Convexity

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A Course in Convexity

Author : Alexander Barvinok
Publisher : American Mathematical Soc.
ISBN 13 : 0821829688
Total Pages : 378 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis A Course in Convexity by : Alexander Barvinok

Download or read book A Course in Convexity written by Alexander Barvinok and published by American Mathematical Soc.. This book was released on 2002-11-19 with total page 378 pages. Available in PDF, EPUB and Kindle. Book excerpt: Convexity is a simple idea that manifests itself in a surprising variety of places. This fertile field has an immensely rich structure and numerous applications. Barvinok demonstrates that simplicity, intuitive appeal, and the universality of applications make teaching (and learning) convexity a gratifying experience. The book will benefit both teacher and student: It is easy to understand, entertaining to the reader, and includes many exercises that vary in degree of difficulty. Overall, the author demonstrates the power of a few simple unifying principles in a variety of pure and applied problems. The prerequisites are minimal amounts of linear algebra, analysis, and elementary topology, plus basic computational skills. Portions of the book could be used by advanced undergraduates. As a whole, it is designed for graduate students interested in mathematical methods, computer science, electrical engineering, and operations research. The book will also be of interest to research mathematicians, who will find some results that are recent, some that are new, and many known results that are discussed from a new perspective.

Convexity

Author : Roger Webster
Publisher : Oxford University Press on Demand
ISBN 13 : 9780198531470
Total Pages : 444 pages
Book Rating : 4.5/5 (314 download)

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Book Synopsis Convexity by : Roger Webster

Download or read book Convexity written by Roger Webster and published by Oxford University Press on Demand. This book was released on 1994 with total page 444 pages. Available in PDF, EPUB and Kindle. Book excerpt: Convexity provides a wide-ranging introduction for final year undergraduates and graduate students. Convex sets and functions are studied in the Euclidean space IRn, thus allowing an exposition demanding only an elementary knowledge of analysis and linear algebra, and enabling concepts to bemotivated through simple geometric examples. The fundemental ideas of convexity are natural and appealing, and does not have to travel far along its path, before meeting significant, aesthetically pleasing results. It develops geometric intuition, and is a showcase for displaying interconnections amongst different parts of mathematics, inaddition to have ties with economics, science and engineering. Despite being an active research field, it abounds in unsolved problems having an instant intuitive appeal. One distinctive feature of the book is the diverse applications that it highlights: number theory, geometric extremum problems, combinatorial geometry, linear programming, game theory, polytopes, bodies of constant width, the gamma function, minimax approximation, and linear, classical and matrixinequalities. Several topics make their first appearance in a general introduction to convexity, while a few have not appeared outside research journals. The account has a self-contained treatment of volume, thus permitting a rigorous discussion of mixed volumes, is operimetry and Brunn-Minkowskitheory. Full solutions to most of the 241 exercises are provided and detailed suggestions for further reading are given.

Convexity

Author : Barry Simon
Publisher : Cambridge University Press
ISBN 13 : 1139497596
Total Pages : 357 pages
Book Rating : 4.1/5 (394 download)

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Book Synopsis Convexity by : Barry Simon

Download or read book Convexity written by Barry Simon and published by Cambridge University Press. This book was released on 2011-05-19 with total page 357 pages. Available in PDF, EPUB and Kindle. Book excerpt: Convexity is important in theoretical aspects of mathematics and also for economists and physicists. In this monograph the author provides a comprehensive insight into convex sets and functions including the infinite-dimensional case and emphasizing the analytic point of view. Chapter one introduces the reader to the basic definitions and ideas that play central roles throughout the book. The rest of the book is divided into four parts: convexity and topology on infinite-dimensional spaces; Loewner's theorem; extreme points of convex sets and related issues, including the Krein–Milman theorem and Choquet theory; and a discussion of convexity and inequalities. The connections between disparate topics are clearly explained, giving the reader a thorough understanding of how convexity is useful as an analytic tool. A final chapter overviews the subject's history and explores further some of the themes mentioned earlier. This is an excellent resource for anyone interested in this central topic.

