Geometric Methods and Optimization Problems

By: Vladimir Boltyanski, Horst Martini, V. Soltan

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Hardcover | 31 December 1998

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444 Pages

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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.

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Published: 9th January 2014

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Nonclassical Variational Calculus	p. 1
The classical problem of mathematical programming	p. 5
The abstract intersection problem	p. 15
The tents (intuitive approach)	p. 26
The tents (definition and justification of examples)	p. 36
Separability of a system of convex cones	p. 51
The Topological Lemma	p. 63
The Kuhn-Tucker Theorem	p. 78
The Maximum Principle (Mayer's problem)	p. 92
The Maximum Principle (Lagrange's and Bolza's problems)	p. 110
Classical variational calculus	p. 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 Principle	p. 181
Minimax extremal problems	p. 197
The maximum principle - how it came to be?	p. 204
Median problems in location science	p. 231
On location science	p. 231
The classical Fermat-Torricelli problem	p. 235
On the location of p[subscript min]	p. 253
The problem in Minkowski spaces	p. 270
Median k-flats in Euclidean n-space	p. 279
Median k-flats in Minkowski spaces	p. 295
Historical survey	p. 312
Minimum Convex Partitions of Polygonal Domains	p. 357
Preliminaries	p. 358
Polygonal Domains	p. 368
Minimum Convex Guillotine F-Partition	p. 375
Minimum F-Partition into Trapezoids	p. 387
Minimum Convex F-Partition	p. 398
Complexity Status of the Minimum Convex F-Partition Problem	p. 409
Table of Contents provided by Blackwell. All Rights Reserved.

Geometric Methods and Optimization Problems

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