Algorithms and Differential Game Representations for Exploring Nonconvex Pareto Fronts in High Dimensions

TL;DR

This paper introduces a novel Hamilton-Jacobi and differential game framework for efficiently exploring high-dimensional nonconvex Pareto fronts in multi-objective optimization, overcoming the curse of dimensionality.

Contribution

It develops a new approach embedding MOO problems into zero-sum games, providing a dense approximation of the Pareto front and a primal-dual algorithm for high-dimensional problems.

Findings

Algorithm scales polynomially with dimension

Able to expose continuous Pareto front curves in 100D

Mitigates curse of dimensionality in MOO

Abstract

We develop a new Hamiton-Jacobi (HJ) and differential game approach for exploring the Pareto front of (constrained) multi-objective optimization (MOO) problems. Given a preference function, we embed the scalarized MOO problem into the value function of a parameterized zero-sum game, whose upper value solves a first-order HJ equation that admits a Hopf-Lax representation formula. For each parameter value, this representation yields an inner minimizer that can be interpreted as an approximate solution to a shifted scalarization of the original MOO problem. Under mild assumptions, the resulting family of solutions maps to a dense subset of the weak Pareto front. Finally, we propose a primal-dual algorithm based on this approach for solving the corresponding optimality system. Numerical experiments show that our algorithm mitigates the curse of dimensionality (scaling polynomially with the dimension of the decision and objective spaces) and is able to expose continuous curves along nonconvex Pareto fronts in 100D in just $\sim$100 seconds.