A nonmonotone proximal quasi-Newton method for multiobjective optimization

TL;DR

This paper introduces a nonmonotone proximal quasi-Newton algorithm tailored for unconstrained convex multiobjective composite optimization, demonstrating convergence to Pareto optimality and superlinear convergence rate under certain conditions.

Contribution

It develops a novel nonmonotone proximal quasi-Newton method with proven convergence and superlinear rate for multiobjective optimization problems.

Findings

Converges to Pareto optimal solutions under strong convexity.

Achieves local superlinear convergence rate.

Numerical experiments confirm effectiveness on test problems.

Abstract

This paper proposes a nonmonotone proximal quasi-Newton algorithm for unconstrained convex multiobjective composite optimization problems. To design the search direction, we minimize the max-scalarization of the variations of the Hessian approximations and nonsmooth terms. Subsequently, a nonmonotone line search is used to determine the step size, we allow for the decrease of a convex combination of recent function values. Under the assumption of strong convexity of the objective function, we prove that the sequence generated by this method converges to a Pareto optimal. Furthermore, based on the strong convexity, Hessian continuity and Dennis-Mor\'{e} criterion, we use a basic inequality to derive the local superlinear convergence rate of the proposed algorithm. Numerical experiments results demonstrate the feasibility and effectiveness of the proposed algorithm on a set of test problems.