A Duality Theorem for a Class of Continuous Linear Programming Problems

Abstract

Introduction. In his book Dynamic Programming [1], R. Bellman discusses a class of continuous linear programming problems which he calls bottleneck The discussion introduces and analyzes several examples of dynamic linear models of production allocation. Moreover, it is shown that to each (primal) program can be associated a dual program. Such pairs of primal and dual programs are then shown to satisfy the optimality condition (compare Theorem 2) and the equilibrium conditions (compare Theorem 3) enjoyed by pairs of (finite) linear programming problems. Others who have discussed these problems are P. Wolfe [2], R. S. Lehman [3], and T. C. Koopmans [4]. In [1] and [3] the authors' principal concern was in finding methods of solutions. They did not treat the problems with complete mathematical rigor. In this paper under suitable hypotheses we shall prove an analogue of the fundamental duality theorem of linear programming which is valid for this class of continuous linear programming problems. Furthermore, several related results will be stated, and the duality theorem will be applied to a dynamic Leontief model of production. The reader is referred to [5] for a discussion of the fundamentals of the theory of linear programming and linear models of production. Moreover, the reader may wish to examine the economic model introduced in ?9 for a motivation of the definition of these continuous linear programming problems.