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An Introduction to Nonsmooth Analysis

An Introduction to Nonsmooth Analysis
  • Author : Juan Ferrera
  • Publisher :Unknown
  • Release Date :2013-11-26
  • Total pages :164
  • ISBN : 0128007311
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Summary : 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 Nonsmooth Analysis

An Introduction to Nonsmooth Analysis
  • Author : Juan Ferrera
  • Publisher :Unknown
  • Release Date :2013-11-26
  • Total pages :164
  • ISBN : 9780128008256
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Summary : 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

Nonsmooth Analysis

Nonsmooth Analysis
  • Author : Winfried Schirotzek
  • Publisher :Unknown
  • Release Date :2007-05-26
  • Total pages :378
  • ISBN : 9783540713333
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Summary : This book treats various concepts of generalized derivatives and subdifferentials in normed spaces, their geometric counterparts and their application to optimization problems. It starts with the subdifferential of convex analysis, passes to corresponding concepts for locally Lipschitz continuous functions and then presents subdifferentials for general lower semicontinuous functions. All basic tools are presented where they are needed: this concerns separation theorems, variational and extremal principles as well as relevant parts of multifunction theory. Each chapter ends with bibliographic notes and exercises.

Introduction to Nonsmooth Optimization

Introduction to Nonsmooth Optimization
  • Author : Adil Bagirov,Napsu Karmitsa,Marko M. Mäkelä
  • Publisher :Unknown
  • Release Date :2014-08-12
  • Total pages :372
  • ISBN : 9783319081144
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Summary : 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.

Nonsmooth Optimization

Nonsmooth Optimization
  • Author : Marko M Mäkelä,Pekka Neittaanmäki
  • Publisher :Unknown
  • Release Date :1992-05-07
  • Total pages :268
  • ISBN : 9789814522410
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Summary : 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

Nonsmooth Analysis and Control Theory

Nonsmooth Analysis and Control Theory
  • Author : Francis H. Clarke,Yuri S. Ledyaev,Ronald J. Stern,Peter R. Wolenski
  • Publisher :Unknown
  • Release Date :1997-12-19
  • Total pages :278
  • ISBN : 0387983368
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Summary : 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.

Optimization and Nonsmooth Analysis

Optimization and Nonsmooth Analysis
  • Author : Frank H. Clarke
  • Publisher :Unknown
  • Release Date :1990-01-01
  • Total pages :308
  • ISBN : 9780898712568
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Summary : Mathematical Reviews said of this book that it was 'destined to become a classical reference.' This book has appeared in Russian translation and has been praised both for its lively exposition and its fundamental contributions. The author first develops a general theory of nonsmooth analysis and geometry which, together with a set of associated techniques, has had a profound effect on several branches of analysis and optimization. Clarke then applies these methods to obtain a powerful, unified approach to the analysis of problems in optimal control and mathematical programming. Examples are drawn from economics, engineering, mathematical physics, and various branches of analysis in this reprint volume.

An Easy Path to Convex Analysis and Applications

An Easy Path to Convex Analysis and Applications
  • Author : Boris S. Mordukhovich,Nguyen Mau Nam
  • Publisher :Unknown
  • Release Date :2013-12-01
  • Total pages :218
  • ISBN : 9781627052382
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Summary : 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

An Introduction to Nonlinear Analysis: Theory

An Introduction to Nonlinear Analysis: Theory
  • Author : Zdzislaw Denkowski,Stanislaw Migórski,Nikolaos S. Papageorgiou
  • Publisher :Unknown
  • Release Date :2013-12-01
  • Total pages :689
  • ISBN : 9781441991584
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Summary : 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.

Functional Analysis, Calculus of Variations and Optimal Control

Functional Analysis, Calculus of Variations and Optimal Control
  • Author : Francis Clarke
  • Publisher :Unknown
  • Release Date :2013-02-06
  • Total pages :591
  • ISBN : 9781447148203
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Summary : 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.

Convex Analysis and Nonlinear Optimization

Convex Analysis and Nonlinear Optimization
  • Author : Jonathan Borwein,Adrian S. Lewis
  • Publisher :Unknown
  • Release Date :2010-05-05
  • Total pages :310
  • ISBN : 9780387312569
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Summary : 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.

Regularity Concepts in Nonsmooth Analysis

Regularity Concepts in Nonsmooth Analysis
  • Author : Messaoud Bounkhel
  • Publisher :Unknown
  • Release Date :2011-11-12
  • Total pages :264
  • ISBN : 9781461410195
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Summary : The results presented in this book are a product of research conducted by the author independently and in collaboration with other researchers in the field. In this light, this work encompasses the most recent collection of various concepts of regularity and nonsmooth analysis into one monograph. The first part of the book attempts to present an accessible and thorough introduction to nonsmooth analysis theory. Main concepts and some useful results are stated and illustrated through examples and exercises. The second part gathers the most prominent and recent results of various regularity concepts of sets, functions, and set-valued mappings in nonsmooth analysis. The third and final section contains six different application, with comments in relation to the existing literature.

