  Submodular Functions are a special class of set functions, which generalize
several information-theoretic quantities such as entropy and mutual information
[1]. Submodular functions have subgradients and subdifferentials [2] and admit
polynomial-time algorithms for minimization, both of which are fundamental
characteristics of convex functions. Submodular functions also show signs
similar to concavity. Submodular function maximization, though NP-hard, admits
constant-factor approximation guarantees, and concave functions composed with
modular functions are submodular. In this paper, we try to provide a more
complete picture of the relationship between submodularity with concavity. We
characterize the super-differentials and polyhedra associated with upper bounds
and provide optimality conditions for submodular maximization using the-super
differentials. This paper is a concise and shorter version of our longer
preprint [3].
