Discrete convex analysis

A theory of ‘discrete convex analysis’ is developed for integer-valued functions defined on integer lattice points. The theory parallels the ordinary convex analysis, covering discrete analogues of the fundamental concepts such as conjugacy, subgradients, the Fenchel min-max duality, separation theorems and the Lagrange duality framework for convex/nonconvex optimization. The technical development is based on matroid-theoretic concepts, in particular, submodular functions and exchange axioms. Sections 1–4 extend the conjugacy relationship between submodularity and exchange ability, deepening our understanding of the relationship between convexity and submodularity investigated in the eighties by Frank, Fujishige, Lovász and others. Sections 5 and 6 establish duality theorems for M- and L-convex functions, namely, the Fenchel min–max duality and separation theorems. These are the generalizations of the discrete separation theorem for submodular functions due to Frank and the optimality criteria for the submodular flow problem due to Iri–Tomizawa, Fujishige, and Frank. A novel Lagrange duality framework is also developed in integer programming. We follow Rockafellar's conjugate duality approach to convex/nonconvex programs in nonlinear optimization, while technically relying on the fundamental theorems of matroid-theoretic nature.