Duration, Convexity, and Other Bond Risk Measures

Author : Frank J. Fabozzi
Publisher : John Wiley & Sons
ISBN 13 : 9781883249632
Total Pages : 270 pages
Book Rating : 4.2/5 (496 download)

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Book Synopsis Duration, Convexity, and Other Bond Risk Measures by : Frank J. Fabozzi

Download or read book Duration, Convexity, and Other Bond Risk Measures written by Frank J. Fabozzi and published by John Wiley & Sons. This book was released on 1999-05-15 with total page 270 pages. Available in PDF, EPUB and Kindle. Book excerpt: Duration, Convexity and other Bond Risk Measures offers the most comprehensive coverage of bond risk measures available. Financial expert Frank Fabozzi walks you through every aspect of bond risk measures from the price volatility characteristics of option-free bonds and bonds with embedded options to the proper method for calculating duration and convexity. Whether you're a novice trader or experienced money manager, if you need to understand the interest rate risk of a portfolio Duration, Convexity and other Bond Risk Measures is the only book you'll need.

Convexity and Optimization in Rn

Author : Leonard D. Berkovitz
Publisher : John Wiley & Sons
ISBN 13 : 0471461660
Total Pages : 283 pages
Book Rating : 4.4/5 (714 download)

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Book Synopsis Convexity and Optimization in Rn by : Leonard D. Berkovitz

Download or read book Convexity and Optimization in Rn written by Leonard D. Berkovitz and published by John Wiley & Sons. This book was released on 2003-04-14 with total page 283 pages. Available in PDF, EPUB and Kindle. Book excerpt: A comprehensive introduction to convexity and optimization inRn This book presents the mathematics of finite dimensionalconstrained optimization problems. It provides a basis for thefurther mathematical study of convexity, of more generaloptimization problems, and of numerical algorithms for the solutionof finite dimensional optimization problems. For readers who do nothave the requisite background in real analysis, the author providesa chapter covering this material. The text features abundantexercises and problems designed to lead the reader to a fundamentalunderstanding of the material. Convexity and Optimization in Rn provides detailed discussionof: * Requisite topics in real analysis * Convex sets * Convex functions * Optimization problems * Convex programming and duality * The simplex method A detailed bibliography is included for further study and an indexoffers quick reference. Suitable as a text for both graduate andundergraduate students in mathematics and engineering, thisaccessible text is written from extensively class-tested notes.

Convexity and Well-Posed Problems

Author : Roberto Lucchetti
Publisher : Springer Science & Business Media
ISBN 13 : 0387287191
Total Pages : 308 pages
Book Rating : 4.3/5 (872 download)

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Book Synopsis Convexity and Well-Posed Problems by : Roberto Lucchetti

Download or read book Convexity and Well-Posed Problems written by Roberto Lucchetti and published by Springer Science & Business Media. This book was released on 2005-11-16 with total page 308 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book deals mainly with the study of convex functions and their behavior from the point of view of stability with respect to perturbations. We shall consider convex functions from the most modern point of view: a function is de?ned to be convex whenever its epigraph, the set of the points lying above the graph, is a convex set. Thus many of its properties can be seen also as properties of a certain convex set related to it. Moreover, we shall consider extended real valued functions, i. e. , functions taking possibly the values?? and +?. The reason for considering the value +? is the powerful device of including the constraint set of a constrained minimum problem into the objective function itself (by rede?ning it as +? outside the constraint set). Except for trivial cases, the minimum value must be taken at a point where the function is not +?, hence at a point in the constraint set. And the value ?? is allowed because useful operations, such as the inf-convolution, can give rise to functions valued?? even when the primitive objects are real valued. Observe that de?ning the objective function to be +? outside the closed constraint set preserves lower semicontinuity, which is the pivotal and mi- mal continuity assumption one needs when dealing with minimum problems. Variational calculus is usually based on derivatives.

Notions of Convexity

Author : Lars Hörmander
Publisher : Springer Science & Business Media
ISBN 13 : 0817645853
Total Pages : 424 pages
Book Rating : 4.8/5 (176 download)

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Book Synopsis Notions of Convexity by : Lars Hörmander

Download or read book Notions of Convexity written by Lars Hörmander and published by Springer Science & Business Media. This book was released on 2007-06-25 with total page 424 pages. Available in PDF, EPUB and Kindle. Book excerpt: The first two chapters of this book are devoted to convexity in the classical sense, for functions of one and several real variables respectively. This gives a background for the study in the following chapters of related notions which occur in the theory of linear partial differential equations and complex analysis such as (pluri-)subharmonic functions, pseudoconvex sets, and sets which are convex for supports or singular supports with respect to a differential operator. In addition, the convexity conditions which are relevant for local or global existence of holomorphic differential equations are discussed.

