Optimality Conditions for Convex Stochastic Optimization Problems in Banach Spaces with Almost Sure State Constraints

TL;DR

This paper develops necessary and sufficient optimality conditions for convex stochastic optimization problems in Banach spaces, with integrable vector-valued Lagrange multipliers, applicable to PDE-constrained optimization under uncertainty.

Contribution

It introduces a novel framework for optimality conditions in Banach space stochastic optimization with almost sure constraints, emphasizing integrable multipliers over measures.

Findings

Optimality conditions are necessary and sufficient under certain models.

Lagrange multipliers are integrable vector-valued functions.

Application demonstrated in PDE-constrained optimization under uncertainty.

Abstract

We analyze a convex stochastic optimization problem where the state is assumed to belong to the Bochner space of essentially bounded random variables with images in a reflexive and separable Banach space. For this problem, we obtain optimality conditions that are, with an appropriate model, necessary and sufficient. Additionally, the Lagrange multipliers associated with optimality conditions are integrable vector-valued functions and not only measures. A model problem is given demonstrating the application to PDE-constrained optimization under uncertainty with an outlook for further applications.