Discrete convex analysis is a novel paradigm for discrete optimization that combines the ideas in continuous optimization convex analysis and combinatorial. Discrete convex analysis, proposed by murota 25, 26, is a unified frame work of discrete optimization. The key of our analysis is to associate the preference of the hospitals to a mathematical concept called m. An approach from discrete convex analysis munich personal. Discrete convex analysis and its applications in operations. Mathematical engineering technical reports a framework of. Recent developments in discrete convex analysis research trends in combinatorial optimization, bonn 2008, springer, 2009, 219260. This classical result can be derived geometrically via the polyhedral split decomposition of a tree metric. February, 2007 abstract a tree metric is known to be representable as the sum of split metrics. Also the vector sum of two closed convex sets need not be closed. It is shown in 8 that l convex functions are the same as submodular integrally convex. M\concave functions are a class of discrete concave functions introduced by murota and shioura 1999, and contain the class of the sum of weighted rank functions as a proper subclass. As a powerful framework, discrete convex analysis is becoming.
Fundamental results in continuous convex analysis, in particular 1 biconjugacy and 2 sub. Discrete convex analysis, integrally convex function, lconvex function, m convex. Discrete convex analysis is aimed at establishing a novel theoretical framework for solvable discrete optimization problems by means of a combination of the ideas in continuous optimization and combinatorial optimization. We selected examples using one period, multi period and dynamic programming models to emphasize that tools in convex analysis, in particular, convex duality is indispensable in dealing with. This paper presents discrete convex analysis as a tool for use in economics and game theory. Discrete dc programming by discrete convex analysis. Recent developments in discrete convex analysis request pdf. The present paper investigates the polyhedral split decomposition from the viewpoint of discrete. Mathematical engineering technical reports polyhedral split. Convex analysis approach to discrete optimization, i iccopt.
A polyhedral convex set is characterized in terms of a. Discrete convex analysis provides the information that professionals in optimization will need to catch up with this new theoretical development. The theoretical framework of convex analysis is adapted to discrete settings and the mathematical results in matroidsubmodular function theory are generalized. Continuous and discrete dynamics for online learning and. Although discrete convex analysis is inspired by concepts and results in convex analysis and combinatorial optimization, familiarity with these areas is not necessary in reading this book. As a powerful framework, discrete convex analysis is. Submodular function minimization and maximization in discrete. Discrete convex analysis society for industrial and applied. Recently, applications of discrete convex analysis. A theory of discrete convex analysis is developed for integervalued functions defined on integer lattice points.
A theoretical framework of difference of discrete convex functions discrete dc functions and optimization problems for discrete dc functions is established. Chapter vii discrete convex analysis pages 315363 download pdf. Discrete optimization, discrete convex function, steepest descent algorithm, analysis of algorithm, iterative auction 1. The theory parallels the ordinary convex analysis, covering discrete analogues of the fundamental concepts such as conjugacy, subgradients, the fenchel minmax duality, separation theorems and the lagrange duality framework for convex nonconvex optimization. On the pipage rounding algorithm for submodular function. Discrete convex analysis 18, 40, 43, 47 aims to establish a general theoretical framework for solvable discrete optimization problems by means of a combination of the ideas in continuous optimization and combinatorial optimization. To prove our main result, we develop a theory of discrete convex functions on orientable modular graphs, analogous to discrete convex analysis by murota, and utilize a recent result of thapper and zivny on valued csp. Convex analysis in continuous optimization generalization of theory of. Applications of discrete convex analysis to mathematical economics. Discrete convex analysis, by kazuo murota, siam monographs on. Discrete convex analysis monographs on discrete mathematics. We first point out the close connection between discrete convex analysis and various research fields such as discrete optimization, auction theory, and computer. M convex and l convex functions1, introduced respectively by murotashioura 22 and fujishigemurota 8, are variants of m convex and l convex functions. Discrete convex analysis journal of mechanism and institution.
Siam monographs on discrete mathematics and applications includes bibliographical references and index. Viewed from the discrete side, it is a theory of discrete functions f. Discrete convex analysis is a new framework of discrete mat. Pdf fundamentals in discrete convex analysis semantic scholar. Discrete convex analysis is a new framework of discrete mathematics and optimization, developed during the last two decades. Convex analysis approach to discrete optimization, i concepts. Discrete convexity and its application to convex optimization. The dual problem provides important information for the solution of the discrete primal e. Convex analysis and duality over discrete domains 193 3. Annals of discrete mathematics submodular functions and. Discrete convex analysis 27, 29, 30, 32 is a general theoretical framework for solvable discrete optimization problems by means of a combination of the ideas in continuous optimization and combinatorial optimization. Standard results in continuous dc theory are exported to the discrete dc theory with the use of discrete convex analysis. May 20, 2014 this completes the classification of graphs g for which 0extg is tractable.
Submodularityand convexity lovasz, frank, fujishige 1992. Convex closure hx is convex and it is the greatest convex extension of h. Classical discrete geometry is a close relative of convex geometry with strong ties to the geometry of numbers, a branch of number theory. Minimization algorithms for discrete convex functions. Discrete convex analysis 59, 62, 63 is aimed at establishing a general theoretical framework for solvable discrete optimization problems by means of a combination of the ideas in continuous optimization and combinatorial optimization. Drawing upon techniques from discrete convex analysis, we. Discrete convex analysis murota1996 two types of discrete convexity. Matroidsubmodular function in discrete opitmization. The real convex optimization and the discrete or combinatorial convex optimization. Discrete convex analysis is a branch of discrete mathematics, and studies several types of convex functions on discrete domains.
Z that enjoy certain nice properties comparable to convexity. A framework of discrete dc programming by discrete convex. Thus the reader may conclude that the book presents a new theory. Bertsekas and john tsitsiklis, 2002, isbn 188652940x, 430 pages 3. Discrete convexity and logconcave distributions in higher. In discrete convex analysis 22, 23, 24,25, a variety of discrete convex functions are considered. A discrete dc algorithm, which is a discrete analogue of the continuous dc algorithm concave convex procedure in. Gruber 1 introduction convex geometry is an area of mathematics between geometry, analysis and discrete mathematics. Convex programs, discrete version of topics in analysis, time scales calculus. On steepest descent algorithms for discrete convex functions. 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.
In the rst part of the thesis, we study online learning dynamics for a class of games called nonatomic convex potential games, which are used for example to model congestion in transportation and communication networks. In discrete convex analysis 5,6, for example, certain combinatorial properties of the discrete hessian matrices are known 1,3 to characterize m. Mathematical engineering technical reports submodular. The dual problem of a discrete problem is continuous convex. The framework of convex analysis is adapted to discrete set tings and the mathematical results in matroidsubmodular function theory are generalized. Discrete convex analysis is a novel paradigm for discrete optimization that combines the ideas in continuous optimization convex analysis and combinatorial optimization matroidsubmodular function theory to establish a unified theoretical framework for nonlinear discrete optimization. Discrete convex analysis murota,1998,2003 is a general theoretical framework constructed through a combination of convex analysis and combinatorial mathematics.
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