Discrete Geometry And Optimization

Author : Károly Bezdek,Antoine Deza,Yinyu Ye
Publisher : Springer Science & Business Media
Page : 341 pages
File Size : 47,6 Mb
Release : 2013-07-09
Category : Mathematics
ISBN : 9783319002002

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Discrete Geometry and Optimization by Károly Bezdek,Antoine Deza,Yinyu Ye Pdf

​Optimization has long been a source of both inspiration and applications for geometers, and conversely, discrete and convex geometry have provided the foundations for many optimization techniques, leading to a rich interplay between these subjects. The purpose of the Workshop on Discrete Geometry, the Conference on Discrete Geometry and Optimization, and the Workshop on Optimization, held in September 2011 at the Fields Institute, Toronto, was to further stimulate the interaction between geometers and optimizers. This volume reflects the interplay between these areas. The inspiring Fejes Tóth Lecture Series, delivered by Thomas Hales of the University of Pittsburgh, exemplified this approach. While these fields have recently witnessed a lot of activity and successes, many questions remain open. For example, Fields medalist Stephen Smale stated that the question of the existence of a strongly polynomial time algorithm for linear optimization is one of the most important unsolved problems at the beginning of the 21st century. The broad range of topics covered in this volume demonstrates the many recent and fruitful connections between different approaches, and features novel results and state-of-the-art surveys as well as open problems.

Author : Jesus A. De Loera,Raymond Hemmecke,Matthias K?ppe
Publisher : SIAM
Page : 320 pages
File Size : 44,5 Mb
Release : 2013-01-31
Category : Mathematics
ISBN : 9781611972436

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Algebraic and Geometric Ideas in the Theory of Discrete Optimization by Jesus A. De Loera,Raymond Hemmecke,Matthias K?ppe Pdf

In recent years, many new techniques have emerged in the mathematical theory of discrete optimization that have proven to be effective in solving a number of hard problems. This book presents these recent advances, particularly those that arise from algebraic geometry, commutative algebra, convex and discrete geometry, generating functions, and other tools normally considered outside of the standard curriculum in optimization. These new techniques, all of which are presented with minimal prerequisites, provide a transition from linear to nonlinear discrete optimization. This book can be used as a textbook for advanced undergraduates or first-year graduate students in mathematics, computer science or operations research. It is also appropriate for mathematicians, engineers, and scientists engaged in computation who wish to gain a deeper understanding of how and why algorithms work.

Author : Csaba D. Toth,Joseph O'Rourke,Jacob E. Goodman
Publisher : CRC Press
Page : 1928 pages
File Size : 49,8 Mb
Release : 2017-11-22
Category : Computers
ISBN : 9781498711425

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Handbook of Discrete and Computational Geometry by Csaba D. Toth,Joseph O'Rourke,Jacob E. Goodman Pdf

The Handbook of Discrete and Computational Geometry is intended as a reference book fully accessible to nonspecialists as well as specialists, covering all major aspects of both fields. The book offers the most important results and methods in discrete and computational geometry to those who use them in their work, both in the academic world—as researchers in mathematics and computer science—and in the professional world—as practitioners in fields as diverse as operations research, molecular biology, and robotics. Discrete geometry has contributed significantly to the growth of discrete mathematics in recent years. This has been fueled partly by the advent of powerful computers and by the recent explosion of activity in the relatively young field of computational geometry. This synthesis between discrete and computational geometry lies at the heart of this Handbook. A growing list of application fields includes combinatorial optimization, computer-aided design, computer graphics, crystallography, data analysis, error-correcting codes, geographic information systems, motion planning, operations research, pattern recognition, robotics, solid modeling, and tomography.

Author : Jiri Matousek
Publisher : Springer Science & Business Media
Page : 491 pages
File Size : 53,8 Mb
Release : 2013-12-01
Category : Mathematics
ISBN : 9781461300397

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Lectures on Discrete Geometry by Jiri Matousek Pdf

The main topics in this introductory text to discrete geometry include basics on convex sets, convex polytopes and hyperplane arrangements, combinatorial complexity of geometric configurations, intersection patterns and transversals of convex sets, geometric Ramsey-type results, and embeddings of finite metric spaces into normed spaces. In each area, the text explains several key results and methods.

Author : Marston D. E. Conder,Antoine Deza,Asia Ivić Weiss
Publisher : Springer
Page : 333 pages
File Size : 44,6 Mb
Release : 2018-06-11
Category : Mathematics
ISBN : 9783319784342

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Discrete Geometry and Symmetry by Marston D. E. Conder,Antoine Deza,Asia Ivić Weiss Pdf

This book consists of contributions from experts, presenting a fruitful interplay between different approaches to discrete geometry. Most of the chapters were collected at the conference “Geometry and Symmetry” in Veszprém, Hungary from 29 June to 3 July 2015. The conference was dedicated to Károly Bezdek and Egon Schulte on the occasion of their 60th birthdays, acknowledging their highly regarded contributions in these fields. While the classical problems of discrete geometry have a strong connection to geometric analysis, coding theory, symmetry groups, and number theory, their connection to combinatorics and optimization has become of particular importance. The last decades have seen a revival of interest in discrete geometric structures and their symmetry. The rapid development of abstract polytope theory has resulted in a rich theory featuring an attractive interplay of methods and tools from discrete geometry, group theory and geometry, combinatorial group theory, and hyperbolic geometry and topology. This book contains papers on new developments in these areas, including convex and abstract polytopes and their recent generalizations, tiling and packing, zonotopes, isoperimetric inequalities, and on the geometric and combinatorial aspects of linear optimization. The book is a valuable resource for researchers, both junior and senior, in the field of discrete geometry, combinatorics, or discrete optimization. Graduate students find state-of-the-art surveys and an open problem collection.

