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

TL;DR

We address stochastic optimization with probabilistic feasibility without requiring distributional assumptions by partitioning the uncertainty domain into K regions and solving a discrete, data-driven approximation PP_ε. The key idea yields a priori feasibility guarantees, while explicit performance bounds are obtained by introducing a relaxed RP_ε problem and tightening the PP_ε constraints; all bounds are controlled by partition quality and Lipschitz properties. The framework is shown to be tractable for MPC problems with piecewise-affine dynamics and logical constraints, and concrete strategies for constructing uncertainty partitions (clustering, iterative gridding, and adaptive splitting) are provided. Numerical results on a PWA MPC case illustrate the trade-offs between conservatism and computation and compare against randomized approaches, highlighting practical viability for online control of hybrid 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.

Figures (3)

• Figure 1: A 2-dimensional polytope $\mathcal{S}:=\{x:Ax\leq b\}$, where $\sigma(A)$ is small. A small tightening (black dashed line) originates $\Tilde{\mathcal{S}}$, and it may still lead to high distance between $\mathcal{S}$ and $\Tilde{\mathcal{S}}$ (pink dashed line), and thus a potentially large optimality gap.
• Figure 2: Empirical constraint violation for different choices of $K,\delta$, and compared with randomized algorithms (RA). For each case, the bold line is the mean value, whereas the shaded area is a confidence interval of one standard deviation, both computed empirically over 500 simulations.
• Figure 3: Closed-loop cost for the adaptive splitting (ADPT), and $K$-means (KMNS) with Voronoi partitioning. In both cases we test $K=4$ and $K=8$. For each case, the bold line is the mean value, whereas the shaded area is a confidence interval of one standard deviation, both computed empirically over 500 simulations.

Theorems & Definitions (17)

• Lemma 1
• proof
• Theorem 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Theorem 2
• proof