Introducing nonpolyhedral cones to multiobjective programming

Abstract

The nondominated set of a multiobjective program is investigated with respect to a class of nonpolyhedral cones, that are defined in direct generalization of Pareto, polyhedral, second order and general p-th order cones. Properties of these cones are derived using the concept of positively homogeneous functions, and two approaches to generating the associated nondominated points are presented. In Particular, it is shown how a well known relationship between the nondominated points with respect to a polyhedral cone and Pareto points can be generalized for a non-polyhedral cone. In addition, several scalarization methods that have originally been formulated for finding Pareto points can be modified to also allow for a general (polyhedral or nonpolyhedral) cone. The results are illustrated on examples and discussed for a specific class of nonpolyhedral cones. © 2009 Springer Berlin Heidelberg.