A globally convergent stochastic semismooth Newton method for strongly convex problems in Hilbert spaces
Abstract
We propose and analyze a globalized stochastic semismooth Newton method for stochastic optimization problems in Hilbert spaces where the objective is strongly convex with Lipschitz continuous, Newton differentiable gradient. Gradient and generalized Hessian information are only available through mini-batching. In each iteration, a full semismooth Newton trial step is accepted or rejected based on a stochastic residual test and rejected steps are replaced by damped steps in the same Newton direction. Throughout, we admit gradient estimators with affinely growing variance, as arises in optimization problems with random partial differential equations as constraints. We prove non-asymptotic global R-linear convergence in expectation as well as almost sure convergence. For sufficiently fast-growing batch sizes, the method transitions almost surely into a regime of pure Newton steps and converges R-superlinearly almost surely. The method is applied to the optimal control of a random elliptic equation featuring a Huber tracking term as well as a random parabolic equation with a Moreau--Yosida penalization of an almost sure state constraint.
Versions
Related
- A Preconditioned Semismooth Newton Method for Box-Constrained Elliptic Optimal Control Problems with Random Coefficients
- Globally Convergent Block Coordinate Descent Algorithms for Semi-Nonnegative Independent Component Analysis
- Globally Convergent Jacobi-Type Algorithms for Robust Approximate Diagonalization of Higher-Order Cumulant Tensors
- A Globally Convergent Newton Method for Nonsmooth Optimization in the absence of Smooth Surrogates
- Existence of Kraus decomposition in infinite dimension via strongly-convergent direct process tomography
BibTeX
@misc{dvurechensky2026globally,
title = {{A globally convergent stochastic semismooth Newton method for strongly convex problems in Hilbert spaces}},
author = {Pavel Dvurechensky and Caroline Geiersbach and Michael Hintermüller and Felix Sauer},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.08061},
eprint = {2611.08061},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.08061},
}
RIS
TY - UNPB
TI - A globally convergent stochastic semismooth Newton method for strongly convex problems in Hilbert spaces
AU - Pavel Dvurechensky
AU - Caroline Geiersbach
AU - Michael Hintermüller
AU - Felix Sauer
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.08061
UR - https://arxiv.org/abs/2611.08061
AB - We propose and analyze a globalized stochastic semismooth Newton method for stochastic optimization problems in Hilbert spaces where the objective is strongly convex with Lipschitz continuous, Newton differentiable gradient. Gradient and generalized Hessian information are only available through mini-batching. In each iteration, a full semismooth Newton trial step is accepted or rejected based on a stochastic residual test and rejected steps are replaced by damped steps in the same Newton direction. Throughout, we admit gradient estimators with affinely growing variance, as arises in optimization problems with random partial differential equations as constraints. We prove non-asymptotic global R-linear convergence in expectation as well as almost sure convergence. For sufficiently fast-growing batch sizes, the method transitions almost surely into a regime of pure Newton steps and converges R-superlinearly almost surely. The method is applied to the optimal control of a random elliptic equation featuring a Huber tracking term as well as a random parabolic equation with a Moreau--Yosida penalization of an almost sure state constraint.
ER -