Consistency of sample-based stationary points for infinite-dimensional stochastic optimization

TL;DR

This paper proves the consistency of sample-based stationary points in infinite-dimensional stochastic optimization problems, including applications to risk-averse and risk-neutral PDE-constrained optimization.

Contribution

It establishes the theoretical foundation for the convergence of approximate stationary points in complex stochastic optimization in Banach spaces, extending to PDE-constrained problems.

Findings

Consistency of approximate Clarke stationary points proven

Applicable to risk-averse PDE-constrained optimization

Applicable to risk-neutral PDE-constrained optimization

Abstract

We consider stochastic optimization problems with possibly nonsmooth integrands posed in Banach spaces and approximate these stochastic programs via a sample-based approaches. We establish the consistency of approximate Clarke stationary points of the sample-based approximations. Our framework is applied to risk-averse semilinear PDE-constrained optimization using the average value-at-risk and to risk-neutral bilinear PDE-constrained optimization.