Weak continuity of risk functionals with applications to stochastic programming

TL;DR

This paper investigates the stability of risk functionals in stochastic programming, establishing conditions for their continuity under probability measure perturbations, and extends robustness results to complex recourse models.

Contribution

It provides a unified framework for the continuity of risk measures in stochastic programming, including models with mixed-integer quadratic and convex recourse.

Findings

Established sufficient conditions for risk functional continuity.

Unified previous stability results and extended to complex recourse models.

Covered a broad class of risk measures and recourse models.

Abstract

Measuring and managing risk has become crucial in modern decision making under stochastic uncertainty. In two-stage stochastic programming, mean risk models are essentially defined by a parametric recourse problem and a quantification of risk. From the perspective of qualitative robustness theory, we discuss sufficient conditions for continuity of the resulting objective functions with respect to perturbation of the underlying probability measure. Our approach covers a fairly comprehensive class of both stochastic-programming related risk measures and relevant recourse models. Not only this unifies previous approaches but also extends known stability results for two-stage stochastic programs to models with mixed-integer quadratic recourse and mixed-integer convex recourse, respectively.