Dominators for multiple-objective quasiconvex maximization problems

In this paper we address the problem of finding a dominator for a multiple-objective maximization problem with quasiconvex functions. The one-dimensional case is discussed in some detail, showing how a Branch-and-Bound procedure leads to a dominator with certain minimality properties. Then, the well-known result stating that the set of vertices of a polytope S contains an optimal solution for single-objective quasiconvex maximization problems is extended to multiple-objective problems, showing that, under upper-semicontinuity assumptions, the set of (k - 1)-dimensional faces is a dominator for k-objective problems. In particular, for biobjective quasiconvex problems on a polytope S, the edges of S constitute a dominator, from which a dominator with minimality properties can be extracted by Branch-and Bound methods.