Author : Li M. Chen
Publisher : Springer
Page : 325 pages
File Size : 42,7 Mb
Release : 2014-12-12
Category : Computers
ISBN : 9783319120997

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Digital and Discrete Geometry by Li M. Chen Pdf

This book provides comprehensive coverage of the modern methods for geometric problems in the computing sciences. It also covers concurrent topics in data sciences including geometric processing, manifold learning, Google search, cloud data, and R-tree for wireless networks and BigData. The author investigates digital geometry and its related constructive methods in discrete geometry, offering detailed methods and algorithms. The book is divided into five sections: basic geometry; digital curves, surfaces and manifolds; discretely represented objects; geometric computation and processing; and advanced topics. Chapters especially focus on the applications of these methods to other types of geometry, algebraic topology, image processing, computer vision and computer graphics. Digital and Discrete Geometry: Theory and Algorithms targets researchers and professionals working in digital image processing analysis, medical imaging (such as CT and MRI) and informatics, computer graphics, computer vision, biometrics, and information theory. Advanced-level students in electrical engineering, mathematics, and computer science will also find this book useful as a secondary text book or reference. Praise for this book: This book does present a large collection of important concepts, of mathematical, geometrical, or algorithmical nature, that are frequently used in computer graphics and image processing. These concepts range from graphs through manifolds to homology. Of particular value are the sections dealing with discrete versions of classic continuous notions. The reader finds compact definitions and concise explanations that often appeal to intuition, avoiding finer, but then necessarily more complicated, arguments... As a first introduction, or as a reference for professionals working in computer graphics or image processing, this book should be of considerable value." - Prof. Dr. Rolf Klein, University of Bonn.

Author : Christer Oscar Kiselman
Publisher : World Scientific
Page : 488 pages
File Size : 46,8 Mb
Release : 2022-01-06
Category : Mathematics
ISBN : 9789811248313

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Elements Of Digital Geometry, Mathematical Morphology, And Discrete Optimization by Christer Oscar Kiselman Pdf

The author presents three distinct but related branches of science in this book: digital geometry, mathematical morphology, and discrete optimization. They are united by a common mindset as well as by the many applications where they are useful. In addition to being useful, each of these relatively new branches of science is also intellectually challenging.The book contains a systematic study of inverses of mappings between ordered sets, and so offers a uniquely helpful organization in the approach to several phenomena related to duality.To prepare the ground for discrete convexity, there are chapters on convexity in real vector spaces in anticipation of the many challenging problems coming up in digital geometry. To prepare for the study of new topologies introduced to serve in discrete spaces, there is also a chapter on classical topology.The book is intended for general readers with a modest background in mathematics and for advanced undergraduate students as well as beginning graduate students.

Author : Peter M. Gruber
Publisher : Springer Science & Business Media
Page : 590 pages
File Size : 45,9 Mb
Release : 2007-05-17
Category : Mathematics
ISBN : 9783540711339

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Convex and Discrete Geometry by Peter M. Gruber Pdf

Convex and Discrete Geometry is an area of mathematics situated between analysis, geometry and discrete mathematics with numerous relations to other subdisciplines. This book provides a comprehensive overview of major results, methods and ideas of convex and discrete geometry and its applications. Besides being a graduate-level introduction to the field, it is a practical source of information and orientation for convex geometers, and useful to people working in the applied fields.

Author : Vladimir Boltyanski,Horst Martini,V. Soltan
Publisher : Springer Science & Business Media
Page : 438 pages
File Size : 51,6 Mb
Release : 2013-12-11
Category : Mathematics
ISBN : 9781461553199

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Geometric Methods and Optimization Problems by Vladimir Boltyanski,Horst Martini,V. Soltan Pdf

VII Preface In many fields of mathematics, geometry has established itself as a fruitful method and common language for describing basic phenomena and problems as well as suggesting ways of solutions. Especially in pure mathematics this is ob vious and well-known (examples are the much discussed interplay between lin ear algebra and analytical geometry and several problems in multidimensional analysis). On the other hand, many specialists from applied mathematics seem to prefer more formal analytical and numerical methods and representations. Nevertheless, very often the internal development of disciplines from applied mathematics led to geometric models, and occasionally breakthroughs were b~ed on geometric insights. An excellent example is the Klee-Minty cube, solving a problem of linear programming by transforming it into a geomet ric problem. Also the development of convex programming in recent decades demonstrated the power of methods that evolved within the field of convex geometry. The present book focuses on three applied disciplines: control theory, location science and computational geometry. It is our aim to demonstrate how methods and topics from convex geometry in a wider sense (separation theory of convex cones, Minkowski geometry, convex partitionings, etc.) can help to solve various problems from these disciplines.

