Convex Analysis (Princeton Landmarks in Mathematics and Physics)

Book Description

Available for the first time in paperback, R. Tyrrell Rockafellar's classic study presents readers with a coherent branch of nonlinear mathematical analysis that is especially suited to the study of optimization problems. Rockafellar's theory differs from classical analysis in that differentiability assumptions are replaced by convexity assumptions. The topics treated in this volume include: systems of inequalities, the minimum or maximum of a convex function over a convex set, Lagrange multipliers, minimax theorems and duality, as well as basic results about the structure of convex sets and the continuity and differentiability of convex functions and saddle- functions.

This book has firmly established a new and vital area not only for pure mathematics but also for applications to economics and engineering. A sound knowledge of linear algebra and introductory real analysis should provide readers with sufficient background for this book. There is also a guide for the reader who may be using the book as an introduction, indicating which parts are essential and which may be skipped on a first reading.

Customer Reviews:

Very Good.......2007-05-09

This book is a classic. It is probably the best reference book although it is tough to read from the beginning untill the end. The style is heavy and you need strong mathematical background to understand it.
Anyway, if you need a result on convex functions or convex analysis it is very likely that you will find it in ths book.

the only thing it lacks is diagrams.......2007-01-12

This book perhaps ranks with Halmos' "Finite Dimensional Vector Spaces" as an unusually clear description of its subject. Rockafellar's book has been through numerous printings in 40 years. The theorem proofs can be intricate. But the thread of logical development makes reading it worthwhile. Certainly, it is beautiful how the crucial assumption of convexity makes all the derivations possible.

The level of discussion is suitable for a 3rd year undergrad [or higher], who is majoring in maths.

By current standards of maths texts, it does lack diagrams. In fact, there doesn't seem to be a single one! Something to get used to, if you are a current undergrad weaned on recent texts. [Since the author is still alive, perhaps he might consider adding diagrams to a future edition.]

a beauty mummified .......2005-12-02

convex programming is a beautiful topic which admits amazing geometric interpretation.

books like this manage to destroy one's appreciation of the topic by not providing even one (gasp!) figure. damn Bourbaki style.

Excellent introduction to Topology........2001-05-10

This is a good book for the first year in PhD studies. I recommend amply this book, it's very clear in the explanation, if you have any doubts about topology, Rockafellar explained in this book very simple the theory and all you need about Topology.

Infinite Dimensional Analysis: A Hitchhiker's Guide

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An excellent treatment of mathematical methods for economist

Infinite Dimensional Analysis: A Hitchhiker's Guide
Charalambos D. Aliprantis , and Kim C. Border
Manufacturer: Springer
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Topological Spaces: Including a Treatment of Multi-Valued Functions, Vector Spaces and Convexity

ASIN: 3540326960

Book Description

This monograph presents a complete and rigorous study of modern functional analysis. It is intended for the student or researcher who could benefit from functional analytic methods, but does not have an extensive background and does not plan to make a career as a functional analyst. It develops the topological structures in connection with measure theory, convexity, Banach lattices, integration, correspondences (multifunctions), and the analytic approach to Markov processes. Many of the results were previously available only in works scattered throughout the literature. The choice of material was motivated from problems in control theory and economics, although the material is more applicable than applied.

Customer Reviews:

An excellent treatment of mathematical methods for economist.......1998-10-20

The monograph covers advanced mathematical methods for economists. It includes chapters on general topology, topological vector spaces, Riesz spaces and Banach lattices, measure and integration, etc. While the book does not contain (hardly) any economics, the mathematics covered is selected under the aspect of later applications to economics. The book contains for example a long chapter on correspondences, a topic which is hardly covered by any standard math book. The presentation of the mathematics is throughout clear and precise. The advantage of the book is that it covers a wide range of mathematical topics, which could not be found together in a book before. Graduate students in economic theory can use it as a text book, but it can also be used as a reference book. The only lacks of the book are that there are no exercises and that not all math areas important to economics (e.g. differential topology) are covered. Overall, this is an excellent book and should become part of the library of everybody interested in mathematical economics.

