"Real continuous submodular functions, as a generalization of the
corresponding discrete notion to the continuous domain, gained considerable
attention recently. The analog notion for entropy functions requires additional
properties: a real function defined on the non-negative orthant of $\R^n$ is
entropy-like (EL) if it is submodular, takes zero at zero, non-decreasing,
and has the Diminishing Returns property.
Motivated by problems concerning the Shannon complexity of multipartite
secret sharing, a special case of the following general optimization problem
is considered: find the minimal cost of those EL functions which satisfy certain
constraints.
In our special case the cost of an EL function is the maximal value of the
$n$ partial derivatives at zero. Another possibility could be the supremum
of the function range. The constraints are specified by a smooth bounded
surface $S$ cutting off a downward closed subset. An EL function is feasible
if at the internal points of $S$ the left and right partial derivatives of
the function differ by at least one.
A general lower bound for the minimal cost is given in terms of the normals
of the surface $S$. The bound is tight when $S$ is linear. In the
two-dimensional case the same bound is tight for convex or concave $S$. It
is shown that the optimal EL function is not necessarily unique. The paper
concludes with several open problems."
Generalized Differentiable -Invex Functions and Their Applications in Optimization