Projected proximal gradient trust-region algorithm for nonsmooth optimization

TL;DR

This paper introduces a new trust-region algorithm for nonsmooth, nonconvex optimization problems, extending convergence theory and providing a simple, effective subproblem solver with promising numerical results.

Contribution

It extends global convergence and complexity bounds for trust-region methods to cases with unbounded Hessian growth and proposes a novel subproblem solver combining proximal gradient steps and projection.

Findings

Extended convergence theory to unbounded Hessian growth scenarios.

Developed a simple, effective subproblem solver for nonsmooth trust-region methods.

Demonstrated promising numerical results with the new approach.

Abstract

We consider trust-region methods for solving optimization problems where the objective is the sum of a smooth, nonconvex function and a nonsmooth, convex regularizer. We extend the global convergence theory of such methods to include worst-case complexity bounds in the case of unbounded model Hessian growth, and introduce a new, simple nonsmooth trust-region subproblem solver based on combining several iterations of proximal gradient descent with a single projection into the trust region, which meets the sufficient descent requirements for algorithm convergence and has promising numerical results.