An Inexact Weighted Proximal Trust-Region Method
This work addresses a specific bottleneck in optimization for researchers dealing with nonsmooth convex functions lacking analytical proximity operators, representing an incremental extension of prior methods.
The authors tackled the limitation of existing trust-region methods that require analytical proximity operators for nonsmooth convex functions by extending the definition of the inexact proximity operator using the δ-Fréchet subdifferential, enabling application to a broader class of functions and demonstrating it on an optimal control problem constrained by Burgers' equation.
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 $δ$-Fréchet 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.