  A min-max formula is proved for the minimum of an integer-valued separable
discrete convex function where the minimum is taken over the set of integral
elements of a box total dual integral (box-TDI) polyhedron. One variant of the
theorem uses the notion of conjugate function (a fundamental concept in
non-linear optimization) but we also provide another version that avoids
conjugates, and its spirit is conceptually closer to the standard form of
classic min-max theorems in combinatorial optimization. The presented framework
provides a unified background for separable convex minimization over the set of
integral 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 of
inverse combinatorial optimization problems can be covered by this new
framework.
