An Inexact Weighted Proximal Trust-Region Method

TL;DR

This paper extends a trust-region optimization method to handle inexact proximity operators for nonsmooth functions, broadening its applicability to complex problems like PDE-constrained control.

Contribution

It introduces an inexact proximity operator using the $ ext{delta}$-Fréchet subdifferential and integrates it into the trust-region framework, with convergence analysis and practical implementation.

Findings

Successfully applied to Burgers' equation control problem.

Extended convergence theory to inexact proximity operators.

Demonstrated effectiveness in complex nonsmooth optimization scenarios.

Abstract

In [R. J. Baraldi and D. P. Kouri, Math. Program., 201:1 (2023), pp. 559-598], the authors introduced a trust-region method for minimizing the sum of a smooth nonconvex and a nonsmooth convex function, the latter of which has an analytical proximity operator. While many functions satisfy this criterion, e.g., the $\ell_1$-norm defined on $\ell_2$, many others are precluded by either the topology or the nature of the nonsmooth term. Using the $\delta$-Fr\'echet subdifferential, we extend the definition of the inexact proximity operator and enable its use within the aforementioned trust-region algorithm. Moreover, we augment the analysis for the standard trust-region convergence theory to handle proximity operator inexactness with weighted inner products. We first introduce an algorithm to generate a point in the inexact proximity operator and then apply the algorithm within the trust-region method to solve an optimal control problem constrained by Burgers' equation.