Scalarizations for Maximization with Respect to Polyhedral Cones

Abstract

Abstract. Efficient points are obtained for cone-ordered maximizations in n R using the method of scalarization. Various scalarizations are presented for ordering cones in general and then for the important special case of polyhedral cones. For polyhedral cones, it is shown how to find vectors in the positive dual cone that are needed for a scalarized objective function. Instructive examples are presented. Keywords:Multiobjective optimization, polyhedral cones, cone maximization, Pareto maximization, scalarization. IntroductionMultiobjective optimization is applied in various fields of science, engineering, and economics when decisions involve two or more conflicting objectives. For example, a pharmaceutical company may wish to determine a dose for a new drug that would maximize its therapeutic benefits while minimizing its deleterious side effects. The result would be a multiobjective optimization problem. In many other situations, however, a decision may not be adequately characterized by such standard criteria as Pareto optimality, goal programming, or lexicographic maximization. For this reason, cone-ordered maximization was developed, of which the previously mentioned criteria are special cases. . Such results are difficult to implement, however. For this reason, scalarization is the method of choice for actually obtaining a numerical solution to a cone-ordered maximization. A scalarization is an optimization problem with a single scalar objective function for which an optimal solution solves the cone-ordered problem. Examples are the scalarizations of Borwein [6] and Jahn [10] in terms of the dual cone and that of Soland [11] for Pareto maximization involving both the dual cone and aspiration levels for the individual objective functions. References [12][13][14][15][16][17][18] consolidate the theory and solution techniques for cone-ordered maximizations. However, specific results for polyhedral cones are scarce. For such cones, Yu [5] proposes a specialized scalarization based on finding cone extreme points. Sawaragi et al. [1985] reduce the problem to finding Pareto efficient points of a set in the objective function's image space. More recently, for example, Gutierrez, et al. [19] define a scalarization by perturbing the same matrix.We consider a decision problem with an n-dimensional objective function, where the optimization criterion is maximization with respect to a partial order induced by a pointed convex cone in .n R The purpose here is to present scalarizations for such problems in general, with the emphasis on polyhedral cones. Methods are also developed for finding dual vectors of a polyhedral cone as required for scalarized objective functions. One such approach involves linear programming. The paper is organized as follows. In Section 2 the requisite preliminaries concerning cones are established, while the notion of coneordered maximization is defined in Section 3. In Section 4 various scalarizations are presented. Section 5 then addresses the problem of obtaining the required dual vector...