Nonlinear Activity Analysis

From a narrow point of view, linear programming is a method for calculating optimal operations plans for various corporate activities. From a broader point of view, however, the theory of linear programming provides a basis for the study of the economics of corporate activity. See, e.g., [Baumol, W. S. 1961. Economic Theory and Operations Analysis. Prentice-Hall, Inc., Englewood Cliffs, N. J.] For example, by providing existence theorems it is possible in many instances to say whether a proposed operation is possible at all; by providing characterization theorems it is possible to identify which variables are important and which are not, and what their relationships must be for optimal operation. The case of nonlinear convex programming is similar. Narrowly viewed, it provides a basis for the calculation of optimal operations plans. Broadly viewed, the theory ought to provide an even broader insight into the economics of corporate activity, but until recently no really satisfactory mathematical theory has been available. The work of R. T. Rockafellar [Rockafellar, R. T. 1967. Convex programming and systems of elementary monotonic relations. J. Math. Anal. Appl. 19(3) 543-564; Rockafellar, R. T. 1967. Duality and stability in extremum problems involving convex functions. Pacific J. Math. 21(1) 167-187.] now appears to fill this need, and in this paper we begin the development of a more general theory of the economics of corporate activity, from the activity analysis [Koopmans, T. C. 1951. Analysis of production as an efficient combination of activities. T. C. Koopmans, ed. Activity Analysis of Production and Allocation. John Wiley and Sons, Inc., New York, 33-97.] point of view.