The Nested Event Tree Model with Application to Combating Terrorism

In this paper, we model and solve the strategic problem of minimizing the expected loss inflicted by a hostile terrorist organization. An appropriate allocation of certain capability-related, intent-related, vulnerability-related, and consequence-related resources is used to reduce the probabilities of success in the respective attack-related actions and to ameliorate losses in case of a successful attack. We adopt a nested event tree optimization framework and formulate the problem as a specially structured nonconvex factorable program. We develop two branch-and-bound schemes based, respectively, on utilizing a convex nonlinear relaxation and a linear outer approximation, both of which are proven to converge to a global optimal solution. We also design an alternative direct mixed-integer programming model representation for this case, and we investigate a fundamental special-case variant for this scheme that provides a relaxation and affords an optimality gap measure. Several range reduction, partitioning, and branching strategies are proposed, and extensive computational results are presented to study the efficacy of different compositions of these algorithmic ingredients, including comparisons with the commercial software BARON. A sensitivity analysis is also conducted to explore the effect of certain key model parameters.