Classification Schemes for the Strong Duality of Linear Programming Over Cones

Even in a finite-dimensional setting for two dual linear programming problems where the positive orthant is replaced by a closed convex cone, it is possible to have a duality gap, i.e., the infimum of one problem greater than the supremum of its dual. The absence of duality gaps is characterized in this paper by using a class of parameterized subsidiary problems of the original pair of dual problems. Characterizations are thereby obtained of “normality”—where the infimum of one problem equate the supremum of its dual. Heretofore only sufficient conditions for normality have been given. In this setting the linear programming characterizations obtained distinguish the four cases of normality where the supremum may or may not be a maximum, and the infimum may or may not be a minimum. An example is given in six dimensions of a dual pair of problems, neither of which is stable, neither of which has an optimal solution, but both of which are normal.