On the Convexity of Level-sets of Probability Functions

TL;DR

This paper investigates conditions under which the feasible set defined by probabilistic constraints remains convex, especially when the safety level exceeds a certain threshold, extending previous results to Banach spaces.

Contribution

It establishes new convexity guarantees for probabilistic feasible sets at high safety levels, including in Banach space settings, using auxiliary transformations.

Findings

Convexity of probabilistic feasible sets is guaranteed above a computable safety threshold.

Results extend to decision spaces in Banach spaces.

Examples illustrate the theoretical convexity conditions.

Abstract

In decision-making problems under uncertainty, probabilistic constraints are a valuable tool to express safety of decisions. They result from taking the probability measure of a given set of random inequalities depending on the decision vector. Even if the original set of inequalities is convex, this favourable property is not immediately transferred to the probabilistically constrained feasible set and may in particular depend on the chosen safety level. In this paper, we provide results guaranteeing the convexity of feasible sets to probabilistic constraints when the safety level is greater than a computable threshold. Our results extend all the existing ones and also cover the case where decision vectors belong to Banach spaces. The key idea in our approach is to reveal the level of underlying convexity in the nominal problem data (e.g., concavity of the probability function) by auxiliary transforming functions. We provide several examples illustrating our theoretical developments.