An Introduction to Nonsmooth Analysis

An Introduction to Nonsmooth Analysis
Author: Juan Ferrera
Publsiher: Academic Press
Total Pages: 164
Release: 2013-11-26
Genre: Mathematics
ISBN: 9780128008256

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Nonsmooth Analysis is a relatively recent area of mathematical analysis. The literature about this subject consists mainly in research papers and books. The purpose of this book is to provide a handbook for undergraduate and graduate students of mathematics that introduce this interesting area in detail. Includes different kinds of sub and super differentials as well as generalized gradients Includes also the main tools of the theory, as Sum and Chain Rules or Mean Value theorems Content is introduced in an elementary way, developing many examples, allowing the reader to understand a theory which is scattered in many papers and research books

An Introduction to Nonlinear Analysis Theory

An Introduction to Nonlinear Analysis  Theory
Author: Zdzislaw Denkowski,Stanislaw Migórski,Nikolaos S. Papageorgiou
Publsiher: Springer Science & Business Media
Total Pages: 690
Release: 2013-12-01
Genre: Mathematics
ISBN: 9781441991584

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An Introduction to Nonlinear Analysis: Theory is an overview of some basic, important aspects of Nonlinear Analysis, with an emphasis on those not included in the classical treatment of the field. Today Nonlinear Analysis is a very prolific part of modern mathematical analysis, with fascinating theory and many different applications ranging from mathematical physics and engineering to social sciences and economics. Topics covered in this book include the necessary background material from topology, measure theory and functional analysis (Banach space theory). The text also deals with multivalued analysis and basic features of nonsmooth analysis, providing a solid background for the more applications-oriented material of the book An Introduction to Nonlinear Analysis: Applications by the same authors. The book is self-contained and accessible to the newcomer, complete with numerous examples, exercises and solutions. It is a valuable tool, not only for specialists in the field interested in technical details, but also for scientists entering Nonlinear Analysis in search of promising directions for research.

Nonsmooth Analysis and Control Theory

Nonsmooth Analysis and Control Theory
Author: Francis H. Clarke,Yuri S. Ledyaev,Ronald J. Stern,Peter R. Wolenski
Publsiher: Springer Science & Business Media
Total Pages: 278
Release: 2008-01-10
Genre: Mathematics
ISBN: 9780387226255

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A clear and succinct presentation of the essentials of this subject, together with some of its applications and a generous helping of interesting exercises. Following an introductory chapter with a taste of what is to come, the next three chapters constitute a course in nonsmooth analysis and identify a coherent and comprehensive approach to the subject, leading to an efficient, natural, and powerful body of theory. The whole is rounded off with a self-contained introduction to the theory of control of ordinary differential equations. The authors have incorporated a number of new results which clarify the relationships between the different schools of thought in the subject, with the aim of making nonsmooth analysis accessible to a wider audience. End-of-chapter problems offer scope for deeper understanding.

Nonsmooth Optimization

Nonsmooth Optimization
Author: Marko M Mäkelä,Pekka Neittaanmäki
Publsiher: World Scientific
Total Pages: 268
Release: 1992-05-07
Genre: Mathematics
ISBN: 9789814522410

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This book is a self-contained elementary study for nonsmooth analysis and optimization, and their use in solution of nonsmooth optimal control problems. The first part of the book is concerned with nonsmooth differential calculus containing necessary tools for nonsmooth optimization. The second part is devoted to the methods of nonsmooth optimization and their development. A proximal bundle method for nonsmooth nonconvex optimization subject to nonsmooth constraints is constructed. In the last part nonsmooth optimization is applied to problems arising from optimal control of systems covered by partial differential equations. Several practical problems, like process control and optimal shape design problems are considered. Contents: Part I: Nonsmooth Analysis:IntroductionConvex AnalysisNonsmooth Differential TheoryNonsmooth GeometryNonsmooth Optimization TheoryPart II: Nonsmooth Optimization:IntroductionA Survey of Bundle MethodsProximal Bundle Method for Nonconvex Constrained OptimizationNumerical ExperimentsPart III: Nonsmooth Optimal Control:IntroductionPreliminariesDistributed Parameter Control Problems Optimal Shape Design Boundary Control for Stefan Type Problems Readership: Applied mathematicians, mathematicians, operations researchers, engineers, economists and mathematical physicists. keywords:Nonsmooth Optimization;Nondifferentiable Programming;Bundle Methods;Convex Analysis;Nonconvexity;Subgradients;Tangent and Normal Cones;Optimal Control;Optimal Shape Design;Continuous Casting

Introduction to Functional Analysis

Introduction to Functional Analysis
Author: Christian Clason
Publsiher: Springer Nature
Total Pages: 170
Release: 2020-11-30
Genre: Mathematics
ISBN: 9783030527846

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Functional analysis has become one of the essential foundations of modern applied mathematics in the last decades, from the theory and numerical solution of differential equations, from optimization and probability theory to medical imaging and mathematical image processing. This textbook offers a compact introduction to the theory and is designed to be used during one semester, fitting exactly 26 lectures of 90 minutes each. It ranges from the topological fundamentals recalled from basic lectures on real analysis to spectral theory in Hilbert spaces. Special attention is given to the central results on dual spaces and weak convergence.

