Duality theorems for continuous linear programming problems

Abstract

Continuous linear programmings were first considered by W.F. Tyndall [7~] as a generalization of "bottle-neck problems" in dynamic programming. N. Levinson Q6], M. A. Hanson Q3] and M. A. Hanson and B. Mond Q4] generalized the results in [7]. In this paper we shall apply the theory of infinite linear programming studied by K.S. Kretschmer [J5Γ\ and M. Yamasaki [8] to the investigation of the continuous linear programmings. Our main purpose is to improve the duality theorems in Q6[] and [7J obtained by approximation from the classical finite duality theorem. In order to state the continuous linear programmings, we shall introduce some notation. If D(t) is a matrix on the interval [0, TJ (0< Γo) in the real line with entries dij(t) and g(t) is a scalar on [0, T~} such that every entry satisfies