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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Bibliographic Details
Published in:arXiv.org
Main Authors: Geiersbach, Caroline, Scarinci, Teresa
Format: Paper
Language:English
Published: Ithaca Cornell University Library, arXiv.org 13.01.2021
Subjects:
Algorithms
Approximation
Computer simulation
Constraints
Continuity (mathematics)
Convergence
Hilbert space
Nonlinear programming
Optimization
Partial differential equations
ISSN:2331-8422
Online Access:Get full text
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Summary: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 \(L^1\)-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.
Bibliography:SourceType-Working Papers-1
ObjectType-Working Paper/Pre-Print-1
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ISSN:2331-8422
DOI:10.48550/arxiv.2001.01329