Proximal Quasi-Newton Methods for Multiobjective Optimization Problems

TL;DR

This paper introduces new proximal quasi-Newton algorithms for unconstrained multiobjective optimization, demonstrating their convergence to Pareto stationary points and effectiveness through numerical experiments.

Contribution

It proposes novel proximal BFGS, self-scaling BFGS, and Huang BFGS methods for multiobjective problems, including applications to constrained and robust cases.

Findings

Algorithms converge to Pareto stationary points under mild conditions.

Numerical experiments confirm the effectiveness of the proposed methods.

Subproblems in robust optimization are quadratic with quadratic constraints.

Abstract

We introduce some new proximal quasi-Newton methods for unconstrained multiobjective optimization problems (in short, UMOP), where each objective function is the sum of a twice continuously differentiable strongly convex function and a proper lower semicontinuous convex but not necessarily differentiable function. We propose proximal BFGS method, proximal self-scaling BFGS method, and proximal Huang BFGS method for (UMOP) with both line searches and without line searches cases. Under mild assumputions, we show that each accumulation point of the sequence generated by these algorithms, if exists, is a Pareto stationary point of the (UMOP). Moreover, we present their applications in both constrained multiobjective optimization problems and robust multiobjective optimization problems. In particular, for robust multiobjective optimization problems, we show that the subproblems of proximal quasi-Newton algorithms can be regarded as quadratic minimization problems with quadratic inequality constraints. Numerical experiments are also carried out to verify the effectiveness of the proposed proximal quasi-Newton methods.