General Imbedding Theory

Abstract

Almost any optimization problem can be viewed as a search for the extreme filial value of a single state variable. The numerical solution of such problems can be undertaken by two different methods—classical solution and dynamic programming solution. Through the concept of invariant imbedding, a problem that is originally boundary valued in nature is viewed as imbedded in a class of more general problems of which the boundary problem is simply a special limiting case. In the classical solution, using variational principles, the primary difficulty is the two-point boundary problem. In the dynamic programming solution, the primary difficulty is the multidimensionality of the return function. The most difficult problem in the computational solution by dynamic programming stems from the multidimensionality of the functional recurrence formulas. There are at least three known methods of reducing dimensionality. These include coarsening the grid, polynomial approximation, and linearization, and successive approximation. The method used for any particular problem depends on the particular problem, the ingenuity of the analyst and his familiarity with either method.