Application of Lemke's method to a class of Markov decision problems

This paper presents an application of Lemke's method to a class of Markov decision problems, appearing in the optimal stopping problems, and other well-known optimization problems. We consider a special case of the Markov decision problems with finitely many states, where the agent can choose one of the alternatives: getting a fixed reward immediately or paying the penalty for one term. We show that the problem can be reduced to a linear complementarity problem that can be solved by Lemke's method with the number of iterations less than the number of states. The reduced linear complementarity problem does not necessarily satisfy the copositive-plus condition. Nevertheless we show that the Lemke's method succeeds in solving the problem by proving that the problem satisfies a necessary and sufficient condition for the extended Lemke's method to compute a solution in the piecewise linear complementarity problem.