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Authors: LASZLO CSIRMAZ.
DOI: 10.24193/subbmath.2021.1.17
Published Online: 2021-03-20
Published Print: 2021-03-30
pp. 211-222
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ABSTRACT.
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.
Mathematics Subject Classification (2010): 90C26, 46N10, 49Q10.
Keywords: Continuous submodular optimization, entropy method, secret sharing.
