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Analytic Aspects Of Convexity
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Book Synopsis Analytic Aspects of Convexity by : Gabriele Bianchi
Download or read book Analytic Aspects of Convexity written by Gabriele Bianchi and published by Springer. This book was released on 2018-02-28 with total page 125 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents the proceedings of the international conference Analytic Aspects in Convexity, which was held in Rome in October 2016. It offers a collection of selected articles, written by some of the world’s leading experts in the field of Convex Geometry, on recent developments in this area: theory of valuations; geometric inequalities; affine geometry; and curvature measures. The book will be of interest to a broad readership, from those involved in Convex Geometry, to those focusing on Functional Analysis, Harmonic Analysis, Differential Geometry, or PDEs. The book is a addressed to PhD students and researchers, interested in Convex Geometry and its links to analysis.
Book Synopsis Convex Analysis by : Steven G. Krantz
Download or read book Convex Analysis written by Steven G. Krantz and published by CRC Press. This book was released on 2014-10-20 with total page 174 pages. Available in PDF, EPUB and Kindle. Book excerpt: Convexity is an ancient idea going back to Archimedes. Used sporadically in the mathematical literature over the centuries, today it is a flourishing area of research and a mathematical subject in its own right. Convexity is used in optimization theory, functional analysis, complex analysis, and other parts of mathematics.Convex Analysis introduces
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 172 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.
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.
Book Synopsis Convex Analysis by : Ralph Tyrell Rockafellar
Download or read book Convex Analysis written by Ralph Tyrell Rockafellar and published by Princeton University Press. This book was released on 2015-04-29 with total page 470 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Convex Analysis and Beyond by : Boris S. Mordukhovich
Download or read book Convex Analysis and Beyond written by Boris S. Mordukhovich and published by Springer Nature. This book was released on 2022-04-24 with total page 597 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents a unified theory of convex functions, sets, and set-valued mappings in topological vector spaces with its specifications to locally convex, Banach and finite-dimensional settings. These developments and expositions are based on the powerful geometric approach of variational analysis, which resides on set extremality with its characterizations and specifications in the presence of convexity. Using this approach, the text consolidates the device of fundamental facts of generalized differential calculus to obtain novel results for convex sets, functions, and set-valued mappings in finite and infinite dimensions. It also explores topics beyond convexity using the fundamental machinery of convex analysis to develop nonconvex generalized differentiation and its applications. The text utilizes an adaptable framework designed with researchers as well as multiple levels of students in mind. It includes many exercises and figures suited to graduate classes in mathematical sciences that are also accessible to advanced students in economics, engineering, and other applications. In addition, it includes chapters on convex analysis and optimization in finite-dimensional spaces that will be useful to upper undergraduate students, whereas the work as a whole provides an ample resource to mathematicians and applied scientists, particularly experts in convex and variational analysis, optimization, and their applications.
Book Synopsis Advances in Convex Analysis and Global Optimization by : Nicolas Hadjisavvas
Download or read book Advances in Convex Analysis and Global Optimization written by Nicolas Hadjisavvas and published by Springer Science & Business Media. This book was released on 2013-12-01 with total page 601 pages. Available in PDF, EPUB and Kindle. Book excerpt: There has been much recent progress in global optimization algo rithms for nonconvex continuous and discrete problems from both a theoretical and a practical perspective. Convex analysis plays a fun damental role in the analysis and development of global optimization algorithms. This is due essentially to the fact that virtually all noncon vex optimization problems can be described using differences of convex functions and differences of convex sets. A conference on Convex Analysis and Global Optimization was held during June 5 -9, 2000 at Pythagorion, Samos, Greece. The conference was honoring the memory of C. Caratheodory (1873-1950) and was en dorsed by the Mathematical Programming Society (MPS) and by the Society for Industrial and Applied Mathematics (SIAM) Activity Group in Optimization. The conference was sponsored by the European Union (through the EPEAEK program), the Department of Mathematics of the Aegean University and the Center for Applied Optimization of the University of Florida, by the General Secretariat of Research and Tech nology of Greece, by the Ministry of Education of Greece, and several local Greek government agencies and companies. This volume contains a selective collection of refereed papers based on invited and contribut ing talks presented at this conference. The two themes of convexity and global optimization pervade this book. The conference provided a forum for researchers working on different aspects of convexity and global opti mization to present their recent discoveries, and to interact with people working on complementary aspects of mathematical programming.
