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 (formula presented) such that the maximizers of ((formula presented)+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.
CITATION STYLE
Murota, K. (1996). Convexity and Steinitz’s exchange property: (Extended abstract). In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 1084, pp. 260–274). Springer Verlag. https://doi.org/10.1007/3-540-61310-2_20
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