Author : Ji?í Matoušek
Publisher : Springer
Page : 486 pages
File Size : 45,9 Mb
Release : 2002-05-02
Category : Mathematics
ISBN : 0387953744

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Lectures on Discrete Geometry by Ji?í Matoušek Pdf

The main topics in this introductory text to discrete geometry include basics on convex sets, convex polytopes and hyperplane arrangements, combinatorial complexity of geometric configurations, intersection patterns and transversals of convex sets, geometric Ramsey-type results, and embeddings of finite metric spaces into normed spaces. In each area, the text explains several key results and methods.

Author : J. Matou Ek
Publisher : Unknown
Page : 504 pages
File Size : 45,8 Mb
Release : 2014-09-01
Category : Electronic
ISBN : 1461300401

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Lectures on Discrete Geometry by J. Matou Ek Pdf

Author : Ullrich Köthe,Annick Montanvert,Pierre Soille
Publisher : Springer
Page : 175 pages
File Size : 51,7 Mb
Release : 2012-07-30
Category : Computers
ISBN : 9783642323133

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Applications of Discrete Geometry and Mathematical Morphology by Ullrich Köthe,Annick Montanvert,Pierre Soille Pdf

This book constitutes the refereed proceedings of the first Workshop on Applications of Discrete Geometry and Mathematical Morphology, WADGMM 2010, held at the International Conference on Pattern Recognition in Istanbul, Turkey, in August 2010. The 11 revised full papers presented were carefully reviewed and selected from 25 submissions. The book was specifically designed to promote interchange and collaboration between experts in discrete geometry/mathematical morphology and potential users of these methods from other fields of image analysis and pattern recognition.

Author : Martin Grötschel,Laszlo Lovasz,Alexander Schrijver
Publisher : Springer Science & Business Media
Page : 374 pages
File Size : 43,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783642978814

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Geometric Algorithms and Combinatorial Optimization by Martin Grötschel,Laszlo Lovasz,Alexander Schrijver Pdf

Historically, there is a close connection between geometry and optImization. This is illustrated by methods like the gradient method and the simplex method, which are associated with clear geometric pictures. In combinatorial optimization, however, many of the strongest and most frequently used algorithms are based on the discrete structure of the problems: the greedy algorithm, shortest path and alternating path methods, branch-and-bound, etc. In the last several years geometric methods, in particular polyhedral combinatorics, have played a more and more profound role in combinatorial optimization as well. Our book discusses two recent geometric algorithms that have turned out to have particularly interesting consequences in combinatorial optimization, at least from a theoretical point of view. These algorithms are able to utilize the rich body of results in polyhedral combinatorics. The first of these algorithms is the ellipsoid method, developed for nonlinear programming by N. Z. Shor, D. B. Yudin, and A. S. NemirovskiI. It was a great surprise when L. G. Khachiyan showed that this method can be adapted to solve linear programs in polynomial time, thus solving an important open theoretical problem. While the ellipsoid method has not proved to be competitive with the simplex method in practice, it does have some features which make it particularly suited for the purposes of combinatorial optimization. The second algorithm we discuss finds its roots in the classical "geometry of numbers", developed by Minkowski. This method has had traditionally deep applications in number theory, in particular in diophantine approximation.

Author : William Kocay,Donald L. Kreher
Publisher : CRC Press
Page : 504 pages
File Size : 51,7 Mb
Release : 2017-09-20
Category : Mathematics
ISBN : 9781351989121

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Graphs, Algorithms, and Optimization by William Kocay,Donald L. Kreher Pdf

Graph theory offers a rich source of problems and techniques for programming and data structure development, as well as for understanding computing theory, including NP-Completeness and polynomial reduction. A comprehensive text, Graphs, Algorithms, and Optimization features clear exposition on modern algorithmic graph theory presented in a rigorous yet approachable way. The book covers major areas of graph theory including discrete optimization and its connection to graph algorithms. The authors explore surface topology from an intuitive point of view and include detailed discussions on linear programming that emphasize graph theory problems useful in mathematics and computer science. Many algorithms are provided along with the data structure needed to program the algorithms efficiently. The book also provides coverage on algorithm complexity and efficiency, NP-completeness, linear optimization, and linear programming and its relationship to graph algorithms. Written in an accessible and informal style, this work covers nearly all areas of graph theory. Graphs, Algorithms, and Optimization provides a modern discussion of graph theory applicable to mathematics, computer science, and crossover applications.

Author : Shmuel Onn
Publisher : Unknown
Page : 200 pages
File Size : 48,5 Mb
Release : 1992
Category : Electronic
ISBN : CORNELL:31924063643591

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Discrete Geometry, Group Representations and Combinatorial Optimization by Shmuel Onn Pdf