Multiobjective Optimization, Scalarization, and Maximal Elements of Preorders

Abstract

We characterize the existence of (weak) Pareto optimal solutions to the classical multiobjective optimization problem by referring to the naturally associated preorders and their finite (Richter-Peleg) multiutility representation. The case of a compact design space is appropriately considered by using results concerning the existence of maximal elements of preorders. The possibility of reformulating the multiobjective optimization problem for determining the weak Pareto optimal solutions by means of a scalarization procedure is finally characterized.

Representing preorders with injective monotones

“…Finally, the uncertainty preorder is also an instance of multicriteria optimization (Ehrgott, 2005), also known as vector optimization (Jahn, 2009), since it is concerned with the simultaneous optimization of a finite number of objective functions (1). Notice strict and injective monotones belong to the scalarization techniques (Bevilacqua et al, 2018b;Ehrgott, 2005;Jahn, 2009) in vector optimization and always exist, again by Proposition 5. Preordered spaces from the next general class, the ones with strict monotones, include general relativity.…”

“…For any preordered space ðX; "Þ, thus, the existence of a strict monotone implies the existence of an optimization principle and the existence of an injective monotone is equivalent to the existence of an injective optimization principle. One can contrast the global injective representation of maximal elements which characterizes injective monotones in Proposition 3 with local approaches, for some specific B X, present in the literature (Bevilacqua et al, 2018b;White, 1980).…”

“…For example, the uncertainty preorder " U is defined in (1) by a countable (finite) multi-utility. Also, many applications in multicriteria optimization (Bevilacqua et al, 2018b;Ehrgott 2005) rely on preordered spaces defined by countable multi-utilities. It turns out that for the existence of strict monotones, such as entropy for " U , it is sufficient to have a countable multi-utility (Alcantud et al, 2016, Section 4).…”

Multi-Objective Optimization by Means of Pairs of Functions

“…Needless to say, this bi-multi-utility representation generalizes to the nontransitive case the classical finite multi-utility representation (u i } i=1,...m of a preorder on X (see e.g. Bevilacqua et al 2018 andKaminski 2007), according to which…”

“…Fliege andWerner 2014 andXidonas et al 2017). Recently, Bevilacqua et al 2018 approached the classical multi-objective optimization problem by referring to the naturally associated preorders, in such a way that, since the Pareto optimal elements are precisely the maximal elements of such preorders, the classical results concerning for example the existence of maximal elements for not necessarily total preorders on compact topological spaces can be used (see e.g. the famous results of Bergstrom 1975, Ward 1954and Rodríguez-Palmero and García-Lapresta 2002, and the recent results in Bosi and Zuanon 2017).…”