Generalized Leibniz rules and Lipschitzian stability for expected-integral mappings

TL;DR

This paper develops calculus rules for expected-integral multifunctions, crucial for stochastic programming, by analyzing their coderivatives and second-order subdifferentials, enhancing the mathematical tools for stochastic optimization problems.

Contribution

It introduces generalized Leibniz rules and stability results for expected-integral mappings, advancing the calculus of set-valued functions in stochastic programming.

Findings

Derived calculus rules for coderivatives of expected-integral multifunctions.

Established second-order subdifferential formulas for expected-integral functionals.

Applied results to constraint systems in stochastic programming.

Abstract

This paper is devoted to the study of the expected-integral multifunctions given in the form \begin{equation*} \operatorname{E}_\Phi(x):=\int_T\Phi_t(x)d\mu, \end{equation*} where $\Phi\colon T\times\mathbb{R}^n \rightrightarrows \mathbb{R}^m$ is a set-valued mapping on a measure space $(T,\mathcal{A},\mu)$. Such multifunctions appear in applications to stochastic programming, which require developing efficient calculus rules of generalized differentiation. Major calculus rules are developed in this paper for coderivatives of multifunctions $\operatorname{E}_\Phi$ and second-order subdifferentials of the corresponding expected-integral functionals with applications to constraint systems arising in stochastic programming. The paper is self-contained with presenting in the preliminaries some needed results on sequential first-order subdifferential calculus of expected-integral functionals taken from the first paper of this series.