Some Demonstrations about the Cardinality of Important Sets of Non–dominated Sets

Abstract

In multiobjective optimization, a very important element is the space of objective functions, usually called Z. The set of Ω of all non–dominated sets that we can generate with elements of Z is especially interesting, because it represent all possible output from an evolutionary multiobjective algorithm. In this study, we make some theoretical demonstrations about the cardinality of Ω and others important sets of non–dominated sets. After, we use these demonstrations to prove some theorems in the area of performance measures for evolutionary multiobjective algorithms.