Simple Problems: The Simplicial Gluing Structure of Pareto Sets and Pareto Fronts
This work addresses a theoretical gap in understanding the topology of Pareto sets and fronts for researchers in optimization, but it is incremental as it builds on prior observations.
The paper provides a theoretical justification for the observed gluing structure of Pareto sets and fronts in multi-objective optimization problems, proving that subproblems of simple problems exhibit this structure and analyzing the simplicity of standard benchmarks.
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.