Uncertainty Partitioning with Probabilistic Feasibility and Performance Guarantees for Chance-Constrained Optimization

TL;DR

This paper introduces a distribution-free method for chance-constrained optimization that partitions uncertainty to balance conservatism and computational effort, with theoretical guarantees and practical applications.

Contribution

It presents a novel uncertainty partitioning scheme with probabilistic feasibility guarantees, improving flexibility and tractability over traditional sampling methods.

Findings

Provides sufficient conditions for feasibility of the approximated problem.

Quantifies the performance gap between original and approximated solutions.

Demonstrates applicability to model predictive control of piecewise affine systems.

Abstract

We propose a novel distribution-free scheme to solve optimization problems where the goal is to minimize the expected value of a cost function subject to probabilistic constraints. Unlike standard sampling-based methods, our idea consists of partitioning the uncertainty domain in a user-defined number of sets, enabling more flexibility in the trade-off between conservatism and computational complexity. We provide sufficient conditions to ensure that our approximated problem is feasible for the original stochastic program, in terms of chance constraint satisfaction. In addition, we perform a rigorous performance analysis, by quantifying the distance between the optimal values of the original and the approximated problem. We show that our approach is tractable for optimization problems that include model predictive control of piecewise affine systems, and we demonstrate the benefits of our approach, in terms of the trade-off between conservatism and computational complexity, on a numerical example.