Pansystems optimization, generalized principles of optimality, and fundamental equations of dynamic programming

Introduction In the theory of dynamic programming (Bellman, 1957; Bellman and Dreyfus, 1962), the Bellman principle of optimality was not called into question until Rudd and Blum found a wrong solution for a chemical reaction system (Jackson, 1963; Rudd and Blum, 1962). Although the Bellman principle of optimality is true for almost all practical problems, there are still some problems, both theoretical and practical, for which the principle is not valid (Denardo, 1967; Hu, 1982; Wu and Wu, 1984, 1986). Thus the recursive fundamental (functional) equation based on the Bellman principle of optimality cannot be unconditionally reliable. The importance of dynamic programming in practice lies in its technique of finding the recursive functional equation. It is therefore necessary to find conditions under which the recursive equation holds (Boltyanski, 1966; Wu, 1980). A variety of such conditions have been found since the 1960s. The present paper was inspired by the questions asked by Hu (1982) and the work of Qin, Wu and others (Furukawa, 1980; Henig, 1983; Hu, 1982; Qin, 1981a, 1981b, 1983; Wu, 1980, 1984a, 1984b; Wu and Wu, 1984, 1986) on pansystems methodology. We consider the foundational problems of dynamic programming in view of whole-part relations and body-shadow relations of pansystems analysis (Wu, 1982, 1983, 1984a). While much of the analysis was completed many years ago (1984), the results are still new today. Our main objective is to clarify the logical relation between the Bellman principle of optimality and the recursive equation of dynamic programming. In the next section, we introduce the concept of optimum operators, which can be maximization, minimization, Pareto optimization, and symbolic optimization, etc. With the concept of optimum operators we establish the global-local principle and transformation principle of optimality. The two principles form a conservation principle of