Generalized Differentiation of Expected-Integral Mappings with Applications to Stochastic Programming, II: Leibniz Rules and Lipschitz Stability

Boris S. Mordukhovich, Pedro Pérez-Aros

Published 2021-01-17Version 1

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.