Consistency of Monte Carlo Estimators for Risk-Neutral PDE-Constrained Optimization

TL;DR

This paper investigates the consistency of Monte Carlo-based sample average approximation methods for solving risk-neutral PDE-constrained optimization problems with random inputs, ensuring reliable solutions.

Contribution

It introduces a novel analysis leveraging PDE problem structure to establish the consistency of SAA solutions in risk-neutral PDE-constrained optimization.

Findings

Consistency of SAA optimal values and solutions is proven.

Framework verified on three nonlinear PDE optimization problems.

Deterministic subsets of feasible sets are constructed for analysis.

Abstract

We apply the sample average approximation (SAA) method to risk-neutral optimization problems governed by nonlinear partial differential equations (PDEs) with random inputs. We analyze the consistency of the SAA optimal values and SAA solutions. Our analysis exploits problem structure in PDE-constrained optimization problems, allowing us to construct deterministic, compact subsets of the feasible set that contain the solutions to the risk-neutral problem and eventually those to the SAA problems. The construction is used to study the consistency using results established in the literature on stochastic programming. The assumptions of our framework are verified on three nonlinear optimization problems under uncertainty.