In the financial system, bailout strategies play a pivotal role in mitigating substantial losses resulting from systemic risk. However, the lack of a closed-form objective function to the optimal bailout problem poses significant challenges in its resolution. This paper conceptualizes the optimal bailout (capital injection) problem as a black-box optimization task, where the black box is modeled as a fixed-point system consistent with the E-N framework for measuring systemic risk in the financia...