Sample-Based Consistency in Infinite-Dimensional Conic-Constrained Stochastic Optimization

TL;DR

This paper establishes the theoretical consistency of sample-based methods for solving infinite-dimensional stochastic optimization problems with conic constraints, supporting their use in various applications.

Contribution

It proves the consistency of sample average approximation and regularized solutions, including KKT conditions, in infinite-dimensional stochastic optimization with conic constraints.

Findings

Consistency of optimal values and solutions established

Applicability demonstrated in diverse fields like optimal transport and PDEs

Theoretical justification for numerical methods provided

Abstract

This paper is concerned with a class of stochastic optimization problems defined on a Banach space with almost sure conic-type constraints. For this class of problems, we investigate the consistency of optimal values and solutions corresponding to sample average approximation. Consistency is also shown in the case where a Moreau--Yosida-type regularization of the constraint is used. Additionally, the consistency of Karush--Kuhn--Tucker conditions is shown under mild conditions. This work provides theoretical justification for the numerical computation of solutions frequently used in the literature. Several applications are explored showing the flexibility of the framework. We cover nonparametric regression over Sobolev balls, operator learning, optimal transport, optimization with dynamical systems under uncertainty, and optimization with partial differential equations under uncertainty.