Lower Bounding Linear Program for the Perimeter Patrol Optimization Problem

摘要

In this article, a stochastic optimal control problem involving an unmanned aerial vehicle flying patrols around a perimeter is considered. To determine the optimal control policy, one has to solve a Markov decision problem, whose large size renders exact dynamic programming methods intractable. Therefore, a state aggregation based approximate linear programming method is used instead, to construct provably good suboptunal patrol policies. The state space is partitioned and the optimal cost-to-go or value function is restricted to be a constant over each partition. The resulting restricted system of linear inequalities embeds a family of Markov chains of lower dimension, one of which can be used to construct a lower bound on the optimal value function. In general, the construction of a lower bound requires the solution to a combinatorial problem. But the perimeter patrol problem exhibits a special structure that enables tractable linear programming formulation for the lower bound. This is demonstrated and numerical results that corroborate the efficacy of the proposed methodology are also provided.