Solving the Defender-Attacker-Defender Constrained Shortest Path Problem with Lagrangian Relaxation

Motivated by problems in mission planning and routing for both manned and unmanned military vehicles, we present a defender-attacker-defender (DAD) formulation of the constrained shortest path (CSP) problem. We show how to decompose the problem, and then combine the attacker-defender (AD) subproblem with a Lagrangian relaxation (LR) of the CSP problem into a single for-mulation that is easier to solve. We show how this AD subproblem can enable fast heuristic solutions, and how we can then use the LR heuristic solution as inputs to a modified nested decomposition algorithm to obtain optimal or near-optimal solutions for various instances of the problem. We test our algorithm on me- dium-and large-scale networks, and our results show that it discovers initial feasible solutions that are closer to optimal than those found by a traditional nested decomposition algorithm, and that it has much faster rates of convergence for solving DAD problems than a traditional algorithm.