Complexity-based modulation of the data-set in scenario optimization

The scenario approach is a broad methodology for data-driven optimization that has found numerous applications in systems and control design. It consists in making a decision that is optimal with respect to a given criterion, while also being consistent with a sample of observations that are called the “scenarios”. More precisely, each scenario corresponds to a constraint and the solution is sought in the domain of feasibility of all scenario constraints. The level of robustness of the scenario solution is quantified by the “risk”, which is the probability that the scenario solution is not consistent with a new, out-of-sample, scenario. Recent studies have unveiled a profound link between the risk and the complexity of the solution (defined as the minimum amount of scenarios that is needed to reconstruct the solution). In this work, we leverage these results to introduce a new learning scheme where the size of the scenario sample is iteratively learned during optimization as a function of the complexity of the current solution. This new scheme implies a better exploitation of the information, so that one achieves a prescribed level of risk while saving many data as compared to standard scenario schemes. This paper presents the theoretical study that proves this result and illustrates it through a numerical example.