Generalized Differentiation of Expected-Integral Mappings with Applications to Stochastic Programming, Part~I: Sequential Calculus for Expected-Integral Functionals

Boris S. Mordukhovich, Pedro Pérez-Aros

Published 2020-09-26Version 1

Motivated by applications to stochastic programming, we introduce and study the {\em expected-integral functionals} in the form \begin{align*} \mathbb{R}^n\times \operatorname{L}^1(T,\mathbb{R}^m)\ni(x,y)\to\operatorname{E}_\varphi(x,y):=\int_T\varphi_t(x,y(t))d\mu \end{align*} defined for extended-real-valued normal integrand functions $\varphi:T\times\mathbb{R}^n\times\mathbb{R}^m\to[-\infty,\infty]$ on complete finite measure spaces $(T,\mathcal{A},\mu)$. The main goal of this paper is to establish sequential versions of Leibniz's rule for regular subgradients by employing and developing appropriate tools of variational analysis.