Book Synopsis Convex Analysis and Optimization by : Dimitri Bertsekas
Download or read book Convex Analysis and Optimization written by Dimitri Bertsekas and published by Athena Scientific. This book was released on 2003-03-01 with total page 560 pages. Available in PDF, EPUB and Kindle. Book excerpt: A uniquely pedagogical, insightful, and rigorous treatment of the analytical/geometrical foundations of optimization. The book provides a comprehensive development of convexity theory, and its rich applications in optimization, including duality, minimax/saddle point theory, Lagrange multipliers, and Lagrangian relaxation/nondifferentiable optimization. It is an excellent supplement to several of our books: Convex Optimization Theory (Athena Scientific, 2009), Convex Optimization Algorithms (Athena Scientific, 2015), Nonlinear Programming (Athena Scientific, 2016), Network Optimization (Athena Scientific, 1998), and Introduction to Linear Optimization (Athena Scientific, 1997). Aside from a thorough account of convex analysis and optimization, the book aims to restructure the theory of the subject, by introducing several novel unifying lines of analysis, including: 1) A unified development of minimax theory and constrained optimization duality as special cases of duality between two simple geometrical problems. 2) A unified development of conditions for existence of solutions of convex optimization problems, conditions for the minimax equality to hold, and conditions for the absence of a duality gap in constrained optimization. 3) A unification of the major constraint qualifications allowing the use of Lagrange multipliers for nonconvex constrained optimization, using the notion of constraint pseudonormality and an enhanced form of the Fritz John necessary optimality conditions. Among its features the book: a) Develops rigorously and comprehensively the theory of convex sets and functions, in the classical tradition of Fenchel and Rockafellar b) Provides a geometric, highly visual treatment of convex and nonconvex optimization problems, including existence of solutions, optimality conditions, Lagrange multipliers, and duality c) Includes an insightful and comprehensive presentation of minimax theory and zero sum games, and its connection with duality d) Describes dual optimization, the associated computational methods, including the novel incremental subgradient methods, and applications in linear, quadratic, and integer programming e) Contains many examples, illustrations, and exercises with complete solutions (about 200 pages) posted at the publisher's web site http://www.athenasc.com/convexity.html
Book Synopsis Analytic and Algebraic Aspects of Convexity by : Liran Rotem
Download or read book Analytic and Algebraic Aspects of Convexity written by Liran Rotem and published by . This book was released on 2016 with total page 166 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Convex Analysis and Optimization in Hadamard Spaces by : Miroslav Bacak
Download or read book Convex Analysis and Optimization in Hadamard Spaces written by Miroslav Bacak and published by Walter de Gruyter GmbH & Co KG. This book was released on 2014-10-29 with total page 217 pages. Available in PDF, EPUB and Kindle. Book excerpt: In the past two decades, convex analysis and optimization have been developed in Hadamard spaces. This book represents a first attempt to give a systematic account on the subject. Hadamard spaces are complete geodesic spaces of nonpositive curvature. They include Hilbert spaces, Hadamard manifolds, Euclidean buildings and many other important spaces. While the role of Hadamard spaces in geometry and geometric group theory has been studied for a long time, first analytical results appeared as late as in the 1990s. Remarkably, it turns out that Hadamard spaces are appropriate for the theory of convex sets and convex functions outside of linear spaces. Since convexity underpins a large number of results in the geometry of Hadamard spaces, we believe that its systematic study is of substantial interest. Optimization methods then address various computational issues and provide us with approximation algorithms which may be useful in sciences and engineering. We present a detailed description of such an application to computational phylogenetics. The book is primarily aimed at both graduate students and researchers in analysis and optimization, but it is accessible to advanced undergraduate students as well.
Book Synopsis An Easy Path to Convex Analysis and Applications by : Boris Mordukhovich
Download or read book An Easy Path to Convex Analysis and Applications written by Boris Mordukhovich and published by Springer Nature. This book was released on 2022-05-31 with total page 202 pages. Available in PDF, EPUB and Kindle. Book excerpt: Convex optimization has an increasing impact on many areas of mathematics, applied sciences, and practical applications. It is now being taught at many universities and being used by researchers of different fields. As convex analysis is the mathematical foundation for convex optimization, having deep knowledge of convex analysis helps students and researchers apply its tools more effectively. The main goal of this book is to provide an easy access to the most fundamental parts of convex analysis and its applications to optimization. Modern techniques of variational analysis are employed to clarify and simplify some basic proofs in convex analysis and build the theory of generalized differentiation for convex functions and sets in finite dimensions. We also present new applications of convex analysis to location problems in connection with many interesting geometric problems such as the Fermat-Torricelli problem, the Heron problem, the Sylvester problem, and their generalizations. Of course, we do not expect to touch every aspect of convex analysis, but the book consists of sufficient material for a first course on this subject. It can also serve as supplemental reading material for a course on convex optimization and applications.
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.
