Abstract
We consider submodular programs which are problems of minimizing submodular functions on distributive lattices with or without constraints. We define a convex (or concave) conjugate function of a submodular (or supermodular) function and show a Fenchel-type min-max theorem for submodular and supermodular functions. We also define a subgradient of a submodular function and derive a necessary and sufficient condition for a feasible solution of a submodular program to be optimal, which is a counterpart of the Karush-Kuhn-Tucker condition for convex programs.
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This work is supported by the Alexander von Humboldt fellowship (1982/83), West Germany.
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Fujishige, S. Theory of submodular programs: A fenchel-type min-max theorem and subgradients of submodular functions. Mathematical Programming 29, 142–155 (1984). https://doi.org/10.1007/BF02592218
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DOI: https://doi.org/10.1007/BF02592218