Theorems on the decomposition of a large nonlinear convex separable economic system in the dual direction

Large mathematical programming problems often arise as the result of the economic planning process. When such a problem is not only large, but nonlinear as well, there is a need to make it more manageable by breaking it down into several smaller and more easily handled subproblems. The subproblems are solved separately with the coordination activity carried out by a master problem. A decomposition method can be seen as a dialogue between the master problem and the subproblems, where the flow of information back and forth between the former and the latters results in a series of approximations converging to the solution of the overall problem. Such a decomposition method was elaborated by Benders 1] for linear programmes and generalized to nonlinear convex separable programmes by Kronsjö 4] and by Geoffrion 3]. After considering our basic nonlinear programming problem from a two-stage minimization point of view, we review the Kronsjö nonlinear decomposition algorithm. Then we establish some properties of a function related to this algorithm.I am grateful to Professor T.O.M. Kronsjö for inspiring this work and giving valuable help.