A class of infinite dimensional linear programming problems

AbstractWe study the infinite dimensional linear programming problem. The previous work done on this subject defined the dual problem in a small space and derived duality results for such pairs of problems. But because of that and of the strong requirements on the functions involved, those theorems do not actually hold in many applications. With our formulation, we define the dual problem in a larger space and obtain new duality results under, generally, mild assumptions. Furthermore, the solutions turn out to be extreme points of the unbounded, but w∗-locally compact, feasibility set. For this purpose, we did not try a constructive proof of our duality results, but instead we examine the problem from a more abstract point of view and derive results using general ideas from the theory of convex analysis in normed spaces [R. T. Rockafellar, “Conjugate Duality and Optimization,” SIAM, Philadelphia, Penn., 1973, and R. Holmes, “Geometric Functional Analysis,” Springer-Verlag, New York, 1975]. Our work extends previous results in this area, which appeared in [N. Levinson, J. Math. Anal. Appl. 16 (1965) 73–83, and W. Tundall, SIAM J. Appl. Math. 13 (1965), 644–666]