Polynomial Convexity

Author : Edgar Lee Stout
Publisher : Springer Science & Business Media
ISBN 13 : 0817645373
Total Pages : 454 pages
Book Rating : 4.8/5 (176 download)

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Book Synopsis Polynomial Convexity by : Edgar Lee Stout

Download or read book Polynomial Convexity written by Edgar Lee Stout and published by Springer Science & Business Media. This book was released on 2007-05-03 with total page 454 pages. Available in PDF, EPUB and Kindle. Book excerpt: This comprehensive monograph details polynomially convex sets. It presents the general properties of polynomially convex sets with particular attention to the theory of the hulls of one-dimensional sets. Coverage examines in considerable detail questions of uniform approximation for the most part on compact sets but with some attention to questions of global approximation on noncompact sets. The book also discusses important applications and motivates the reader with numerous examples and counterexamples, which serve to illustrate the general theory and to delineate its boundaries.

Generalized Convexity and Optimization

Author : Alberto Cambini
Publisher : Springer Science & Business Media
ISBN 13 : 3540708766
Total Pages : 252 pages
Book Rating : 4.5/5 (47 download)

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Book Synopsis Generalized Convexity and Optimization by : Alberto Cambini

Download or read book Generalized Convexity and Optimization written by Alberto Cambini and published by Springer Science & Business Media. This book was released on 2008-10-14 with total page 252 pages. Available in PDF, EPUB and Kindle. Book excerpt: The authors have written a rigorous yet elementary and self-contained book to present, in a unified framework, generalized convex functions. The book also includes numerous exercises and two appendices which list the findings consulted.

Infinite-Dimensional Optimization and Convexity

Author : Ivar Ekeland
Publisher : University of Chicago Press
ISBN 13 : 0226199886
Total Pages : 175 pages
Book Rating : 4.2/5 (261 download)

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Book Synopsis Infinite-Dimensional Optimization and Convexity by : Ivar Ekeland

Download or read book Infinite-Dimensional Optimization and Convexity written by Ivar Ekeland and published by University of Chicago Press. This book was released on 1983-09-15 with total page 175 pages. Available in PDF, EPUB and Kindle. Book excerpt: The caratheodory approach; Infinite-dimensional optimization; Duality theory.

Convexity

Author : H. G. Eggleston
Publisher : CUP Archive
ISBN 13 : 9780521077347
Total Pages : 160 pages
Book Rating : 4.0/5 (773 download)

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Book Synopsis Convexity by : H. G. Eggleston

Download or read book Convexity written by H. G. Eggleston and published by CUP Archive. This book was released on 1958 with total page 160 pages. Available in PDF, EPUB and Kindle. Book excerpt: This account of convexity includes the basic properties of convex sets in Euclidean space and their applications, the theory of convex functions and an outline of the results of transformations and combinations of convex sets. It will be useful for those concerned with the many applications of convexity in economics, the theory of games, the theory of functions, topology, geometry and the theory of numbers.

Complex Convexity and Analytic Functionals

Author : Mats Andersson
Publisher : Birkhäuser
ISBN 13 : 3034878710
Total Pages : 164 pages
Book Rating : 4.0/5 (348 download)

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Book Synopsis Complex Convexity and Analytic Functionals by : Mats Andersson

Download or read book Complex Convexity and Analytic Functionals written by Mats Andersson and published by Birkhäuser. This book was released on 2012-12-06 with total page 164 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book puts the modern theory of complex linear convexity on a solid footing, and gives a thorough and up-to-date survey of its current status. Applications include the Fantappié transformation of analytic functionals, integral representation formulas, polynomial interpolation, and solutions to linear partial differential equations.

Convexity and Optimization in Finite Dimensions I

Author : Josef Stoer
Publisher : Springer Science & Business Media
ISBN 13 : 3642462162
Total Pages : 306 pages
Book Rating : 4.6/5 (424 download)

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Book Synopsis Convexity and Optimization in Finite Dimensions I by : Josef Stoer

Download or read book Convexity and Optimization in Finite Dimensions I written by Josef Stoer and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 306 pages. Available in PDF, EPUB and Kindle. Book excerpt: Dantzig's development of linear programming into one of the most applicable optimization techniques has spread interest in the algebra of linear inequalities, the geometry of polyhedra, the topology of convex sets, and the analysis of convex functions. It is the goal of this volume to provide a synopsis of these topics, and thereby the theoretical back ground for the arithmetic of convex optimization to be treated in a sub sequent volume. The exposition of each chapter is essentially independent, and attempts to reflect a specific style of mathematical reasoning. The emphasis lies on linear and convex duality theory, as initiated by Gale, Kuhn and Tucker, Fenchel, and v. Neumann, because it represents the theoretical development whose impact on modern optimi zation techniques has been the most pronounced. Chapters 5 and 6 are devoted to two characteristic aspects of duality theory: conjugate functions or polarity on the one hand, and saddle points on the other. The Farkas lemma on linear inequalities and its generalizations, Motzkin's description of polyhedra, Minkowski's supporting plane theorem are indispensable elementary tools which are contained in chapters 1, 2 and 3, respectively. The treatment of extremal properties of polyhedra as well as of general convex sets is based on the far reaching work of Klee. Chapter 2 terminates with a description of Gale diagrams, a recently developed successful technique for exploring polyhedral structures.

