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

A min-max formula is proved for the minimum of an integer-valued separable discrete convex function in which taken over set integral elements a box total dual polyhedron. One variant theorem uses notion conjugate (a fundamental concept nonlinear optimization), but we also provide another version that avoids conjugates, and its spirit conceptually closer to standard form classic theorems combinatorial optimization. The presented framework provides unified background minimization intersection two base-polyhedra, submodular flows, L-convex sets, polyhedra defined by totally unimodular matrices. As unexpected application, show how wide class inverse optimization problems can be covered this new framework.