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

Maximal covering location problems on networks with regional demand

Covering problems are well studied in the Operations Research literature under the assumption that both the set of users and the set of potential facilities are finite. In this paper we address the following variant, which leads to a Mixed Integer Nonlinear Program (MINLP): locations of p facilities are sought along the edges of a network so that the expected demand covered is maximized, where demand is continuously distributed along the edges. This MINLP has a combinatorial part (which edges of the network are chosen to contain facilities) and a continuous global optimization part (once the edges are chosen, which are the optimal locations within such edges). A branch and bound algorithm is proposed, which exploits the structure of the problem: specialized data structures are introduced to successfully cope with the combinatorial part, inserted in a geometric branch and bound. Computational results are presented, showing the appropriateness of our procedure to solve covering problems for small (but nontrivial) values of p.Unión EuropeaMinisterio de Ciencia e InnovaciónJunta de Andalucí

Máquinas de vector de apoyo: problemas de programación matemática en clasificación

Ministerio de Ciencia y TecnologíaPlan Andaluz de Investigación (Junta de Andalucía

Supervised classification and mathematical optimization

Data Mining techniques often ask for the resolution of optimization problems. Supervised Classification, and, in particular, Support Vector Machines, can be seen as a paradigmatic instance. In this paper, some links between Mathematical Optimization methods and Supervised Classification are emphasized. It is shown that many different areas of Mathematical Optimization play a central role in off-the-shelf Supervised Classification methods. Moreover, Mathematical Optimization turns out to be extremely useful to address important issues in Classification, such as identifying relevant variables, improving the interpretability of classifiers or dealing with vagueness/noise in the data.Ministerio de Ciencia e InnovaciónJunta de Andalucí

Supervised Classification and Mathematical Optimization

Data Mining techniques often ask for the resolution of optimization problems. Supervised Classification, and, in particular, Support Vector Machines, can be seen as a paradigmatic instance. In this paper, some links between Mathematical Optimization methods and Supervised Classification are emphasized. It is shown that many different areas of Mathematical Optimization play a central role in off-the-shelf Supervised Classification methods. Moreover, Mathematical Optimization turns out to be extremely useful to address important issues in Classification, such as identifying relevant variables, improving the interpretability of classifiers or dealing with vagueness/noise in the data