{
  "id": "2009.12492",
  "version": "v1",
  "published": "2020-09-26T01:08:51.000Z",
  "updated": "2020-09-26T01:08:51.000Z",
  "title": "Generalized Differentiation of Expected-Integral Mappings with Applications to Stochastic Programming, Part~I: Sequential Calculus for Expected-Integral Functionals",
  "authors": [
    "Boris S. Mordukhovich",
    "Pedro Pérez-Aros"
  ],
  "comment": "23 pages",
  "categories": [
    "math.OC"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-09-26T01:08:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "49J53",
      "90C15",
      "49J52"
    ],
    "keywords": [
      "expected-integral functionals",
      "expected-integral mappings",
      "stochastic programming",
      "sequential calculus",
      "generalized differentiation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}