Cone superadditivity of discrete convex functions

A function f is said to be cone superadditive if there exists a partition of R ⁿ into a family of polyhedral convex cones such that f(z?+?x) + f(z?+?y) ≤ f(z) + f(z?+?x?+?y) holds whenever x and y belong to the same cone in the family. This concept is useful in nonlinear integer programming in that, if the objective function is cone superadditive, the global minimality can be characterized by local optimality criterion involving Hilbert bases. This paper shows cone superadditivity of L-convex and M-convex functions with respect to conic partitions that are independent of particular functions. L-convex and M-convex functions in discrete variables (integer vectors) as well as in continuous variables (real vectors) are considered.