Infinite-Horizon Deterministic Dynamic Programming in Discrete Time: A Monotone Convergence Principle and a Penalty Method


We consider infinite-horizon deterministic dynamic programming problems in discrete time. We show that the value function of such a problem is always a fixed point of a modified version of the Bell-man operator. We also show that value iteration converges increasingly to the value function if the initial function is dominated by the value function, is mapped upward by the modified Bellman operator, and satisfies a transversality-like condition. These results require no assumption except for the general framework of infinite-horizon deterministic dynamic programming. As an application, we show that the value function can be approximated by computing the value function of an unconstrained version of the problem with the constraint replaced by a penalty function.


Dynamic programming, Bellman operator, Fixed point, Value iteration

AMS Subject Classifications

90C39, 47N10


Research Institute for Economics and Business Administration,
Kobe University
Rokkodai-cho, Nada-ku, Kobe
657-8501 Japan
Phone: +81-78-803-7036
FAX: +81-78-803-7059

Masayuki YAO
Graduate School of Economics, Keio University