Gradients and Subgradients of Buffered Failure Probability

TL;DR

This paper develops a mathematical framework for calculating gradients and subgradients of buffered failure probabilities, aiding optimization and sensitivity analysis in probabilistic models.

Contribution

It provides a novel characterization of subgradients using subdifferential calculus for finite distributions and extends to gradient expressions for general distributions.

Findings

Characterization of subgradients for finite probability distributions

Gradient expressions derived for general distributions

Application examples demonstrating optimality conditions

Abstract

Gradients and subgradients are central to optimization and sensitivity analysis of buffered failure probabilities. We furnish a characterization of subgradients based on subdifferential calculus in the case of finite probability distributions and, under additional assumptions, also a gradient expression for general distributions. Several examples illustrate the application of the results, especially in the context of optimality conditions.