On the risk levels of distributionally robust chance constrained problems

TL;DR

This paper introduces a new discrepancy measure called relative variation distance for distributionally robust chance-constrained problems, enabling better risk level rescaling and providing guarantees for solutions under distributional ambiguity.

Contribution

It proposes the RVD discrepancy functional for PRLs, expanding the tools for handling distributional ambiguity in chance-constrained problems.

Findings

RVD allows effective risk level rescaling even at low risk levels.

Distributionally robust guarantees are derived for randomized methods.

The approach is demonstrated on a model predictive control scenario.

Abstract

In this paper, we discuss the utilization of perturbed risk levels (PRLs) for the solution of chance-constrained problems via sampling-based approaches. PRLs allow the consideration of distributional ambiguity by rescaling the risk level of the nominal chance constraint. Explicit expressions of the PRL exist for some discrepancy-based ambiguity sets. We propose a discrepancy functional not included in previous comparisons of different PRLs based on the likelihood ratio, which we term ,,relative variation distance" (RVD). If the ambiguity set can be described by the RVD, the rescaling of the risk level with the PRL is in contrast to other discrepancy functionals possible even for very low risk levels. We derive distributionally robust one- and two-level guarantees for the solution of chance-constrained problems with randomized methods. We demonstrate the viability of the derived guarantees for a randomized MPC under distributional ambiguity.