Multiobjective Optimization and Phase Transitions

TL;DR

This paper explores the connection between multiobjective optimization and phase transitions, revealing how Pareto front geometry relates to thermodynamic phase behavior and critical points in complex systems.

Contribution

It provides a unified framework linking phase transitions in simple and multiobjective optimization, highlighting the role of Pareto front shape and thermodynamics.

Findings

Pareto front shape determines phase transition locations.

Thermodynamics can be formulated as a multiobjective optimization problem.

Phase transitions in MOO and statistical mechanics share deep similarities.

Abstract

Many complex systems obey to optimality conditions that are usually not simple. Conflicting traits often interact making a Multi Objective Optimization (MOO) approach necessary. Recent MOO research on complex systems report about the Pareto front (optimal designs implementing the best trade-off) in a qualitative manner. Meanwhile, research on traditional Simple Objective Optimization (SOO) often finds phase transitions and critical points. We summarize a robust framework that accounts for phase transitions located through SOO techniques and indicates what MOO features resolutely lead to phase transitions. These appear determined by the shape of the Pareto front, which at the same time is deeply related to the thermodynamic Gibbs surface. Indeed, thermodynamics can be written as an MOO from where its phase transitions can be parsimoniously derived; suggesting that the similarities between transitions in MOO-SOO and Statistical Mechanics go beyond mere coincidence.