Theoretical Aspects of Subset Selection in Multi-Objective Optimisation

Abstract

In multi-objective optimisation, it is common to transform the multi-objective optimisation problem into a (sequence of) single-objective problems, and then compute or approximate the solution(s) of these transformed problems. Scalarisation methods are one such example where a set of solutions is determined by solving a sequence of single-objective problems. Another example are indicator-based methods where the aim is to determine, at once, a set of solutions that maximises a given set-quality indicator, i.e., a single-objective function. The aim of this chapter is to explore the connections between set-quality indicators and scalarisations, and discuss the corresponding theoretical properties. In particular, the connection between the optimal solutions of the original multi-objective problem and the optimal solutions of the single-objective problems into which it is transformed is considered.