Subdifferentials of Convex Operators Valued in the Space of Integrable Functions with Application to Risk-Averse Optimization

TL;DR

This paper develops new theoretical tools to analyze the differentiability and subdifferentials of convex operators valued in integrable function spaces, with applications to risk-averse stochastic optimization.

Contribution

It introduces a novel approach to study convex operators with the local property and describes their subdifferentials, extending classical theories and enabling new optimality conditions.

Findings

Derived new subdifferential formulas for convex operators in $ ext{L}^p$ spaces.

Established optimality conditions for risk-averse stochastic optimization problems.

Extended the theory beyond classical normal integrands and lattice-valued operators.

Abstract

We study differentiability properties of convex operators defined on a Banach space with values in an $\Lc_p$ space and of their compositions with monotonic convex functionals on this space. We develop new tools for operators enjoying an additional feature known as the local property. The new approach and results go beyond the classical theory of normal integrands and lattice-valued operators. We further describe the subdifferentials of compositions of such operators with convex monotonic functionals. The new results are applied to obtain novel optimality conditions in the subdifferential form for a broad class of risk-averse stochastic optimization problems with risk functionals as objectives, with partial information, and with stochastic dominance constraints. While our analysis is motivated by the theory and methods of risk-averse optimization, it addresses problems of a more general structure and has potential for further applications.