A Lower Bounding Linear Programming approach to the Perimeter Patrol Stochastic Control Problem
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One encounters the curse of dimensionality in the application of dynamic programming to determine optimal policies for large scale controlled Markov chains. In this article, we consider a perimeter patrol stochastic optimal control problem. To determine the optimal control policy, one has to solve a Markov decision problem, whose large size renders exact dynamic programming methods intractable. So, we propose a state aggregation based approximate linear programming method to construct provably good sub-optimal policies instead. The state-space is partitioned and the optimal cost-to-go or value function is restricted to be a constant over each partition. We show that 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 tight lower bound on the optimal value function. In general, the construction of the 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. We demonstrate this and also provide numerical results that corroborate the efficacy of the proposed methodology.
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author list (cited authors)
Kalyanam, K., Darbha, S., Park, M., Pachter, M., Chandler, P., & Casbeer, D.
complete list of authors
Kalyanam, Krishnamoorthy||Darbha, Swaroop||Park, Myoungkuk||Pachter, Meir||Chandler, Phil||Casbeer, David