Decision-Scaled Scenario Approach for Rare Chance-Constrained Optimization

TL;DR

This paper introduces a decision-scaling approach for rare chance-constrained optimization that reduces sample size requirements and guarantees feasibility, enabling efficient solutions for safety-critical applications.

Contribution

The paper proposes a novel decision-scaling method that simplifies handling rare-event chance constraints using only original data and a single hyperparameter, improving computational feasibility.

Findings

Achieves polynomial reduction in sample size compared to classical scenario approach.

Guarantees asymptotic feasibility in the rare-event regime.

Demonstrates effectiveness in finance and engineering applications.

Abstract

Chance-constrained optimization is a suitable modeling framework for safety-critical applications where violating constraints is nearly unacceptable. The scenario approach is a popular solution method for these problems, due to its straightforward implementation and ability to preserve problem structure. However, in the rare-event regime where constraint violations must be kept extremely unlikely, the scenario approach becomes computationally infeasible due to the excessively large sample sizes it demands. We address this limitation with a new yet straightforward decision-scaling method that relies exclusively on original data samples and a single scalar hyperparameter that scales the constraints in a way amenable to standard solvers. Our method leverages large deviation principles under mild nonparametric assumptions satisfied by commonly used distribution families in practice. For a broad class of problems satisfying certain practically verifiable structural assumptions, the method achieves a polynomial reduction in sample size requirements compared to the classical scenario approach, while also guaranteeing asymptotic feasibility in the rare-event regime. Numerical experiments spanning finance and engineering applications show that our decision-scaling method significantly expands the scope of problems that can be solved both efficiently and reliably.