Introduction to Nonsmooth Optimization

Introduction to Nonsmooth Optimization
Author: Adil Bagirov,Napsu Karmitsa,Marko M. Mäkelä
Publsiher: Springer
Total Pages: 372
Release: 2014-08-12
Genre: Business & Economics
ISBN: 9783319081144

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This book is the first easy-to-read text on nonsmooth optimization (NSO, not necessarily differentiable optimization). Solving these kinds of problems plays a critical role in many industrial applications and real-world modeling systems, for example in the context of image denoising, optimal control, neural network training, data mining, economics and computational chemistry and physics. The book covers both the theory and the numerical methods used in NSO and provide an overview of different problems arising in the field. It is organized into three parts: 1. convex and nonconvex analysis and the theory of NSO; 2. test problems and practical applications; 3. a guide to NSO software. The book is ideal for anyone teaching or attending NSO courses. As an accessible introduction to the field, it is also well suited as an independent learning guide for practitioners already familiar with the basics of optimization.

An Introduction to Nonsmooth Analysis

An Introduction to Nonsmooth Analysis
Author: Juan Ferrera
Publsiher: Unknown
Total Pages: 164
Release: 2013-11-26
Genre: Mathematics
ISBN: 0128007311

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Nonsmooth Analysis is a relatively recent area of mathematical analysis. The literature about this subject consists mainly in research papers and books. The purpose of this book is to provide a handbook for undergraduate and graduate students of mathematics that introduce this interesting area in detail. Includes different kinds of sub and super differentials as well as generalized gradients Includes also the main tools of the theory, as Sum and Chain Rules or Mean Value theorems Content is introduced in an elementary way, developing many examples, allowing the reader to understand a theory which is scattered in many papers and research books

Functional Analysis Calculus of Variations and Optimal Control

Functional Analysis  Calculus of Variations and Optimal Control
Author: Francis Clarke
Publsiher: Springer Science & Business Media
Total Pages: 591
Release: 2013-02-06
Genre: Mathematics
ISBN: 9781447148203

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Functional analysis owes much of its early impetus to problems that arise in the calculus of variations. In turn, the methods developed there have been applied to optimal control, an area that also requires new tools, such as nonsmooth analysis. This self-contained textbook gives a complete course on all these topics. It is written by a leading specialist who is also a noted expositor. This book provides a thorough introduction to functional analysis and includes many novel elements as well as the standard topics. A short course on nonsmooth analysis and geometry completes the first half of the book whilst the second half concerns the calculus of variations and optimal control. The author provides a comprehensive course on these subjects, from their inception through to the present. A notable feature is the inclusion of recent, unifying developments on regularity, multiplier rules, and the Pontryagin maximum principle, which appear here for the first time in a textbook. Other major themes include existence and Hamilton-Jacobi methods. The many substantial examples, and the more than three hundred exercises, treat such topics as viscosity solutions, nonsmooth Lagrangians, the logarithmic Sobolev inequality, periodic trajectories, and systems theory. They also touch lightly upon several fields of application: mechanics, economics, resources, finance, control engineering. Functional Analysis, Calculus of Variations and Optimal Control is intended to support several different courses at the first-year or second-year graduate level, on functional analysis, on the calculus of variations and optimal control, or on some combination. For this reason, it has been organized with customization in mind. The text also has considerable value as a reference. Besides its advanced results in the calculus of variations and optimal control, its polished presentation of certain other topics (for example convex analysis, measurable selections, metric regularity, and nonsmooth analysis) will be appreciated by researchers in these and related fields.

Introduction to Piecewise Differentiable Equations

Introduction to Piecewise Differentiable Equations
Author: Stefan Scholtes
Publsiher: Springer Science & Business Media
Total Pages: 133
Release: 2012-08-01
Genre: Mathematics
ISBN: 9781461443407

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​​​​​​​ This brief provides an elementary introduction to the theory of piecewise differentiable functions with an emphasis on differentiable equations. In the first chapter, two sample problems are used to motivate the study of this theory. The presentation is then developed using two basic tools for the analysis of piecewise differentiable functions: the Bouligand derivative as the nonsmooth analogue of the classical derivative concept and the theory of piecewise affine functions as the combinatorial tool for the study of this approximation function. In the end, the results are combined to develop inverse and implicit function theorems for piecewise differentiable equations. This Introduction to Piecewise Differentiable Equations will serve graduate students and researchers alike. The reader is assumed to be familiar with basic mathematical analysis and to have some familiarity with polyhedral theory.

