Optimization Under Probabilistic Envelope Constraints

Chance constraints are an important modeling tool in stochastic optimization, providing probabilistic guarantees that a solution “succeeds” in satisfying a given constraint. While they control the probability of “success,” they provide no control whatsoever in the event of a “failure.” That is, they do not distinguish between a slight overor under-shoot of the bounds, and more catastrophic violation. In short, they do not capture the magnitude of violation of the bounds. This paper addresses precisely this topic. We show that the problem of requiring different probabilistic guarantees at each level of constraint violation can be reformulated as a semi-infinite optimization problem. We provide conditions — surprisingly flexible and mild, given the nature of the problem — that guarantee polynomial-time solvability of the resulting semi-infinite formulation. We show further that this resulting problem is what has been called a comprehensive robust optimization problem in the literature. As a byproduct, we provide tight probabilistic bounds for comprehensive robust optimization. Thus, analogously to the connection between chance constraints and robust optimization, we provide a broader connection between probabilistic envelope constraints and comprehensive robust optimization.