A Mathematical View of Interior-Point Methods in Convex Optimization (MPS-SIAM Series on Optimization)

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A simplifying perspective of IPM's in Convex Optimization
A simplifying perspective of IPM's in Convex Optimization

A Mathematical View of Interior-Point Methods in Convex Optimization (MPS-SIAM Series on Optimization)
James Renegar
Manufacturer: Society for Industrial Mathematics
ProductGroup: Book
Binding: Paperback

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Convex Analysis (Princeton Landmarks in Mathematics and Physics)
Convex Optimization

ASIN: 0898715024

Book Description

This compact book, through the simplifying perspective it presents, will take a reader who knows little of interior-point methods to within sight of the research frontier, developing key ideas that were over a decade in the making by numerous interior-point method researchers. It aims at developing a thorough understanding of the most general theory for interior-point methods, a class of algorithms for convex optimization problems. The study of these algorithms has dominated the continuous optimization literature for nearly 15 years. In that time, the theory has matured tremendously, but much of the literature is difficult to understand, even for specialists. By focusing only on essential elements of the theory and emphasizing the underlying geometry, A Mathematical View of Interior-Point Methods in Convex Optimization makes the theory accessible to a wide audience, allowing them to quickly develop a fundamental understanding of the material.

Customer Reviews:

A simplifying perspective of IPM's in Convex Optimization.......2001-11-09

At last, we find a book which develops a thorough understanding
of the most general theory of interior point methods for
convex optimization, and is easily accessible. As the author
himself remarks "Much of the literature on the general
theory of interior point methods is difficult to understand,
even for specialists. My hope is that this book will make
the most general theory accessible to a wide audience -
especially Ph.D. students, the next generation of optimizers".
The book covers basic interior point theory including
the theory of self concordant functionals. There is a chapter
on conic programming covering the relationship between interior point
methods and duality theory, and the development
of primal dual interior point algorithms for solving conic
optimization problems (Conic programming includes linear,
semidefinite and second order cone programming as special
cases!).

One can then "perhaps" take on Nesterov and Nemirovskii's
seminal treatise on Interior Point Polynomial Algorithms
in Convex Programming, one of the most widely cited references
in optimization, which I must confess is not exactly
an easy read.

A simplifying perspective of IPM's in Convex Optimization.......2001-11-09

Book Description

This book is an abridged version of the two volumes "Convex Analysis and Minimization Algorithms I and II" (Grundlehren der mathematischen Wissenschaften Vol. 305 and 306), which presented an introduction to the basic concepts in convex analysis and a study of convex minimization problems. The "backbone" of both volumes was extracted, some material deleted that was deemed too advanced for an introduction, or too closely related to numerical algorithms. Some exercises were included and finally the index has been considerably enriched.
The main motivation of the authors was to "light the entrance" of the monument Convex Analysis. This book is not a reference book to be kept on the shelf by experts who already know the building and can find their way through it; it is far more a book for the purpose of learning and teaching.

Convexity and Well-Posed Problems (CMS Books in Mathematics)

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Convexity and Well-Posed Problems (CMS Books in Mathematics)
Roberto Lucchetti
Manufacturer: Springer
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Binding: Hardcover

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ASIN: 0387287191

Book Description

This book deals with the study of convex functions and of their behavior from the point of view of stability with respect to perturbations. Convex functions are considered from the modern point of view that underlines the geometrical aspect: thus a function is defined as convex whenever its graph is a convex set. A primary goal of this book is to study the problems of stability and well-posedness, in the convex case. Stability means that the basic parameters of a minimum problem do not vary much if we slightly change the initial data. On the other hand, well-posedness means that points with values close to the value of the problem must be close to actual solutions. In studying this, one is naturally led to consider perturbations of functions and of sets. This approach fits perfectly with the idea of regarding functions as sets. Thus the second part of the book starts with a short, yet rather complete, overview of the so-called hypertopologies, i.e. topologies in the closed subsets of a metric space. While there exist numerous classic texts on the issue of stability, there only exists one book on hypertopologies [Beer 1993]. The current book differs from Beer’s in that it contains a much more condensed explication of hypertopologies and is intended to help those not familiar with hypertopologies learn how to use them in the context of optimization problems.

