Abstract
A theory of “convex analysis” is developed for functions defined on integer lattice points. We investigate the class of functions which enjoy a variant of Steinitz's exchange property. It includes linear functions on matroids, valuations on matroids, and separable concave functions on the integral base polytope. It is shown that a function ω has the exchange property if and only if it can be extended to a concave function \(\bar \omega\)such that the maximizers of (\(\bar \omega\)+any linear function) form an integral base polytope. A Fenchel-type min-max theorem and discrete separation theorems are given, which contain, e.g., Frank's discrete separation theorem for submodular functions, and also Frank's weight splitting theorem for weighted matroid intersection.
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The author thanks András Frank, Satoru Fujishige, Satoru Iwata, András Sebö and Akiyoshi Shioura for valuable discussions.
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Murota, K. (1996). Convexity and Steinitz's exchange property. In: Cunningham, W.H., McCormick, S.T., Queyranne, M. (eds) Integer Programming and Combinatorial Optimization. IPCO 1996. Lecture Notes in Computer Science, vol 1084. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-61310-2_20
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DOI: https://doi.org/10.1007/3-540-61310-2_20
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