Ensuring timely patient discharges is central to managing a hospital's patient flow; however, discharges are dependent on the coordination of multiple care teams and thus are highly decentralized in nature. Many large hospitals have established capacity management centers to centrally direct and inform flow and support clinical teams across the hospital system, but they often lack transparency into what are the actionable, high-yield barriers to discharge that they need to focus on to be most effective. Moreover, these barriers are patient-specific and context-dependent, i.e., a patient's clinical-operational context determines what issues must be resolved and with which urgency. In this study, we leverage a machine learning model that predicts which patients are likely to be discharged in the next 24 hours together with a mixed-integer prescriptive optimization model to identify a subset of issues called minimal barriers that stand in the way of discharging a patient. Such barriers balance two aims: a high likelihood that the patient will be discharged from the hospital in the next 24 hours if these barriers are resolved; and a high likelihood that these barriers will indeed be resolved. We empirically demonstrate the efficacy of the proposed formulation and solution methodology in identifying a small number of minimal barriers using real data from a large academic medical center.