Stochastic proximal gradient methods for nonconvex problems in Hilbert spaces

• Published in: Computational optimization and applications Vol. 78; no. 3; pp. 705 - 740
• Main Authors: Geiersbach, Caroline, Scarinci, Teresa
• Format: Journal Article
• Language: English
• Published: New York, NY Springer US 01.04.2021; Springer Nature B.V
• Subjects: 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
• DOI: 10.1007/s10589-020-00259-y
• ISSN: 1573-2894; 0926-6003

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.