A stochastic gradient method for a class of nonlinear PDE-constrained optimal control problems under uncertainty

TL;DR

This paper introduces a stochastic gradient method for solving nonconvex PDE-constrained optimal control problems under uncertainty, demonstrating convergence to stationary points and applicability to complex PDEs.

Contribution

It proposes a novel stochastic gradient approach with convergence guarantees for nonconvex infinite-dimensional stochastic optimization problems involving PDE constraints.

Findings

Convergence of the method to stationary points under suitable assumptions

Handling of measurability and convergence rates for stationarity measures

Successful demonstration on PDE-constrained optimal control under uncertainty

Abstract

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.