This paper studies the class of polyhedrally tight functions in terms of the basic theorems on convex functions over Rn, such as the Fenchel Duality Theorem, Separation Theorem etc.(Polyhedrally tight functions are those for which the inequalities
yT x ≤ f (y ), y ∈ A, x ∈ A*, A, A* ⊆ Rn
can be satisfied as equalities for some vector x, not necessarily simultaneously for all y.) It is shown, using results in \cite{convex functionals}, that the basic theorems hold for polyhedrally tight set functions provided the concerned functions can be extended to convex/concave functionals retaining certain essential features. These essential features carry over only if the functions are compatible in the sense that the normal cone structures of the associated polyhedra are related in a strong way.
|
|
