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Discrete convex analysis

Abstract: "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. This paper extends our understanding of the conjugacy relationship between convex functions and submodular functions investigated in the eighties by A. Frank, S. Fujishige, L. Lovász and others, and also explores a novel duality framework in nonlinear integer programming."

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  • "Abstract: "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. This paper extends our understanding of the conjugacy relationship between convex functions and submodular functions investigated in the eighties by A. Frank, S. Fujishige, L. Lovász and others, and also explores a novel duality framework in nonlinear integer programming.""@en
  • "Discrete Convex Analysis is a novel paradigm for discrete optimization that combines the ideas in continuous optimization (convex analysis) and combinatorial optimization (matroid/submodular function theory) to establish a unified theoretical framework for nonlinear discrete optimization. The study of this theory is expanding with the development of efficient algorithms and applications to a number of diverse disciplines like matrix theory, operations research, and economics. This self-contained book is designed to provide a novel insight into optimization on discrete structures and should reveal unexpected links among different disciplines. It is the first and only English-language monograph on the theory and applications of discrete convex analysis."
  • "Discrete Convex Analysis is a novel paradigm for discrete optimization that combines the ideas in continuous optimization (convex analysis) and combinatorial optimization (matroid/submodular function theory) to establish a unified theoretical framework for nonlinear discrete optimization. The study of this theory is expanding with the development of efficient algorithms and applications to a number of diverse disciplines like matrix theory, operations research, and economics. This self-contained book is designed to provide a novel insight into optimization on discrete structures and should reveal unexpected links among different disciplines. It is the first and only English-language monograph on the theory and applications of discrete convex analysis."@en
  • "Abstract: "This is a survey of the theory of 'discrete convex analysis' that has been developed recently by the author 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, and separation theorems. The technical development is based on matroid- theoretic concepts, in particular, submodular functions and exchange axioms. The results extend the relationship investigated in the eighties between convex functions and submodular functions. This paper puts stress on conjugacy and duality for discrete convex functions.""@en

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  • "Discrete convex analysis"
  • "Discrete convex analysis"@en
  • "Discrete convex analysis : exposition on conjugacy and duality"@en
  • "Discrete convex analysis : exposition on conjugacy and duality"
  • "Discrete Convex Analysis : Exposiiton on Conjugacy and Duality"
  • "Discrete Convex Analysis"

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