ABSTRACT
In this chapter I describe, in broad terms, the main methods for solving dynamic programming functional equations. The focus is on the rationale behind these methods and their mode of operation. Over the years numerous solution techniques that capitalize on specific
properties of the functional equation have also been developed. But these will not be discussed here either as my objective in this chapter is to profile the major approaches to solving the equation. Picking up where we left off then, our starting point is the thesis that
once a problem has been brought by the dynamic programming treatment to the form of a functional equation, the equation is solved by whatever means are available. It is handled not as a dynamic programming functional equation as such, but as an equation pure and simple. Commensurate with the equation’s character, it is solved either by standard analytic methods, or by standard numeric methods, or even by simple enumeration techniques. These methods can be classified, both conceptually and technically, in two
groups:
· Direct methods · Successive approximation methods Direct methods are straightforward transliterations of the dynamic pro-
gramming functional equation. That is, they literally activate the statement made by the functional equation, executing it, as it were, “verbatim” by means of an iterative procedure. In contrast, successive approximation methods put into action an iterative approximation scheme where, beginning with a rough approximation, the functional equation is solved iteratively in a manner that gradually updates — improves — the current approximation. It should be noted that in both cases there are a number of variations on
the main theme. Also, often the same functional equation can be solved by both direct and successive approximation methods so that there may be a choice in this matter. But on the whole each approach is tailored to solve equations that are
yielded by either one of these two types of models:
· Finite horizon models · Infinite horizon models Direct methods typically handle functional equations derived from finite
horizon models, whereas successive approximation methods typically handle equations derived from infinite horizon models. The distinction between these two classes of models is induced by a key
feature of the multistage decision model — the value of N , namely the are characterized by a
but models, respectively.
