Please use this identifier to cite or link to this item: https://hdl.handle.net/10419/287443 
Year of Publication: 
2021
Citation: 
[Journal:] Computational Optimization and Applications [ISSN:] 1573-2894 [Volume:] 78 [Issue:] 3 [Publisher:] Springer US [Place:] New York, NY [Year:] 2021 [Pages:] 705-740
Publisher: 
Springer US, New York, NY
Abstract: 
For finite-dimensional problems, stochastic approximation methods have long been used to solve stochastic optimization problems. Their application to infinite-dimensional problems is less understood, particularly for nonconvex objectives. This paper presents convergence results for the stochastic proximal gradient method applied to Hilbert spaces, motivated by optimization problems with partial differential equation (PDE) constraints with random inputs and coefficients. We study stochastic algorithms for nonconvex and nonsmooth problems, where the nonsmooth part is convex and the nonconvex part is the expectation, which is assumed to have a Lipschitz continuous gradient. The optimization variable is an element of a Hilbert space. We show almost sure convergence of strong limit points of the random sequence generated by the algorithm to stationary points. We demonstrate the stochastic proximal gradient algorithm on a tracking-type functional with a L1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L$$\end{document}-penalty term constrained by a semilinear PDE and box constraints, where input terms and coefficients are subject to uncertainty. We verify conditions for ensuring convergence of the algorithm and show a simulation.
Subjects: 
Stochastic programming
Nonsmooth and nonconvex optimization
Differential inclusions
Mathematical programming methods
Partial differential equations with randomness
Optimal control problems involving partial differential equations
Persistent Identifier of the first edition: 
Creative Commons License: 
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Document Type: 
Article
Document Version: 
Published Version

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