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Codifferentials and Quasidifferentials of the Expectation of Nonsmooth Random Integrands and Two-Stage Stochastic Programming

M. V. Dolgopolik

Abstract

This work is devoted to an analysis of exact penalty functions and optimality conditions for nonsmooth two-stage stochastic programming problems. To this end, we first study the co-/quasi-differentiability of the expectation of nonsmooth random integrands and obtain explicit formulae for its co- and quasidifferential under some natural assumptions on the integrand. Then we analyse exact penalty functions for a variational reformulation of two-stage stochastic programming problems and obtain sufficient conditions for the global exactness of these functions with two different penalty terms. In the end of the paper, we combine our results on the co-/quasi-differentiability of the expectation of nonsmooth random integrands and exact penalty functions to derive optimality conditions for nonsmooth two-stage stochastic programming problems in terms of codifferentials.

Codifferentials and Quasidifferentials of the Expectation of Nonsmooth Random Integrands and Two-Stage Stochastic Programming

Abstract

This work is devoted to an analysis of exact penalty functions and optimality conditions for nonsmooth two-stage stochastic programming problems. To this end, we first study the co-/quasi-differentiability of the expectation of nonsmooth random integrands and obtain explicit formulae for its co- and quasidifferential under some natural assumptions on the integrand. Then we analyse exact penalty functions for a variational reformulation of two-stage stochastic programming problems and obtain sufficient conditions for the global exactness of these functions with two different penalty terms. In the end of the paper, we combine our results on the co-/quasi-differentiability of the expectation of nonsmooth random integrands and exact penalty functions to derive optimality conditions for nonsmooth two-stage stochastic programming problems in terms of codifferentials.

Paper Structure

This paper contains 7 sections, 9 theorems, 113 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Codifferentials of the Expectation of Nonsmooth Random Integrands
  4. Nonsmooth Two-Stage Stochastic Programming
  5. Exact Penalty Functions
  6. Optimality Conditions
  7. Conclusions

Key Result

Theorem \oldthetheorem

Let $1 < p \le + \infty$ and Assumption 1 be valid. Then the functional $\mathcal{I}$ is codifferentiable on $\mathbb{R}^d \times \mathop{\mathrm{\textrm{L}}}\nolimits(\Omega, \mathfrak{A}, P; \mathbb{R}^m)$, and for any $(x, y)$ from this space the pair $D \mathcal{I}(x, y) = [\underline{d} \mathca and is a codifferential of $\mathcal{I}$ at $(x, y)$.

Theorems & Definitions (31)

  • Definition 1
  • Remark 1
  • Definition 2
  • Definition 3
  • Theorem \oldthetheorem
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Corollary 1
  • ...and 21 more