Convexity and Optimization in Banach Spaces

Author : Viorel Barbu
Publisher : Springer Science & Business Media
ISBN 13 : 940072246X
Total Pages : 376 pages
Book Rating : 4.4/5 (7 download)

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Book Synopsis Convexity and Optimization in Banach Spaces by : Viorel Barbu

Download or read book Convexity and Optimization in Banach Spaces written by Viorel Barbu and published by Springer Science & Business Media. This book was released on 2012-01-03 with total page 376 pages. Available in PDF, EPUB and Kindle. Book excerpt: An updated and revised edition of the 1986 title Convexity and Optimization in Banach Spaces, this book provides a self-contained presentation of basic results of the theory of convex sets and functions in infinite-dimensional spaces. The main emphasis is on applications to convex optimization and convex optimal control problems in Banach spaces. A distinctive feature is a strong emphasis on the connection between theory and application. This edition has been updated to include new results pertaining to advanced concepts of subdifferential for convex functions and new duality results in convex programming. The last chapter, concerned with convex control problems, has been rewritten and completed with new research concerning boundary control systems, the dynamic programming equations in optimal control theory and periodic optimal control problems. Finally, the structure of the book has been modified to highlight the most recent progression in the field including fundamental results on the theory of infinite-dimensional convex analysis and includes helpful bibliographical notes at the end of each chapter.

Geometry and Convexity

Author : Paul J. Kelly
Publisher :
ISBN 13 : 9780486469805
Total Pages : 0 pages
Book Rating : 4.4/5 (698 download)

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Book Synopsis Geometry and Convexity by : Paul J. Kelly

Download or read book Geometry and Convexity written by Paul J. Kelly and published by . This book was released on 2009 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: This text assumes no prerequisites, offering an easy-to-read treatment with simple notation and clear, complete proofs. From motivation to definition, its explanations feature concrete examples and theorems. 1979 edition.

Metric Spaces, Convexity and Nonpositive Curvature

Author : Athanase Papadopoulos
Publisher : European Mathematical Society
ISBN 13 : 9783037190104
Total Pages : 306 pages
Book Rating : 4.1/5 (91 download)

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Book Synopsis Metric Spaces, Convexity and Nonpositive Curvature by : Athanase Papadopoulos

Download or read book Metric Spaces, Convexity and Nonpositive Curvature written by Athanase Papadopoulos and published by European Mathematical Society. This book was released on 2005 with total page 306 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Abstract Convexity and Global Optimization

Author : Alexander M. Rubinov
Publisher : Springer Science & Business Media
ISBN 13 : 1475732007
Total Pages : 506 pages
Book Rating : 4.4/5 (757 download)

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Book Synopsis Abstract Convexity and Global Optimization by : Alexander M. Rubinov

Download or read book Abstract Convexity and Global Optimization written by Alexander M. Rubinov and published by Springer Science & Business Media. This book was released on 2013-03-14 with total page 506 pages. Available in PDF, EPUB and Kindle. Book excerpt: Special tools are required for examining and solving optimization problems. The main tools in the study of local optimization are classical calculus and its modern generalizions which form nonsmooth analysis. The gradient and various kinds of generalized derivatives allow us to ac complish a local approximation of a given function in a neighbourhood of a given point. This kind of approximation is very useful in the study of local extrema. However, local approximation alone cannot help to solve many problems of global optimization, so there is a clear need to develop special global tools for solving these problems. The simplest and most well-known area of global and simultaneously local optimization is convex programming. The fundamental tool in the study of convex optimization problems is the subgradient, which actu ally plays both a local and global role. First, a subgradient of a convex function f at a point x carries out a local approximation of f in a neigh bourhood of x. Second, the subgradient permits the construction of an affine function, which does not exceed f over the entire space and coincides with f at x. This affine function h is called a support func tion. Since f(y) ~ h(y) for ally, the second role is global. In contrast to a local approximation, the function h will be called a global affine support.