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Geometric Methods and Optimization Problems : Combinatorial Optimization - Vladimir Boltianskii

Geometric Methods and Optimization Problems

Combinatorial Optimization

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Published: 31st December 1998
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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.

Nonclassical Variational Calculusp. 1
The classical problem of mathematical programmingp. 5
The abstract intersection problemp. 15
The tents (intuitive approach)p. 26
The tents (definition and justification of examples)p. 36
Separability of a system of convex conesp. 51
The Topological Lemmap. 63
The Kuhn-Tucker Theoremp. 78
The Maximum Principle (Mayer's problem)p. 92
The Maximum Principle (Lagrange's and Bolza's problems)p. 110
Classical variational calculusp. 126
The Maximum Principle (synthesis of optimal trajectories)p. 141
The Maximum Principle (method of local sections)p. 153
The Maximum Principle (sufficient condition for time-optimality)p. 170
The Robust Maximum Principlep. 181
Minimax extremal problemsp. 197
The maximum principle - how it came to be?p. 204
Median problems in location sciencep. 231
On location sciencep. 231
The classical Fermat-Torricelli problemp. 235
On the location of p[subscript min]p. 253
The problem in Minkowski spacesp. 270
Median k-flats in Euclidean n-spacep. 279
Median k-flats in Minkowski spacesp. 295
Historical surveyp. 312
Minimum Convex Partitions of Polygonal Domainsp. 357
Preliminariesp. 358
Polygonal Domainsp. 368
Minimum Convex Guillotine F-Partitionp. 375
Minimum F-Partition into Trapezoidsp. 387
Minimum Convex F-Partitionp. 398
Complexity Status of the Minimum Convex F-Partition Problemp. 409
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9780792354543
ISBN-10: 0792354540
Series: Combinatorial Optimization
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 432
Published: 31st December 1998
Publisher: Springer
Country of Publication: NL
Dimensions (cm): 23.4 x 15.6  x 2.54
Weight (kg): 1.76