The Volume of Convex Bodies and Banach Space Geometry

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The Volume of Convex Bodies and Banach Space Geometry
Gilles Pisier
Manufacturer: Cambridge University Press
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Asymptotic Theory of Finite Dimensional Normed Spaces: Isoperimetric Inequalities in Riemannian Manifolds (Lecture Notes in Mathematics)

ASIN: 052166635X

Book Description

Now in paperback, this popular book gives a self-contained presentation of a number of recent results, which relate the volume of convex bodies in n-dimensional Euclidean space and the geometry of the corresponding finite-dimensional normed spaces. The methods employ classical ideas from the theory of convex sets, probability theory, approximation theory, and the local theory of Banach spaces. The first part of the book presents self-contained proofs of the quotient of the subspace theorem, the inverse Santalo inequality and the inverse Brunn-Minkowski inequality. In the second part Pisier gives a detailed exposition of the recently introduced classes of Banach spaces of weak cotype 2 or weak type 2, and the intersection of the classes (weak Hilbert space). This text will be a superb choice for courses in analysis and probability theory.

Notions of Convexity (Modern Birkhäuser Classics)

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Notions of Convexity (Modern Birkhäuser Classics)
Lars Hörmander
Manufacturer: Birkhäuser Boston
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ASIN: 0817645845

Book Description

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, leading up to Trépreau’s theorem on sufficiency of condition (capital Greek letter Psi) for microlocal solvability in the analytic category.

At the beginning of the book, no prerequisites are assumed beyond calculus and linear algebra. Later on, basic facts from distribution theory and functional analysis are needed. In a few places, a more extensive background in differential geometry or pseudodifferential calculus is required, but these sections can be bypassed with no loss of continuity. The major part of the book should therefore be accessible to graduate students so that it can serve as an introduction to complex analysis in one and several variables. The last sections, however, are written mainly for readers familiar with microlocal analysis.

Convex Functions and their Applications: A Contemporary Approach (CMS Books in Mathematics)

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Convex Functions and their Applications: A Contemporary Approach (CMS Books in Mathematics)
Constantin Niculescu , and Lars-Erik Persson
Manufacturer: Springer
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Binding: Hardcover

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Convex Analysis and Nonlinear Optimization: Theory and Examples (CMS Books in Mathematics)

ASIN: 0387243003

Book Description

Convex functions play an important role in almost all branches of mathematics as well as other areas of science and engineering. This book is a thorough introduction to contemporary convex function theory addressed to all people whose research or teaching interests intersect with the field of convexity. It covers a large variety of subjects, from the one real variable case (with all its mathematical gems) to some of the most advanced topics such as Choquet's theory, the Prékopa-Leindler type inequalities and their ramifications, as well as the variational approach of partial differential equations and convex programming. Many results are new and the whole book reflects the authors’ own experience, both in teaching and research. The book can serve as a reference and source of inspiration to researchers in several branches of mathematics and engineering and it can also be used for graduate courses.

Convex Analysis and Nonlinear Optimization: Theory and Examples (CMS Books in Mathematics)

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Convex Analysis and Nonlinear Optimization: Theory and Examples (CMS Books in Mathematics)
Jonathan Borwein , and Adrian S. Lewis
Manufacturer: Springer
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ASIN: 0387295704

Book Description

Optimization is a rich and thriving mathematical discipline. The theory underlying current computational optimization techniques grows ever more sophisticated. The powerful and elegant language of convex analysis unifies much of this theory. The aim of this book is to provide a concise, accessible account of convex analysis and its applications and extensions, for a broad audience. It can serve as a teaching text, at roughly the level of first year graduate students. While the main body of the text is self-contained, each section concludes with an often extensive set of optional exercises. The new edition adds material on semismooth optimization, as well as several new proofs that will make this book even more self-contained.

Convex Analysis: Theory and Applications (Translations of Mathematical Monographs)

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Convex Analysis: Theory and Applications (Translations of Mathematical Monographs)
G. G. Magaril-Ilyaev , and V. M. Tikhomirov
Manufacturer: American Mathematical Society
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Applied Nonlinear Analysis

ASIN: 0821835254

Book Description

This book is an introduction to convex analysis and some of its applications. It starts with basic theory, which is explained within the framework of finite-dimensional spaces. The only prerequisites are basic analysis and simple geometry. The second chapter presents some applications of convex analysis, including problems of linear programming, geometry, and approximation. Special attention is paid to applications of convex analysis to Kolmogorov-type inequalities for derivatives of functions in one variable. Chapter 3 collects some results on geometry and convex analysis in infinite-dimensional spaces. A comprehensive introduction written "for beginners" illustrates the fundamentals of convex analysis in finite-dimensional spaces.

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