Backward, forward and backward-forward dynamic programming models under commutativity conditions

Several authors (Denardo [61 Karp and Held [18], and Bertsekas [3]) have proposed abstract dynamic programming models encompassing a wide variety of sequential optimization problems. The unifying purpose of these models is to impose sufficient conditions on the recursive defimtion of the objective function to guarantee the validity of the solution of the optimization problem by a dynamic programming iteration. In this paper we propose a general dynamic programming operator model that includes, but is not restricted to, optimization problems. Any functional satisfying a certain commutativity condition (which reduces to the principle of optimality in extrermzation problems see Section 2, B2) a-ith the generating operator of the objective recursive function, results in a sequential problem solvable by a dynamic programming iteration. Examples of sequential nonextremization problems fitting t h s framework are the derivation of marginal distributions in arbitrary probability spaces, iterative computation of stageseparated functions defined on general algebra~c systems such as additive commutative semi-groups with distributlve products, generation of symbolic transfer functions, and the Chapman-Kolmogorov equations.