Generalized Sequential Differential Calculus for Expected-Integral Functionals

Abstract

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.