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A min-max formula is proved for the minimum of an integer-valued separablediscrete convex function where the minimum is taken over the set of integralelements of a box total dual integral (box-TDI) polyhedron. One variant of thetheorem uses the notion of conjugate function (a fundamental concept innon-linear optimization) but we also provide another version that avoidsconjugates, and its spirit is conceptually closer to the standard form ofclassic min-max theorems in combinatorial optimization. The presented frameworkprovides a unified background for separable convex minimization over the set ofintegral elements of the intersection of two integral base-polyhedra,submodular flows, L-convex sets, and polyhedra defined by totally unimodular(TU) matrices. As an unexpected application, we show how a wide class ofinverse combinatorial optimization problems can be covered by this newframework.

A Discrete Convex Min-Max Formula for Box-TDI Polyhedra