Emilio Carrizosa

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.

Multi-criteria analysis with partial information about the weighting coefficients

Abstract In this paper we address the problem of ranking a set of alternatives with partial information about the weighting coefficients. We introduce a family of quasiorders that are easily interpretable and manageable, which includes among others, the natural quasiorder in R n and other well known preference structures in the literature. The enrichment of the preference structure with respect to the natural quasiorder is measured by means of an absolute measure we introduce.

Optimal location and design of a competitive facility

Abstract.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 favour 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 problems in which optimal solutions might not exist.

Undesirable facility location with minimal covering objectives

An undesirable facility is to be located within some feasible region of any shape in the plane or on a planar network. Population is supposed to be concentrated at a finite number n of points. Two criteria are taken into account: a radius of influence to be maximised, indicating within which distance from the facility population disturbance is taken into consideration, and the total covered population, i.e. lying within the influence radius from the facility, which should be minimised. Low complexity polynomial algorithms are derived to determine all nondominated solutions, of which there are only O(n3) for a fixed feasible region or O(n2) when locating on a planar network. Once obtained, this information allows almost instant answers and a trade-off sensitivity analysis to questions such as minimising the population within a given radius (minimal covering problem) or finding the largest circle not covering more than a given total population.

Gauge Distances and Median Hyperplanes

A median hyperplane in d-dimensional space minimizes the weighted sum of the distances from a finite set of points to it. When the distances from these points are measured by possibly different gauges, we prove the existence of a median hyperplane passing through at least one of the points. When all the gauges are equal, some median hyperplane will pass through at least d-1 points, this number being increased to d when the gauge is symmetric, i.e. the gauge is a norm.Whereas some of these results have been obtained previously by different methods, we show that they all derive from a simple formula for the distance of a point to a hyperplane as measured by an arbitrary gauge.

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.

Semi-obnoxious location models: A global optimization approach

In the last decades there has been an increasing interest in environmental topics. This interest has been reflected in modeling the location of obnoxious facilities, as shown by the number of important papers published in this field. However, a very common drawback of the existing literature is that, as soon as environmental aspects are taken into account, economical considerations (e.g. transportation costs) are ignored, leading to models with dubious practical interest. In this paper we take into account both the environmental impact and the transportation costs caused by the location of an obnoxious facility, and propose a solution method the well-known BSSS, with a new bounding scheme which exploits the structure of the problem.

Continuous location problems and Big Triangle Small Triangle: constructing better bounds

The Big Triangle Small Triangle method has shown to be a powerful global optimization procedure to address continuous location problems. In the paper published in J. Global Optim. (37:305–319, 2007), Drezner proposes a rather general and effective approach for constructing the bounds needed. Such bounds are obtained by using the fact that the objective functions in continuous location models can usually be expressed as a difference of convex functions. In this note we show that, exploiting further the rich structure of such objective functions, alternative bounds can be derived, yielding a significant improvement in computing times, as reported in our numerical experience.

A D.C. biobjective location model

In this paper we address the biobjective problem of locating a semiobnoxious facility, that must provide service to a given set of demand points and, at the same time, has some negative effect on given regions in the plane. In the model considered, the location of the new facility is selected in such a way that it gives answer to these contradicting aims: minimize the service cost (given by a quite general function of the distances to the demand points) and maximize the distance to the nearest affected region, in order to reduce the negative impact. Instead of addressing the problem following the traditional trend in the literature (i.e., by aggregation of the two objectives into a single one), we will focus our attention in the construction of a finite ε-dominating set, that is, a finite feasible subset that approximates the Pareto-optimal outcome for the biobjective problem. This approach involves the resolution of univariate d.c. optimization problems, for each of which we show that a d.c. decomposition of its objective can be obtained, allowing us to use standard d.c. optimization techniques.

Gaussian variable neighborhood search for continuous optimization

Variable Neighborhood Search (VNS) has shown to be a powerful tool for solving both discrete and box-constrained continuous optimization problems. In this note we extend the methodology by allowing also to address unconstrained continuous optimization problems. Instead of perturbing the incumbent solution by randomly generating a trial point in a ball of a given metric, we propose to perturb the incumbent solution by adding some noise, following a Gaussian distribution. This way of generating new trial points allows one to give, in a simple and intuitive way, preference to some directions in the search space, or, contrarily, to treat uniformly all directions. Computational results show some advantages of this new approach.