exploit the strong points of each. Extensive computational tests show that the resulting method performs well

Improved results for the k-centrum straight-line location problem

The k-Centrum problem consists in finding a point that minimises the sum of the distances to the k farthest points out of a set of given points. It encloses as particular cases to two of the most known problems in Location Analysis: the center, also named as the minimum enclosing circle, and the median. In this paper the k-Centrum criteria is applied to obtaining a straight line-shaped facility. A reduced finite dominant set is determined and an algorithm with lower complexity than the previous one obtained.Ministerio de Ciencia y Tecnologí

Assessing Spatial Data Infrastructure Policy Strategies Using the Multi-Actor Multi-Criteria Analysis

Assessments of Spatial Data Infrastructure (SDI) strategies are rather scarce. When evaluations do occur, they usually cover only one aspect of a certain initiative or infrastructure, without taking into account all the other impacts it might have. Moreover, similar methods, such as the Cost-Benefit Analysis, are used in most situations. However, these techniques are only suited for very specific objectives as they require exclusively monetary information and they do not include the objectives or perceptions of the many relevant stakeholders of the SDI environment. This is why a new methodology for assessing SDI strategies is presented in this paper, namely the Multi-Actor Multi-Criteria Analysis (MAMCA). This technique is an extension of the original Multi-Criteria Analysis (MCA) and allows for structured and extensive stakeholder participation during the entire evaluation procedure. The methodology provides a new assessment framework that takes into account all the different criteria and actors of the complex SDI decision making context. In order to illustrate the opportunities and strengths of the MAMCA in the SDI context, a case study will be presented, where possible policy strategies for the SDI in Flanders will be assessed. In this paper, which forms the first section of a two-parted article, the methodology of the MAMCA method and its possible merits for the assessment of an SDI are illustrated, together with the first three steps of the case study. The second paper, which will be published at a later date, will document the four following MAMCA steps of the case study as well as its global results and possible future recommendations