A Piecewise Convexification Method for Non-Convex Multi-Objective Optimization Programs with Box Constraints

TL;DR

This paper introduces a novel piecewise convexification approach for non-convex multi-objective optimization problems with box constraints, enabling efficient approximation of globally efficient solutions through convex relaxations and interval subdivision.

Contribution

It develops a new piecewise convexification method based on interval subdivision and the $ m extalpha BB$ approach, providing an effective algorithm for approximating global solutions.

Findings

The method effectively approximates globally efficient solutions.

The algorithm successfully solves various test instances.

It demonstrates improved efficiency over existing approaches.

Abstract

This paper presents a piecewise convexification method for solving non-convex multi-objective optimization problems with box constraints. Based on the ideas of the $\alpha$-based Branch and Bound (${\rm \alpha BB}$) method of global optimization and the interval subdivision, a series of convex relaxation sub-multiobjective problems for this non-convex multi-objective optimization problem are firstly obtained, and these sub-problems constitute a piecewise convexification problem of the original problem on the whole box. We then construct the (approximate, weakly) efficient solution set of this piecewise convexification problem, and use these sets to approximate the globally (weakly) efficient solution set of the original problem. Furthermore, we propose a piecewise convexification algorithm and show that this algorithm can also obtain approximate globally efficient solutions by calculating a finite subset of the efficient solution set of the multi-objective convex sub-problems only. Finally, its performance is demonstrated with various test instances.