Book Synopsis Handbook of Convex Geometry by : Bozzano G Luisa
Download or read book Handbook of Convex Geometry written by Bozzano G Luisa and published by Elsevier. This book was released on 2014-06-28 with total page 803 pages. Available in PDF, EPUB and Kindle. Book excerpt: Handbook of Convex Geometry, Volume A offers a survey of convex geometry and its many ramifications and relations with other areas of mathematics, including convexity, geometric inequalities, and convex sets. The selection first offers information on the history of convexity, characterizations of convex sets, and mixed volumes. Topics include elementary convexity, equality in the Aleksandrov-Fenchel inequality, mixed surface area measures, characteristic properties of convex sets in analysis and differential geometry, and extensions of the notion of a convex set. The text then reviews the standard isoperimetric theorem and stability of geometric inequalities. The manuscript takes a look at selected affine isoperimetric inequalities, extremum problems for convex discs and polyhedra, and rigidity. Discussions focus on include infinitesimal and static rigidity related to surfaces, isoperimetric problem for convex polyhedral, bounds for the volume of a convex polyhedron, curvature image inequality, Busemann intersection inequality and its relatives, and Petty projection inequality. The book then tackles geometric algorithms, convexity and discrete optimization, mathematical programming and convex geometry, and the combinatorial aspects of convex polytopes. The selection is a valuable source of data for mathematicians and researchers interested in convex geometry.
Book Synopsis Discrete Convex Analysis by : Kazuo Murota
Download or read book Discrete Convex Analysis written by Kazuo Murota and published by SIAM. This book was released on 2003-01-01 with total page 406 pages. Available in PDF, EPUB and Kindle. Book excerpt: Discrete Convex Analysis is a novel paradigm for discrete optimization that combines the ideas in continuous optimization (convex analysis) and combinatorial optimization (matroid/submodular function theory) to establish a unified theoretical framework for nonlinear discrete optimization. The study of this theory is expanding with the development of efficient algorithms and applications to a number of diverse disciplines like matrix theory, operations research, and economics. This self-contained book is designed to provide a novel insight into optimization on discrete structures and should reveal unexpected links among different disciplines. It is the first and only English-language monograph on the theory and applications of discrete convex analysis. Discrete Convex Analysis provides the information that professionals in optimization will need to "catch up" with this new theoretical development. It also presents an unexpected connection between matroid theory and mathematical economics and expounds a deeper connection between matrices and matroids than most standard textbooks.
Book Synopsis Convex Analysis by : R. Tyrrell Rockafellar
Download or read book Convex Analysis written by R. Tyrrell Rockafellar and published by Princeton University Press. This book was released on 1997-01-12 with total page 469 pages. Available in PDF, EPUB and Kindle. Book excerpt: Topics treat systems of inequalities; Lagrange multipliers; minimax theorems and duality; structures of convex sets and functions; and more. Available for the first time in paperback, Rockafellar's classic study has firmly established a vital area not only for pure mathematics but also for applications to economics and engineering. Readers will find sound knowledge of linear algebra and introductory real analysis a major benefit to the assimilation of this work.
Book Synopsis Convex Analysis and Variational Problems by : Ivar Ekeland
Download or read book Convex Analysis and Variational Problems written by Ivar Ekeland and published by SIAM. This book was released on 1999-12-01 with total page 414 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book contains different developments of infinite dimensional convex programming in the context of convex analysis, including duality, minmax and Lagrangians, and convexification of nonconvex optimization problems in the calculus of variations (infinite dimension). It also includes the theory of convex duality applied to partial differential equations; no other reference presents this in a systematic way. The minmax theorems contained in this book have many useful applications, in particular the robust control of partial differential equations in finite time horizon. First published in English in 1976, this SIAM Classics in Applied Mathematics edition contains the original text along with a new preface and some additional references.
Book Synopsis Fundamentals of Convex Analysis by : M.J. Panik
Download or read book Fundamentals of Convex Analysis written by M.J. Panik and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 313 pages. Available in PDF, EPUB and Kindle. Book excerpt: Fundamentals of Convex Analysis offers an in-depth look at some of the fundamental themes covered within an area of mathematical analysis called convex analysis. In particular, it explores the topics of duality, separation, representation, and resolution. The work is intended for students of economics, management science, engineering, and mathematics who need exposure to the mathematical foundations of matrix games, optimization, and general equilibrium analysis. It is written at the advanced undergraduate to beginning graduate level and the only formal preparation required is some familiarity with set operations and with linear algebra and matrix theory. Fundamentals of Convex Analysis is self-contained in that a brief review of the essentials of these tool areas is provided in Chapter 1. Chapter exercises are also provided. Topics covered include: convex sets and their properties; separation and support theorems; theorems of the alternative; convex cones; dual homogeneous systems; basic solutions and complementary slackness; extreme points and directions; resolution and representation of polyhedra; simplicial topology; and fixed point theorems, among others. A strength of this work is how these topics are developed in a fully integrated fashion.
