On the Convergence of Newton-type Proximal Gradient Method for Multiobjective Optimization Problems

TL;DR

This paper analyzes the convergence properties of a Newton-type proximal gradient method for nonlinear multiobjective optimization, establishing quadratic, superlinear, and quadratic convergence under various smoothness and convexity assumptions.

Contribution

The paper provides the first rigorous convergence analysis of NPGMO, demonstrating its quadratic, superlinear, and quadratic convergence under specific conditions.

Findings

NPGMO exhibits quadratic termination for strongly convex problems.

Superlinear convergence is established for general nonlinear problems.

Quadratic convergence is proven for twice Lipschitz continuously differentiable quadratic problems.

Abstract

In a recent study, Ansary (Optim Methods Softw 38(3):570-590,2023) proposed a Newton-type proximal gradient method for nonlinear multiobjective optimization problems (NPGMO). However, the favorable convergence properties typically associated with Newton-type methods were not established for NPGMO in Ansary's work. In response to this gap, we develop a straightforward framework for analyzing the convergence behavior of the NPGMO. Specifically, under the assumption of strong convexity, we demonstrate that the NPGMO enjoys quadratic termination, superlinear convergence, and quadratic convergence for problems that are quadratic, twice continuously differentiable and twice Lipschitz continuously differentiable, respectively.