Stochastic proximal gradient methods for nonconvex problems in Hilbert spaces

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 pr...

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Vydáno v:	Computational optimization and applications Ročník 78; číslo 3; s. 705 - 740
Hlavní autoři:	Geiersbach, Caroline, Scarinci, Teresa
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	New York, NY Springer US 01.04.2021 Springer Nature B.V
Témata:	Algorithms Approximation Constraints Continuity (mathematics) Convergence Convex and Discrete Geometry Differential inclusions Hilbert space Management Science Mathematical programming methods Mathematics Mathematics and Statistics Nonsmooth and nonconvex optimization Operations Research Operations Research/Decision Theory Optimal control problems involving partial differential equations Optimization Partial differential equations Partial differential equations with randomness Statistics Stochastic programming Differential inclusions Optimal control problems involving partial differential equations Partial differential equations with randomness Nonsmooth and nonconvex optimization Mathematical programming methods Stochastic programming
ISSN:	1573-2894, 0926-6003, 1573-2894
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Shrnutí:	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}$$L^1$$\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.
Bibliografie:	ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 content type line 23
ISSN:	1573-2894 0926-6003 1573-2894
DOI:	10.1007/s10589-020-00259-y