  The events of the last few years revealed an acute need for tools to
systematically model and analyze large financial networks. Many applications of
such tools include the forecasting of systemic failures and analyzing probable
effects of economic policy decisions. We consider optimizing the amount and
structure of a bailout in a borrower-lender network: Given a fixed amount of
cash to be injected into the system, how should it be distributed among the
nodes in order to achieve the smallest overall amount of unpaid liabilities or
the smallest number of nodes in default? We develop an exact algorithm for the
problem of minimizing the amount of unpaid liabilities, by showing that it is
equivalent to a linear program. For the problem of minimizing the number of
defaults, we develop an approximate algorithm using a reweighted l1
minimization approach. We illustrate this algorithm using an example with
synthetic data for which the optimal solution can be calculated exactly, and
show through numerical simulation that the solutions calculated by our
algorithm are close to optimal.
