Robust Counterpart Optimization: Uncertainty Sets, Formulations and Probabilistic Guarantees

Robust counterpart optimization techniques are studied in this paper. Different uncertainty sets, including those studied in literature (i.e., interval set; combined interval and ellipsoidal set; combined interval and polyhedral set) and new ones (i.e., adjustable box; pure ellipsoidal; pure polyhedral; combined interval, ellipsoidal, and polyhedral set) are studied in this work and their geometric relationship is discussed. Robust counterpart optimization formulations induced by those different uncertainty sets are derived. For those robust formulations, their corresponding probability bounds on constraint violation are derived based on the distributional information of the uncertainty (i.e., bounded or unbounded uncertainty, with or without known probability distribution function). The tightness of the different probability bounds and the conservatism of different robust counterpart optimization formulations are illustrated through a case study.