Simple Problems: The Simplicial Gluing Structure of Pareto Sets and Pareto Fronts

TL;DR

This paper provides a theoretical foundation for the observed simplex-like topological structure of Pareto sets and fronts in simple multi-objective optimization problems, explaining their gluing structure and analyzing benchmark problems.

Contribution

It proves the gluing structure of Pareto sets and fronts in simple problems, offering a theoretical justification for their simplex-like topology.

Findings

Pareto sets and fronts of simple problems form a gluing structure

Standard benchmark problems exhibit the simplicity property

Theoretical justification for the topological structure of Pareto sets

Abstract

Quite a few studies on real-world applications of multi-objective optimization reported that their Pareto sets and Pareto fronts form a topological simplex. Such a class of problems was recently named the simple problems, and their Pareto set and Pareto front were observed to have a gluing structure similar to the faces of a simplex. This paper gives a theoretical justification for that observation by proving the gluing structure of the Pareto sets/fronts of subproblems of a simple problem. The simplicity of standard benchmark problems is studied.