
Journal of Convex Analysis 23 (2016), No. 4, 11851204 Copyright Heldermann Verlag 2016 Locating a SemiObnoxious Facility  A TolandSinger Duality Based Approach Andrea Wagner Dept. of Finance, Accounting and Statistics, Institute for Statistics and Mathematics, Vienna University of Economics and Business, Welthandelsplatz 1, 1020 Vienna, Austria andrea.wagner@wu.ac.at JuanEnrique MartínezLegaz Dep. d'Economia i d'Histňria Econňmica, Universitat Autónoma de Barcelona, 08193 Bellaterra, Spain JuanEnrique.Martinez.Legaz@uab.cat Christiane Tammer Institute of Mathematics, MartinLutherUniversity HalleWittenberg, TheodorLieser Str. 5, 06120 Halle, Germany christiane.tammer@mathematik.unihalle.de We consider the problem of locating a facility amongst a given collection of attraction and repulsion points. The goal is to find a location x in the Euclidean space R^{n} for a facility, such that the difference between the weighted sum of distances from x to the attraction points and the weighted sum of distances to the repulsion points is minimized. The corresponding objective function constitutes as a D.C. function. Based on the duality theory by Toland and Singer for the class of D.C. programs, we formulate a dual problem to the given location problem. Taking into account the special structure of the location problem, we present geometrical properties of the model, give conditions for the existence of an optimal solution, obtain duality results, describe the relationship between primal and dual elements, and formulate an algorithm which determines exact solutions for the location problem by reducing this nonconvex optimization problem to a finite number of linear programs. Keywords: Locational analysis, conjugate duality, obnoxious facilities, D.C. optimization, geometric duality, linear vector optimization. [ Fulltextpdf (233 KB)] for subscribers only. 