Duality in continuous linear programming

A duality theorem has been proved for these problems under various assumptions. Tyndall in [I] assumes that B and C are constant and 01 and y are continuous, extending the result in [2] to the case where 01 and y are only bounded and measurable. Levinson [3] again assumes that 01 and y are continuous but also allows B and C to be arbitrary continuous matrix-valued functions. Hanson and Mond [4] extend Levinson’s result by requiring of 01 and y only that they be bounded and measurable. All these results require, in addition, certain algebraic properties of y, B and C. These algebraic restrictions are relaxed in Grinold [5] w h ere 01, y, B, and C are required to be only bounded and measurable. Grinold, however, assumes that C(t, S) = 0 if s > t. In this paper a duality theorem is proved under the same hypothesis as in [3] except that 01, y, B, and C are required only to be bounded and measurable. Also a duality theorem without existence of an optimal solution to the primal problem is proved with y only assumed integrable. Finally, it is shown that the values of both problems depend continuously on 01. In proving these results a key role is played by the author’s formulation of linear programming in topological vector spaces [6].