[Paper Review] Chance constrained sets approximation: A probabilistic scaling approach -- EXTENDED VERSION

Martina Mammarella, Victor Mirasierra+3 more|arXiv (Cornell University)|Jan 15, 2021

Probabilistic and Robust Engineering Design48 references4 citations

TL;DR

This paper proposes a probabilistic scaling approach to approximate chance-constrained sets using simple approximating sets (SAS) of controlled complexity. By combining sample-based SAS construction with a probabilistic scaling procedure, the method guarantees desired violation probabilities with high confidence, enabling efficient solutions for stochastic model predictive control and probabilistic set membership estimation under joint chance constraints.

ABSTRACT

In this paper, a sample-based procedure for obtaining simple and computable approximations of chance-constrained sets is proposed. The procedure allows to control the complexity of the approximating set, by defining families of simple-approximating sets of given complexity. A probabilistic scaling procedure then allows to rescale these sets to obtain the desired probabilistic guarantees. The proposed approach is shown to be applicable in several problem in systems and control, such as the design of Stochastic Model Predictive Control schemes or the solution of probabilistic set membership estimation problems.

Motivation & Objective

• To develop a computationally efficient method for approximating chance-constrained sets while ensuring probabilistic feasibility guarantees.
• To allow control over the complexity of the approximating set through predefined families of simple approximating sets (SAS).
• To extend sample-based approximation techniques to handle joint chance constraints, which are more realistic but more complex than individual constraints.
• To provide a unified framework applicable to stochastic model predictive control (SMPC) and probabilistic set membership estimation.
• To mathematically analyze and rigorously prove the probabilistic guarantees of the proposed scaling procedure.

Proposed method

• The method uses a two-step procedure: first, constructing a simple approximating set (SAS) of given complexity from sampled data.
• Second, applying a probabilistic scaling procedure that rescales the SAS to ensure the desired violation probability is met with high confidence.
• The approach leverages order statistics and binomial tail bounds to determine the minimal scaling factor that guarantees probabilistic feasibility.
• It supports two classes of SAS: sampled-polytopes and norm-based sets, including generalized norms for broader applicability.
• The method is grounded in theoretical results from statistical learning and order statistics, ensuring finite-sample probabilistic guarantees.
• The scaling factor is computed as the (r+1)th smallest value among sample-based feasibility measures, where r is the number of allowed constraint violations.

Experimental results

Research questions

• RQ1 How can we construct a computationally tractable inner approximation of a chance-constrained set with guaranteed probabilistic feasibility?
• RQ2 What is the minimal sample size required to ensure a desired violation probability with high confidence in the presence of uncertainty?
• RQ3 How can the complexity of the approximating set be controlled without sacrificing probabilistic guarantees?
• RQ4 Can the proposed method be extended to handle joint chance constraints, which require simultaneous satisfaction of multiple constraints?
• RQ5 How does the probabilistic scaling procedure ensure that the scaled set remains a valid inner approximation of the original chance-constrained set?

Key findings

• The proposed method guarantees that the probability of constraint violation is bounded by ε with confidence at least 1−δ, using a sample size N that depends on ε, δ, and the problem complexity.
• The sample complexity is significantly reduced by selecting low-complexity SAS, such as sampled-polytopes or norm-based sets, which simplify the optimization and scaling steps.
• The method achieves probabilistic guarantees for joint chance constraints by treating the entire constraint set as a single Boolean function, enabling application to more realistic control and estimation problems.
• The scaling factor is derived as the (r+1)th order statistic of feasibility measures, ensuring that at most r samples violate the scaled set with high probability.
• The theoretical analysis confirms that the method maintains feasibility with high confidence even under non-Gaussian and non-parametric uncertainty distributions.
• Numerical validation demonstrates the method’s effectiveness in probabilistic set membership estimation, showing accurate and robust identification under uncertainty.