On set functions that can be extended to convex functionals

A set function f : 2(S) is said to be polyhedrally tight (pt) (dually polyhedrally tight (dpt)) iff in the set polyhedron (dual set polyhedron) denoted by P-f (P-f) defined by x(M) <= f (X) for all X subset of S (x (X) >= f (X) for all X subset of S every inequality can be satisfied as an equality (not necessarily simultaneously). We show that these are precisely the set functions that can be extended to convex (concave) functionals over R-+(S). We characterize such functions and show that if they have certain additional desirable properties, they are forced to become submodular/supermodular. We study pt and dpt functions using the notion of a legal dual generator (LDG) structure which is a refinement of the sets of generator vectors of the dual cones associated with the faces of the set polyhedron. We extend f (g) to convex and concave functionals on R-S by f(cup)(c) equivalent to max(x is an element of Pf) c(T)x, g(cap)(c) equivalent to min (g)(x is an element of P) c(T)x. We then show a refinement (in terms of LDG) of the following discrete separaton theorem. Theorem 0.1. If f is polyhedrally tight, g is dually polyhedrally tight and f >= g and P-f and P-g have the same dual cones associated with their faces, then f(cup) >= g(cap) and there exists a modular function h s.t. f >= h >= g.