Convex Analysis And Optimization In Hadamard Spaces

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Convex Analysis and Optimization in Hadamard Spaces

Miroslav Bacak
Convex Analysis and Optimization in Hadamard Spaces Book PDF

Author : Miroslav Bacak
Publisher : Walter de Gruyter GmbH & Co KG
Page : 194 pages
File Size : 42,7 Mb
Release : 2014-10-29
Category : Mathematics
ISBN : 9783110361629

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Convex Analysis and Optimization in Hadamard Spaces by Miroslav Bacak Pdf

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.

Convex Analysis and Optimization in Hadamard Spaces

Miroslav Bacak
Convex Analysis and Optimization in Hadamard Spaces Book PDF

Author : Miroslav Bacak
Publisher : Walter de Gruyter GmbH & Co KG
Page : 217 pages
File Size : 47,5 Mb
Release : 2014-10-29
Category : Mathematics
ISBN : 9783110391084

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Convex Analysis and Optimization in Hadamard Spaces by Miroslav Bacak Pdf

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.

Convexity and Optimization in Banach Spaces

Viorel Barbu,Teodor Precupanu
Convexity and Optimization in Banach Spaces Book PDF

Author : Viorel Barbu,Teodor Precupanu
Publisher : Springer Science & Business Media
Page : 376 pages
File Size : 41,6 Mb
Release : 2012-01-03
Category : Mathematics
ISBN : 9789400722460

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Convexity and Optimization in Banach Spaces by Viorel Barbu,Teodor Precupanu Pdf

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.

Foundations of Mathematical Optimization

Diethard Ernst Pallaschke,S. Rolewicz
Foundations of Mathematical Optimization Book PDF

Author : Diethard Ernst Pallaschke,S. Rolewicz
Publisher : Springer Science & Business Media
Page : 597 pages
File Size : 52,6 Mb
Release : 2013-03-14
Category : Mathematics
ISBN : 9789401715881

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Foundations of Mathematical Optimization by Diethard Ernst Pallaschke,S. Rolewicz Pdf

Many books on optimization consider only finite dimensional spaces. This volume is unique in its emphasis: the first three chapters develop optimization in spaces without linear structure, and the analog of convex analysis is constructed for this case. Many new results have been proved specially for this publication. In the following chapters optimization in infinite topological and normed vector spaces is considered. The novelty consists in using the drop property for weak well-posedness of linear problems in Banach spaces and in a unified approach (by means of the Dolecki approximation) to necessary conditions of optimality. The method of reduction of constraints for sufficient conditions of optimality is presented. The book contains an introduction to non-differentiable and vector optimization. Audience: This volume will be of interest to mathematicians, engineers, and economists working in mathematical optimization.

Convex Optimization in Normed Spaces

Juan Peypouquet
Convex Optimization in Normed Spaces Book PDF

Author : Juan Peypouquet
Publisher : Springer
Page : 124 pages
File Size : 48,9 Mb
Release : 2015-03-18
Category : Mathematics
ISBN : 9783319137100

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Convex Optimization in Normed Spaces by Juan Peypouquet Pdf

This work is intended to serve as a guide for graduate students and researchers who wish to get acquainted with the main theoretical and practical tools for the numerical minimization of convex functions on Hilbert spaces. Therefore, it contains the main tools that are necessary to conduct independent research on the topic. It is also a concise, easy-to-follow and self-contained textbook, which may be useful for any researcher working on related fields, as well as teachers giving graduate-level courses on the topic. It will contain a thorough revision of the extant literature including both classical and state-of-the-art references.

Convex Analysis and Optimization

Dimitri Bertsekas,Angelia Nedic,Asuman Ozdaglar
Convex Analysis and Optimization Book PDF

Author : Dimitri Bertsekas,Angelia Nedic,Asuman Ozdaglar
Publisher : Athena Scientific
Page : 560 pages
File Size : 54,7 Mb
Release : 2003-03-01
Category : Mathematics
ISBN : 9781886529458

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Convex Analysis and Optimization by Dimitri Bertsekas,Angelia Nedic,Asuman Ozdaglar Pdf

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

Convex Analysis and Nonlinear Optimization

Jonathan M. Borwein,Adrian S. Lewis
Convex Analysis and Nonlinear Optimization Book PDF

Author : Jonathan M. Borwein,Adrian S. Lewis
Publisher : Springer Science & Business Media
Page : 281 pages
File Size : 42,9 Mb
Release : 2013-06-29
Category : Mathematics
ISBN : 9781475798593

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Convex Analysis and Nonlinear Optimization by Jonathan M. Borwein,Adrian S. Lewis Pdf

This book provides 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, since the main body of the text is self-contained, with each section rounded off by 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 and Monotone Operator Theory in Hilbert Spaces