Convex Analysis and Nonlinear Optimization

Convex Analysis and Nonlinear Optimization
Author: Jonathan Borwein,Adrian S. Lewis
Publsiher: Springer Science & Business Media
Total Pages: 310
Release: 2010-05-05
Genre: Mathematics
ISBN: 9780387312569

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Optimization is a rich and thriving mathematical discipline, and the underlying theory of current computational optimization techniques grows ever more sophisticated. This book aims to provide a concise, accessible account of convex analysis and its applications and extensions, for a broad audience. Each section concludes with an often extensive set of optional exercises. This new edition adds material on semismooth optimization, as well as several new proofs.

Lectures on Nonsmooth Differential Geometry

Lectures on Nonsmooth Differential Geometry
Author: Nicola Gigli,Enrico Pasqualetto
Publsiher: Springer Nature
Total Pages: 204
Release: 2020-02-10
Genre: Mathematics
ISBN: 9783030386139

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This book provides an introduction to some aspects of the flourishing field of nonsmooth geometric analysis. In particular, a quite detailed account of the first-order structure of general metric measure spaces is presented, and the reader is introduced to the second-order calculus on spaces – known as RCD spaces – satisfying a synthetic lower Ricci curvature bound. Examples of the main topics covered include notions of Sobolev space on abstract metric measure spaces; normed modules, which constitute a convenient technical tool for the introduction of a robust differential structure in the nonsmooth setting; first-order differential operators and the corresponding functional spaces; the theory of heat flow and its regularizing properties, within the general framework of “infinitesimally Hilbertian” metric measure spaces; the RCD condition and its effects on the behavior of heat flow; and second-order calculus on RCD spaces. The book is mainly intended for young researchers seeking a comprehensive and fairly self-contained introduction to this active research field. The only prerequisites are a basic knowledge of functional analysis, measure theory, and Riemannian geometry.

An Easy Path to Convex Analysis and Applications

An Easy Path to Convex Analysis and Applications
Author: Boris S. Mordukhovich,Nguyen Mau Nam
Publsiher: Morgan & Claypool Publishers
Total Pages: 218
Release: 2013-12-01
Genre: Mathematics
ISBN: 9781627052382

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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 f

Nonsmooth Variational Problems and Their Inequalities

Nonsmooth Variational Problems and Their Inequalities
Author: Siegfried Carl,Vy K. Le,Dumitru Motreanu
Publsiher: Springer Science & Business Media
Total Pages: 395
Release: 2007-06-08
Genre: Mathematics
ISBN: 9780387462523

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This monograph focuses primarily on nonsmooth variational problems that arise from boundary value problems with nonsmooth data and/or nonsmooth constraints, such as multivalued elliptic problems, variational inequalities, hemivariational inequalities, and their corresponding evolution problems. It provides a systematic and unified exposition of comparison principles based on a suitably extended sub-supersolution method.

Mathematics of Optimization Smooth and Nonsmooth Case

Mathematics of Optimization  Smooth and Nonsmooth Case
Author: Giorgio Giorgi,A. Guerraggio,J. Thierfelder
Publsiher: Elsevier
Total Pages: 614
Release: 2004-03-10
Genre: Mathematics
ISBN: 9780080535951

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The book is intended for people (graduates, researchers, but also undergraduates with a good mathematical background) involved in the study of (static) optimization problems (in finite-dimensional spaces). It contains a lot of material, from basic tools of convex analysis to optimality conditions for smooth optimization problems, for non smooth optimization problems and for vector optimization problems. The development of the subjects are self-contained and the bibliographical references are usually treated in different books (only a few books on optimization theory deal also with vector problems), so the book can be a starting point for further readings in a more specialized literature. Assuming only a good (even if not advanced) knowledge of mathematical analysis and linear algebra, this book presents various aspects of the mathematical theory in optimization problems. The treatment is performed in finite-dimensional spaces and with no regard to algorithmic questions. After two chapters concerning, respectively, introductory subjects and basic tools and concepts of convex analysis, the book treats extensively mathematical programming problems in the smmoth case, in the nonsmooth case and finally vector optimization problems. · Self-contained · Clear style and results are either proved or stated precisely with adequate references · The authors have several years experience in this field · Several subjects (some of them non usual in books of this kind) in one single book, including nonsmooth optimization and vector optimization problems · Useful long references list at the end of each chapter

Abstract Convexity and Global Optimization

Abstract Convexity and Global Optimization
Author: Alexander M. Rubinov
Publsiher: Springer Science & Business Media
Total Pages: 490
Release: 2000-05-31
Genre: Mathematics
ISBN: 079236323X

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This book consists of two parts. Firstly, the main notions of abstract convexity and their applications in the study of some classes of functions and sets are presented. Secondly, both theoretical and numerical aspects of global optimization based on abstract convexity are examined. Most of the book does not require knowledge of advanced mathematics. Classical methods of nonconvex mathematical programming, being based on a local approximation, cannot be used to examine and solve many problems of global optimization, and so there is a clear need to develop special global tools for solving these problems. Some of these tools are based on abstract convexity, that is, on the representation of a function of a rather complicated nature as the upper envelope of a set of fairly simple functions. Audience: The book will be of interest to specialists in global optimization, mathematical programming, and convex analysis, as well as engineers using mathematical tools and optimization techniques and specialists in mathematical modelling.