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Extra info for Dynamic Programming and the Calculus of Variations (Mathematics in Science and Engineering, Volume 21)
First, one thinks small, seeking to isolate a property that an optimal decision seqdence satisfies at each decision point. That local property, expressed by the principle of optimality, concerns the optimal value 10 I. DISCRETE DYNAMIC PROGRAMMING associated with starting the process in any state that can be reached in one stage from the initial state. This fact requires several conceptual developments. , rather than analyze a particular problem with specific initial and termihal conditions, one must consider many problems, each with a different initial stage and state.
The Nature of Necessary Conditions The classical variational theory begins by deducing conditions that the minimizing curve must satisfy. These are called necessary conditions. While the minimizing curve must satisfy a necessary condition, other nonminimizing curves may also meet the condition. Hence, the set of curves satisfying any necessary condition is larger than, or equal to, the set of curves solving the problem. ) The situation is diagrammed schematically in Fig. , it includes nonsolution functions).
5. 5 Consider the initial arc of the path of minimum value connecting A and B. The value of the optimal value function at A , the point (0, Oj, is related to the value at C, the point (1, l),by the equation + S(1, 1). 2) S(0, 0) = au(0, 0) Hence, S(1, 1j - S(0, 0) or, for the specific values of our example, 12 - 13 = -1. 4) or, for our example 8 - 12 = -4. 5) If we call the difference between the value of the optimal value function at 12 I. 6 the right end of a n arc and the value a t its left end the forward difference at its left end with respect to the included arc, we can state: Property 1.