Heinz H. Bauschke,Patrick L. Combettes
Convex Analysis and Monotone Operator Theory in Hilbert Spaces Book PDF

Author : Heinz H. Bauschke,Patrick L. Combettes
Publisher : Springer
Page : 624 pages
File Size : 44,9 Mb
Release : 2017-02-28
Category : Mathematics
ISBN : 9783319483115

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Convex Analysis and Monotone Operator Theory in Hilbert Spaces by Heinz H. Bauschke,Patrick L. Combettes Pdf

This reference text, now in its second edition, offers a modern unifying presentation of three basic areas of nonlinear analysis: convex analysis, monotone operator theory, and the fixed point theory of nonexpansive operators. Taking a unique comprehensive approach, the theory is developed from the ground up, with the rich connections and interactions between the areas as the central focus, and it is illustrated by a large number of examples. The Hilbert space setting of the material offers a wide range of applications while avoiding the technical difficulties of general Banach spaces. The authors have also drawn upon recent advances and modern tools to simplify the proofs of key results making the book more accessible to a broader range of scholars and users. Combining a strong emphasis on applications with exceptionally lucid writing and an abundance of exercises, this text is of great value to a large audience including pure and applied mathematicians as well as researchers in engineering, data science, machine learning, physics, decision sciences, economics, and inverse problems. The second edition of Convex Analysis and Monotone Operator Theory in Hilbert Spaces greatly expands on the first edition, containing over 140 pages of new material, over 270 new results, and more than 100 new exercises. It features a new chapter on proximity operators including two sections on proximity operators of matrix functions, in addition to several new sections distributed throughout the original chapters. Many existing results have been improved, and the list of references has been updated. Heinz H. Bauschke is a Full Professor of Mathematics at the Kelowna campus of the University of British Columbia, Canada. Patrick L. Combettes, IEEE Fellow, was on the faculty of the City University of New York and of Université Pierre et Marie Curie – Paris 6 before joining North Carolina State University as a Distinguished Professor of Mathematics in 2016.

Introduction to Optimization and Hadamard Semidifferential Calculus, Second Edition

Michel C. Delfour
Introduction to Optimization and Hadamard Semidifferential Calculus Second Edition Book PDF

Author : Michel C. Delfour
Publisher : SIAM
Page : 445 pages
File Size : 41,7 Mb
Release : 2019-12-19
Category : Mathematics
ISBN : 9781611975963

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Introduction to Optimization and Hadamard Semidifferential Calculus, Second Edition by Michel C. Delfour Pdf

This second edition provides an enhanced exposition of the long-overlooked Hadamard semidifferential calculus, first introduced in the 1920s by mathematicians Jacques Hadamard and Maurice René Fréchet. Hadamard semidifferential calculus is possibly the largest family of nondifferentiable functions that retains all the features of classical differential calculus, including the chain rule, making it a natural framework for initiating a large audience of undergraduates and non-mathematicians into the world of nondifferentiable optimization. Introduction to Optimization and Hadamard Semidifferential Calculus, Second Edition builds upon its prior edition’s foundations in Hadamard semidifferential calculus, showcasing new material linked to convex analysis and nonsmooth optimization. It presents a modern treatment of optimization and Hadamard semidifferential calculus while remaining at a level that is accessible to undergraduate students, and challenges students with exercises related to problems in such fields as engineering, mechanics, medicine, physics, and economics. Answers are supplied in Appendix B. Students of mathematics, physics, engineering, economics, and other disciplines that demand a basic knowledge of mathematical analysis and linear algebra will find this a fitting primary or companion resource for their studies. This textbook has been designed and tested for a one-term course at the undergraduate level. In its full version, it is appropriate for a first-year graduate course and as a reference.

An Introduction to Convexity, Optimization, and Algorithms

Heinz H. Bauschke,Walaa M. Moursi
An Introduction to Convexity Optimization and Algorithms Book PDF

Author : Heinz H. Bauschke,Walaa M. Moursi
Publisher : SIAM
Page : 192 pages
File Size : 48,6 Mb
Release : 2023-12-20
Category : Mathematics
ISBN : 9781611977806

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An Introduction to Convexity, Optimization, and Algorithms by Heinz H. Bauschke,Walaa M. Moursi Pdf

This concise, self-contained volume introduces convex analysis and optimization algorithms, with an emphasis on bridging the two areas. It explores cutting-edge algorithms—such as the proximal gradient, Douglas–Rachford, Peaceman–Rachford, and FISTA—that have applications in machine learning, signal processing, image reconstruction, and other fields. An Introduction to Convexity, Optimization, and Algorithms contains algorithms illustrated by Julia examples and more than 200 exercises that enhance the reader’s understanding of the topic. Clear explanations and step-by-step algorithmic descriptions facilitate self-study for individuals looking to enhance their expertise in convex analysis and optimization. Designed for courses in convex analysis, numerical optimization, and related subjects, this volume is intended for undergraduate and graduate students in mathematics, computer science, and engineering. Its concise length makes it ideal for a one-semester course. Researchers and professionals in applied areas, such as data science and machine learning, will find insights relevant to their work.

