Stochastic proximal gradient methods for nonconvex problems in Hilbert spaces

Caroline Geiersbach () and Teresa Scarinci ()
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Caroline Geiersbach: Weierstrass Institute
Teresa Scarinci: University of L’Aquila

Computational Optimization and Applications, 2021, vol. 78, issue 3, No 2, 705-740

Abstract: 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 $$L^1$$ 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.

Keywords: Stochastic programming; Nonsmooth and nonconvex optimization; Differential inclusions; Mathematical programming methods; Partial differential equations with randomness; Optimal control problems involving partial differential equations (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s10589-020-00259-y

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