An Inexact Trust-Region Method for Structured Nonsmooth Optimization with Application to Risk-Averse Stochastic Programming

TL;DR

This paper introduces an inexact trust-region method tailored for structured nonsmooth optimization problems, with applications in risk-averse stochastic programming and PDE-constrained optimization, demonstrating robustness and efficiency.

Contribution

It presents a novel inexact trust-region algorithm capable of handling complex nonsmooth structures and infinite-dimensional problems, with proven convergence and practical PDE application results.

Findings

Method converges globally to stationary points.

Performance remains stable across different PDE discretizations.

Efficiently solves large-scale PDE-constrained problems.

Abstract

We develop a trust-region method for efficiently minimizing the sum of a smooth function, a nonsmooth convex function, and the composition of a finite-valued support function with a smooth function. Optimization problems with this structure arise in numerous applications including risk-averse stochastic programming and subproblems for nonsmooth penalty nonlinear programming methods. Our method permits the use of inexact value and derivative information, enabling the solution of infinite-dimensional problems governed by, e.g., partial differential equations (PDEs). We prove global convergence of our method and under additional regularity assumptions, demonstrate that the sequence of iterates accumulates at a stationary point of our target problem. We demonstrate our method's efficiency on two PDE-constrained optimization examples, showing that its performance is invariant to the PDE discretization size.