An optimization problem for continuous submodular functions
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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.
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