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

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.

Theorems & Definitions (33)

• Example 1.1: Sobolev regression with pointwise nonnegativity
• Example 1.2: Semilinear PDE-constrained optimization under uncertainty
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Proposition 3.1
• Lemma 3.2
• proof
• proof : Proof of \ref{['prop:consistencyreducedsaa']}