Generalized robust counterparts for constraints with bounded and unbounded uncertain parameters

Abstract Robust optimization has emerged as a powerful and efficient methodology for incorporating uncertain parameters into optimization models. In robust optimization, robust counterparts for uncertain constraints are created by imposing a known set of uncertain parameter realizations onto the new robust constraint. For constraints with all bounded parameters, the interval + ellipsoidal and interval + polyhedral uncertainty sets are well-established in robust optimization literature, while box, ellipsoidal, or polyhedral sets may be used for unbounded parameters. However, there has yet to be any counterparts proposed for constraints that simultaneously contain both bounded and unbounded parameters. This is crucial, as using the traditional box, ellipsoidal, or polyhedral sets with bounded parameters may impose impossible parameter realizations outside of their bounds, unnecessarily increasing the conservatism of results. In this work, robust counterparts for uncertain constraints with both bounded and unbounded uncertain parameters are derived: the generalized interval + box, generalized interval + ellipsoidal, and generalized interval + polyhedral counterparts. These counterparts reduce to the traditional box, ellipsoidal, and polyhedral counterparts if all parameters are unbounded, and reduce to the traditional interval + ellipsoidal and interval + polyhedral counterparts if all parameters are bounded. It is proven that established a priori probabilistic bounds remain valid for these counterparts. The importance of these developments is demonstrated with computational examples, showing the reduction of conservatism that is gained by appropriately limiting the possible realizations of the bounded parameters. The developments increase the scope and applicability of robust optimization as a tool for optimization under uncertainty.