Some sufficient conditions for continuouslinear-programming problems

Abstract

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.