Performance Bounds for the Scenario Approach and an Extension to a Class of Non-convex Programs

TL;DR

This paper develops probabilistic bounds linking the scenario approach's optimal value to that of robust and chance-constrained programs, extending to certain non-convex problems and addressing measurability issues.

Contribution

It introduces a probabilistic framework connecting scenario program solutions to non-convex and infinite-dimensional problems, advancing theoretical understanding and practical applicability.

Findings

Established probabilistic bounds for RCP and CCP from SCP

Extended results to non-convex problems including binary variables

Addressed measurability issues in scenario programs

Abstract

We consider the Scenario Convex Program (SCP) for two classes of optimization problems that are not tractable in general: Robust Convex Programs (RCPs) and Chance-Constrained Programs (CCPs). We establish a probabilistic bridge from the optimal value of SCP to the optimal values of RCP and CCP in which the uncertainty takes values in a general, possibly infinite dimensional, metric space. We then extend our results to a certain class of non-convex problems that includes, for example, binary decision variables. In the process, we also settle a measurability issue for a general class of scenario programs, which to date has been addressed by an assumption. Finally, we demonstrate the applicability of our results on a benchmark problem and a problem in fault detection and isolation.