On Repetitive Scenario Design

TL;DR

Repetitive Scenario Design (RSD) is a probabilistic method that iteratively refines robust solutions by balancing the number of design samples and repetitions, with a full characterization of its convergence properties.

Contribution

This paper provides a complete probabilistic analysis of RSD, establishing the relationship between sample sizes, iterations, and robustness levels, thus expanding scenario design applicability.

Findings

Exact probabilistic characterization of RSD iterations

Tradeoff analysis between sample size and number of repetitions

Enhanced understanding of RSD's convergence behavior

Abstract

Repetitive Scenario Design (RSD) is a randomized approach to robust design based on iterating two phases: a standard scenario design phase that uses $N$ scenarios (design samples), followed by randomized feasibility phase that uses $N_o$ test samples on the scenario solution. We give a full and exact probabilistic characterization of the number of iterations required by the RSD approach for returning a solution, as a function of $N$, $N_o$, and of the desired levels of probabilistic robustness in the solution. This novel approach broadens the applicability of the scenario technology, since the user is now presented with a clear tradeoff between the number $N$ of design samples and the ensuing expected number of repetitions required by the RSD algorithm. The plain (one-shot) scenario design becomes just one of the possibilities, sitting at one extreme of the tradeoff curve, in which one insists in finding a solution in a single repetition: this comes at the cost of possibly high $N$. Other possibilities along the tradeoff curve use lower $N$ values, but possibly require more than one repetition.