Newton Method for Multiobjective Optimization Problems of Interval-Valued Maps

TL;DR

This paper introduces a Newton-based algorithm for multiobjective interval optimization problems, establishing convergence to Pareto critical points and demonstrating effectiveness through numerical experiments and a portfolio optimization application.

Contribution

It develops a novel Newton method tailored for MIOPs, linking weakly Pareto optimal and Pareto critical points, with proven convergence and practical performance.

Findings

Algorithm converges to Pareto critical points under certain conditions.

Numerical experiments show the method's effectiveness.

Application to portfolio optimization demonstrates practical utility.

Abstract

In this article, we propose a Newton-based method for solving multiobjective interval optimization problems (MIOPs). We first provide a connection between weakly Pareto optimal points and Pareto critical points in the context of MIOPs. Introducing this relationship, we develop an algorithm aimed at computing a Pareto critical point. The algorithm incorporates the computation of a descent direction at a non-Pareto critical point and employs an Armijo-like line search strategy to ensure sufficient decrease. Under suitable assumptions, we prove that the sequence generated by our proposed algorithm converges to a Pareto critical point. The effectiveness and performance of the proposed method are demonstrated through a series of numerical experiments on some test problems. Finally, we apply our proposed algorithm in a portfolio optimization problem with interval uncertainty.