Scenario optimization with constraint relaxation in a non-convex setup: a flexible and general framework for data-driven design

The scenario approach, originally developed as a computational tool for robust problems, has through the years developed into a solid, general, framework for data-driven decision making and design. One main driving force that has fostered this process has certainly been the increasing generality of the considered schemes. In this paper, we move a further step forward in this process. By leveraging some recent results in the wake of the so-called wait-and-judge paradigm, we fully develop a scheme for scenario optimization with constraint relaxation in a non-convex setup, so greatly expanding previous achievements valid under a convexity assumption. We show that a purely data-driven, and yet tight and informative, quantification of the solution robustness is possible regardless of the mechanism through which uncertainty is generated. The generality of this new non-convex setup provides an extremely versatile scheme for data-driven design that can be applied to a variety of problems ranging from mixed-integer optimization to design in abstract spaces.