Singular Continuation: Generating Piece-wise Linear Approximations to Pareto Sets via Global Analysis

TL;DR

This paper introduces a novel global analysis-based method for approximating Pareto optimal sets using piece-wise linear approximations that leverage the manifold structure of Pareto sets, with proven quadratic convergence.

Contribution

The paper presents a new approach that exploits the manifold structure of Pareto sets and distinguishes different critical sets, improving approximation accuracy in nonconvex problems.

Findings

Method effectively handles superposition of local Pareto fronts.

Quadratic convergence is proven and demonstrated numerically.

Approximations are constructed using simplicial complexes.

Abstract

We propose a strategy for approximating Pareto optimal sets based on the global analysis framework proposed by Smale (Dynamical systems, New York, 1973, pp. 531-544). The method highlights and exploits the underlying manifold structure of the Pareto sets, approximating Pareto optima by means of simplicial complexes. The method distinguishes the hierarchy between singular set, Pareto critical set and stable Pareto critical set, and can handle the problem of superposition of local Pareto fronts, occurring in the general nonconvex case. Furthermore, a quadratic convergence result in a suitable set-wise sense is proven and tested in a number of numerical examples.