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.

References in corpus (5)

• A Unifying Theory of Exactness of Linear Penalty Functions
• Abstract convex approximations of nonsmooth functions
• Exact Penalty Functions for Optimal Control Problems I: Main Theorem and Free-Endpoint Problems
• A convergence analysis of the method of codifferential descent
• Constrained Nonsmooth Problems of the Calculus of Variations