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On the subdifferential of a submodular function

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Abstract

The author recently introduced a concept of a subdifferential of a submodular function defined on a distributive lattice. Each subdifferential is an unbounded polyhedron. In the present paper we determine the set of all the extreme points and rays of each subdifferential and show the relationship between subdifferentials of a submodular function and subdifferentials, in an ordinary sense of convex analysis, of Lovász's extension of the submodular function. Furthermore, for a modular function on a distributive lattice we give an algorithm for determining which subdifferential contains a given vector and finding a nonnegative linear combination of extreme vectors of the subdifferential which expresses the given vector minus the unique extreme point of the subdifferential.

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Authors and Affiliations

  1. Institute of Socio-Economic Planning, University of Tsukuba, Sakura, 305, Ibaraki, Japan

    Satoru Fujishige

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  1. Satoru Fujishige
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The present research was carried out when the author was on leave at Institut für Ökonometrie und Operations Research, Universität Bonn, and was supported by the Alexander von Humboldt Fellowship (1982/83), West Germany.

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Fujishige, S. On the subdifferential of a submodular function. Mathematical Programming 29, 348–360 (1984). https://doi.org/10.1007/BF02592001

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  • DOI: https://doi.org/10.1007/BF02592001

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