On Conditional Risk Assessments in Scenario Optimization.

Scenario optimization is a data-driven technique in which one optimizes an objective function subject to a set of constraints, each given by a data point. In this article, we show that probabilistic claims on the violation of out-of-sample constraints (risk) conditional on the complexity of the solution (number of elements in the data set by which the solution can be reconstructed) are impossible if one does not use extra information in addition to the data. While this article establishes this fundamental limitation, it also proves that a ``mild"" prior suffices to draw strong conditional conclusions. Precisely, a prior on the distribution of the complexity (which has support in a finite dimensional space) allows one to effectively bound the conditional distribution of the risk. Besides its intrinsic epistemological value, this result is useful for the conditional quantification of the risk of constraints violation in various application endeavors.