2020/01/31 by Caroline Geiersbach, Teresa Scarinci · 1 citation
Computer Science · Decision Sciences · Mathematics · #Convergence (economics) #Convergence of random variables #Hilbert space #Lipschitz continuity #Optimization and Variational Analysis #Proximal Gradient Methods #Risk and Portfolio Optimization #Sequence (biology) #Stationary point #Stochastic Gradient Optimization Techniques #Stochastic optimization #Weak convergence #math.OC
paper · pdf · doi:10.1007/s10589-020-00259-y
published as Computational Optimization and Applications (2021)
openalex created_date 2020/01/10 · openalex publication_date 2021/01/12 · arxiv created 2021/01/13 · arxiv updated 2021/01/14 · openalex updated_date 2026/08/06
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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>L</mml:mi><mml:mn>1</mml:mn></mml:msup></mml:math> -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.