Global convergence of a modified BFGS-type method based on function information for nonconvex multiobjective optimization problems

TL;DR

This paper introduces a modified BFGS-type method for nonconvex multiobjective optimization that uses a common Hessian approximation to reduce computational costs and proves its convergence without convexity assumptions.

Contribution

The paper proposes a novel MFQNMO method that employs a shared Hessian approximation, balancing efficiency and accuracy, with proven convergence and superlinear rate.

Findings

Converges without convexity assumptions.

Achieves local superlinear convergence.

Effective on nonconvex and convex problems.

Abstract

In this paper, based on function information, we propose a modified BFGS-type method for nonconvex multiobjective optimization problems (MFQNMO). In the multiobjective quasi-Newton method (QNMO), each iteration involves separately approximating the Hessian matrix for each component objective function, which results in significant storage and computational burdens. MFQNMO employs a common BFGS-type matrix to approximate the Hessian matrix of all objective functions in each iteration. This matrix is updated using function information from the previous step. This approach strikes a balance between efficacy and computational cost. We confirm the convergence of the method without relying on convexity assumptions, under mild conditions, we establish a local superlinear convergence rate for MFQNMO. Furthermore, we validate its effectiveness through experiments on both nonconvex and convex test problems.