Some Bounds for Discounted Sequential Decision Processes

New bounds are obtained on the optimal return function for what are called discounted sequential decision processes. Such processes are equivalent to ones satisfying the contraction and monotonicity properties (Denardo [Denardo, E. V., 1967. Contraction mappings in the theory underlying dynamic programming. SIAM Review. Vol. 9, pp. 165–177.]). The bounds are useful primarily in the infinite horizon case. Certain subprocesses are exploited, based on the simple notion of taking only those states which are relevant into consideration. Some existing algorithms and some of their obvious extensions are listed. The possibility of identifying nonoptimal decisions, as in MacQueen [MacQueen, J. B., 1966. A Modified dynamic programming method for markovian decision problems. Journal of Mathematical Analysis and Applications. Vol. 14, pp. 38–43.], is included.