A Quasi Newton Method for Uncertain Multiobjective Optimization Problems via Robust Optimization Approach

TL;DR

This paper introduces a quasi Newton method tailored for solving the robust counterpart of uncertain multiobjective optimization problems, ensuring superlinear convergence and demonstrating effectiveness through numerical comparisons.

Contribution

It develops a novel quasi Newton algorithm with a modified BFGS formula for robust multiobjective problems under uncertainty, with proven convergence and practical validation.

Findings

The algorithm converges superlinearly under standard assumptions.

Numerical results show competitive performance compared to weighted sum methods.

The method effectively handles nonsmooth deterministic reformulations of uncertain problems.

Abstract

In this paper, we propose a quasi Newton method to solve the robust counterpart of an uncertain multiobjective optimization problem under an arbitrary finite uncertainty set. Here the robust counterpart of an uncertain multiobjective optimization problem is the minimum of objective-wise worst case, which is a nonsmooth deterministic multiobjective optimization problem. In order to solve this robust counterpart with the help of quasi Newton method, we construct a sub-problem using Hessian approximation and solve it to determine a descent direction for the robust counterpart. We introduce an Armijo-type inexact line search technique to find an appropriate step length, and develop a modified BFGS formula to ensure positive definiteness of the Hessian matrix at each iteration. By incorporating descent direction, step length size, and modified BFGS formula, we write the quasi Newton's descent algorithm for the robust counterpart. We prove the convergence of the algorithm under standard assumptions and demonstrate that it achieves superlinear convergence rate. Furthermore, we validate the algorithm by comparing it with the weighted sum method through some numerical examples by using a performance profile.