Continuous Linear Programming Research Articles

An Extended Duality Theorem for Continuous Linear Programming Problems

An Extended Duality Theorem for Continuous Linear Programming Problems

Some sufficient conditions for continuouslinear-programming problems

This paper treats continuous-linear programming problems with time delays in the constraints, which are expressed in the form of difference-differential inequalities. Lagrange multiplier functions are used to adjoin the constraints to the cost functional and to compute, by explicit algorithm, a candidate solution. It is proved that if this solution is feasible, then it is also optimal. A similar method is developed for dealing with dual problems. When both the primal and the dual problems have solutions, the relationship between their solutions is exhibited and it is shown how, in some cases, solutions to the dual problem can be constructed from the solutions to the primal problem and vice versa. Finally, the differences between continuous linear programming problems and problems of classical calculus of variations and modern optimal control are discussed. The main advantage of the method proposed in this paper, for computing trial solutions, is that it is quite straightforward and simple and that it does not depend on the simultaneous existence of solutions to both the primal and dual problems.

A class of continuous linear programming problems

A class of continuous linear programming problems

A Duality Theorem for a Class of Continuous Linear Programming Problems

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.