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. Discrete Convex Analysis provides the information that professionals in optimization will need to "catch up" with this new theoretical development. It also presents an unexpected connection between matroid theory and mathematical economics and expounds a deeper connection between matrices and matroids than most standard textbooks.
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Aft-convex arg min argmin assume base polyhedron combinatorial concave functions conjugacy conjugate convex analysis convex extension convex hull cost flow problem cost function defined denote discrete convex functions distance function domg domR domz effective domain electrical network equilibrium equivalent exchange axiom exists feasible flow finite follows function g functions section G Rv G supp+(x G Zv Hence implies indicator function inequality infimal convolution integer points integer-valued integrally convex function L-convex polyhedron linear Lovasz extension M-convex set M-convex submodular flow M-EXC[Z M-separation M^-concave matrix maximal minimum cost flow Minkowski sum Murota Murota-Shioura network flow nonempty Note optimality criterion polyhedral M-convex functions polyhedron polynomial positively homogeneous proper convex function Proposition quadratic function RU o0 satisfies submodular flow problem submodular function submodular set function subset supermodular theorem Theorem triangle inequality valuated matroid vertex