171 00255610v83n3p313 eng Murota, Kazuo 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 A. Frank, S. Fujishige, L. LovĂˇsz 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 A. Frank and the optimality criteria for the submodular flow problem due to M. Iri-N. Tomizawa, S. Fujishige, and A. 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.. Article de fons. Convex analysis Combinatorial optimization Discrete separation theorem Integer programming Lagrange duality Submodular function Article info:eu-repo/semantics/article info:eu-repo/semantics/publishedVersion Elsevier vol. 83 n. 3 (1998) p. 313-371 83:3&amp;amp;lt;313 Mathematical Programming 0025-5610 59 2349166 http://ddd.uab.cat/uab/matpro/00255610v83n3p313.pdf 313 371 11 3 83 00255610v83n3 1998 ARTPUB