A Tunneling Method for Nonlinear Multi-objective Optimization Problems

TL;DR

This paper introduces a parameter-free tunneling method for nonlinear multi-objective optimization that iteratively finds better Pareto optimal solutions, effectively approximating the global Pareto front in complex nonconvex problems.

Contribution

It adapts single-objective tunneling ideas to multi-objective problems without requiring prior parameter tuning or objective ordering, providing a new algorithm with proven convergence.

Findings

Successfully approximates the global Pareto front in nonconvex problems

Demonstrates effectiveness through numerical examples

Provides theoretical convergence justification

Abstract

In this paper, a tunneling method is developed for nonlinear multiobjective optimization problems using some ideas of the single objective tunneling method. The proposed method does not require any a priori chosen parameters or ordering information of the objective functions. At any critical point, an auxiliary function is developed to find a different critical point that dominates the previous one. By repeatedly applying the tunneling procedure, it is possible to construct a broader approximation to the global Pareto front in nonconvex multi-objective optimization problems that may contain multiple local Pareto fronts. An algorithm is then designed based on this auxiliary function, and the convergence of this algorithm is justified under some mild assumptions. Finally, several numerical examples are presented to illustrate the effectiveness of the proposed method and to justify the theoretical results.