Non-convex scenario optimization

Scenario optimization is an approach to data-driven decision-making that has been introduced some fifteen years ago and has ever since then grown fast. Its most remarkable feature is that it blends the heuristic nature of data-driven methods with a rigorous theory that allows one to gain factual, reliable, insight in the solution. The usability of the scenario theory, however, has been restrained thus far by the obstacle that most results are standing on the assumption of convexity. With this paper, we aim to free the theory from this limitation. Specifically, we focus on the body of results that are known under the name of "wait-and-judge" and show that its fundamental achievements maintain their validity in a non-convex setup. While optimization is a major center of attention, this paper travels beyond it and into data-driven decision making. Adopting such a broad framework opens the door to building a new theory of truly vast applicability.