2021/08/26 by Caroline Geiersbach, Geiersbach, Caroline, Teresa Scarinci +1 · 1 citation
Decision Sciences · Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Optimization and Control (math.OC) #Risk and Portfolio Optimization #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2108.11782
openalex publication_date 2021/08/26 · openalex created_date 2022/07/19 · openalex updated_date 2026/07/28
The study of optimal control problems under uncertainty plays an important role in scientific numerical simulations. This class of optimization problems is strongly utilized in engineering, biology and finance. In this paper, a stochastic gradient method is proposed for the numerical resolution of a nonconvex stochastic optimization problem on a Hilbert space. We show that, under suitable assumptions, strong or weak accumulation points of the iterates produced by the method converge almost surely to stationary points of the original optimization problem. Measurability and convergence rates of a stationarity measure are handled, filling a gap for applications to nonconvex infinite dimensional stochastic optimization problems. The method is demonstrated on an optimal control problem constrained by a class of elliptic semilinear partial differential equations (PDEs) under uncertainty.