Convex Analysis

Ralph Tyrell Rockafellar
Convex Analysis Book PDF

Author : Ralph Tyrell Rockafellar
Publisher : Princeton University Press
Page : 470 pages
File Size : 40,7 Mb
Release : 2015-04-29
Category : Mathematics
ISBN : 9781400873173

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Convex Analysis by Ralph Tyrell Rockafellar Pdf

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.

Convex Analysis and Optimization

Jean Pierre Aubin,Richard B. Vinter
Convex Analysis and Optimization Book PDF

Author : Jean Pierre Aubin,Richard B. Vinter
Publisher : Pitman Advanced Publishing Program
Page : 228 pages
File Size : 48,8 Mb
Release : 1982
Category : Mathematics
ISBN : UCAL:B4405614

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Convex Analysis and Optimization by Jean Pierre Aubin,Richard B. Vinter Pdf

This book is a collection of invited papers, presented at an international colloquium on convex analysis and its applications, held in honor of the Russian mathematician Alexander D. Ioffe. The possibilities of extending the theory to meet the challenges of potential new applications in mathematical programming, optimal control, econometrics, and modeling provides a common theme to the papers. This book will be of interest to researchers in non-smooth analysis, approximate subdifferentials, quasiconvexity, control theory and mathematical programming.

Fundamentals of Convex Analysis and Optimization

Rafael Correa,Abderrahim Hantoute,Marco A. López
Fundamentals of Convex Analysis and Optimization Book PDF

Author : Rafael Correa,Abderrahim Hantoute,Marco A. López
Publisher : Springer Nature
Page : 451 pages
File Size : 46,9 Mb
Release : 2023-07-11
Category : Business & Economics
ISBN : 9783031295515

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Fundamentals of Convex Analysis and Optimization by Rafael Correa,Abderrahim Hantoute,Marco A. López Pdf

This book aims at an innovative approach within the framework of convex analysis and optimization, based on an in-depth study of the behavior and properties of the supremum of families of convex functions. It presents an original and systematic treatment of convex analysis, covering standard results and improved calculus rules in subdifferential analysis. The tools supplied in the text allow a direct approach to the mathematical foundations of convex optimization, in particular to optimality and duality theory. Other applications in the book concern convexification processes in optimization, non-convex integration of the Fenchel subdifferential, variational characterizations of convexity, and the study of Chebychev sets. At the same time, the underlying geometrical meaning of all the involved concepts and operations is highlighted and duly emphasized. A notable feature of the book is its unifying methodology, as well as the novelty of providing an alternative or complementary view to the traditional one in which the discipline is presented to students and researchers. This textbook can be used for courses on optimization, convex and variational analysis, addressed to graduate and post-graduate students of mathematics, and also students of economics and engineering. It is also oriented to provide specific background for courses on optimal control, data science, operations research, economics (game theory), etc. The book represents a challenging and motivating development for those experts in functional analysis, convex geometry, and any kind of researchers who may be interested in applications of their work.

Convex Analysis and Beyond

Boris S. Mordukhovich,Nguyen Mau Nam
Convex Analysis and Beyond Book PDF

Author : Boris S. Mordukhovich,Nguyen Mau Nam
Publisher : Springer
Page : 0 pages
File Size : 48,9 Mb
Release : 2023-04-25
Category : Mathematics
ISBN : 3030947874

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Convex Analysis and Beyond by Boris S. Mordukhovich,Nguyen Mau Nam Pdf

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.

Constrained Optimization and Image Space Analysis

Franco Giannessi
Constrained Optimization and Image Space Analysis Book PDF

Author : Franco Giannessi
Publisher : Springer Science & Business Media
Page : 404 pages
File Size : 51,8 Mb
Release : 2006-10-27
Category : Mathematics
ISBN : 9780387280202

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Constrained Optimization and Image Space Analysis by Franco Giannessi Pdf

Over the last twenty years, Professor Franco Giannessi, a highly respected researcher, has been working on an approach to optimization theory based on image space analysis. His theory has been elaborated by many other researchers in a wealth of papers. Constrained Optimization and Image Space Analysis unites his results and presents optimization theory and variational inequalities in their light. It presents a new approach to the theory of constrained extremum problems, including Mathematical Programming, Calculus of Variations and Optimal Control Problems. Such an approach unifies the several branches: Optimality Conditions, Duality, Penalizations, Vector Problems, Variational Inequalities and Complementarity Problems. The applications benefit from a unified theory.