On Refinements of Some Duality Theorems in Linear Programming over Cones

Recently Schechter (in a Lehigh University report) formulated two dual linear programming problems over closed convex cones in quite general spaces and showed that, if precisely one of the problems is feasible, then the feasible one has infinite value. In this paper we obtain a refinement by using asymptotic duality states as defined by Ben Israel-Charnes-Kortanek and their classification theorem [Bull. Am. Math. Soc. 74, 318–324 (1969) and 76, 426 (1970)]. The sharpened version shows that in general the feasible-infinite states of the dual problem-pair are not quite as symmetrical as, for example, in the special case of polyhedral cones. Similarly, a sharpened version of Duffin's Theorem 1 [H. W. Kuhn and A. W. Tucker (eds.) Linear Equalities and Related Systems. Princeton University Press, pp. 317–329, 1956] is also developed.