  The bailout strategy is crucial to cushion the massive loss caused by
systemic risk in the financial system. There is no closed-form formulation of
the optimal bailout problem, making solving it difficult. In this paper, we
regard the issue of the optimal bailout (capital injection) as a black-box
optimization problem, where the black box is characterized as a fixed-point
system that follows the E-N framework for measuring the systemic risk of the
financial system. We propose the so-called ``Prediction-Gradient-Optimization''
(PGO) framework to solve it, where the ``Prediction'' means that the objective
function without a closed-form is approximated and predicted by a neural
network, the ``Gradient'' is calculated based on the former approximation, and
the ``Optimization'' procedure is further implemented within a gradient
projection algorithm to solve the problem. Comprehensive numerical simulations
demonstrate that the proposed approach is promising for systemic risk
management.
