On the number of criteria needed to decide Pareto optimality

In this paper we address the question of how many objective functions are really needed to decide whether a given point is Pareto optimal. We prove a reduction result for the case of quasi-convex objective functions and a convex feasible set. This result states that in order to decide whether a point x in the decision space is Pareto optimal it suuces to consider at most n + 1 objectives at a time, where n is the dimension of the decision space. The main theorem is based on a geometric characterization of Pareto, strict Pareto and weak Pareto solutions and Helly's Theorem.