Topological Aspects of Nonsmooth Optimization

Topological Aspects of Nonsmooth Optimization
  • Author : Vladimir Shikhman
  • Publisher :Unknown
  • Release Date :2011-11-18
  • Total pages :196
  • ISBN : 9781461418979
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Summary : This book deals with nonsmooth structures arising within the optimization setting. It considers four optimization problems, namely, mathematical programs with complementarity constraints, general semi-infinite programming problems, mathematical programs with vanishing constraints and bilevel optimization. The author uses the topological approach and topological invariants of corresponding feasible sets are investigated. Moreover, the critical point theory in the sense of Morse is presented and parametric and stability issues are considered. The material progresses systematically and establishes a comprehensive theory for a rather broad class of optimization problems tailored to their particular type of nonsmoothness. Topological Aspects of Nonsmooth Optimization will benefit researchers and graduate students in applied mathematics, especially those working in optimization theory, nonsmooth analysis, algebraic topology and singularity theory. ​ ​

Introduction to Piecewise Differentiable Equations

Introduction to Piecewise Differentiable Equations
  • Author : Stefan Scholtes
  • Publisher :Unknown
  • Release Date :2012-08-01
  • Total pages :133
  • ISBN : 9781461443407
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Summary : ​​​​​​​ 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.

An Introduction to Nonlinear Optimization Theory

An Introduction to Nonlinear Optimization Theory
  • Author : Marius Durea,Radu Strugariu
  • Publisher :Unknown
  • Release Date :2014-01-01
  • Total pages :328
  • ISBN : 9783110427356
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Summary : The goal of this book is to present the main ideas and techniques in the field of continuous smooth and nonsmooth optimization. Starting with the case of differentiable data and the classical results on constrained optimization problems, and continuing with the topic of nonsmooth objects involved in optimization theory, the book concentrates on both theoretical and practical aspects of this field. This book prepares those who are engaged in research by giving repeated insights into ideas that are subsequently dealt with and illustrated in detail.

Derivative-Free and Blackbox Optimization

Derivative-Free and Blackbox Optimization
  • Author : Charles Audet,Warren Hare
  • Publisher :Unknown
  • Release Date :2017-12-02
  • Total pages :302
  • ISBN : 9783319689135
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Summary : This book is designed as a textbook, suitable for self-learning or for teaching an upper-year university course on derivative-free and blackbox optimization. The book is split into 5 parts and is designed to be modular; any individual part depends only on the material in Part I. Part I of the book discusses what is meant by Derivative-Free and Blackbox Optimization, provides background material, and early basics while Part II focuses on heuristic methods (Genetic Algorithms and Nelder-Mead). Part III presents direct search methods (Generalized Pattern Search and Mesh Adaptive Direct Search) and Part IV focuses on model-based methods (Simplex Gradient and Trust Region). Part V discusses dealing with constraints, using surrogates, and bi-objective optimization. End of chapter exercises are included throughout as well as 15 end of chapter projects and over 40 figures. Benchmarking techniques are also presented in the appendix.

Applied Nonlinear Functional Analysis

Applied Nonlinear Functional Analysis
  • Author : Nikolaos S. Papageorgiou,Patrick Winkert
  • Publisher :Unknown
  • Release Date :2018-08-06
  • Total pages :622
  • ISBN : 9783110532982
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Summary : The aim of this book is to provide a concise but complete introduction to the main mathematical tools of nonlinear functional analysis, which are also used in the study of concrete problems in economics, engineering, and physics. This volume gathers the mathematical background needed in order to conduct research or to deal with theoretical problems and applications using the tools of nonlinear functional analysis.

Optimal Control Via Nonsmooth Analysis

Optimal Control Via Nonsmooth Analysis
  • Author : Philip Daniel Loewen
  • Publisher :Unknown
  • Release Date :1993
  • Total pages :153
  • ISBN : 0821869965
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Summary : This book provides a complete and unified treatment of deterministic problems of dynamic optimization, from the classical themes of the calculus of variations to the forefront of modern research in optimal control. At the heart of the presentation is nonsmooth analysis, a theory of local approximation developed over the last twenty years to provide useful first-order information about sets and functions lying beyond the reach of classical analysis. The book includes an intuitive and geometrically transparent approach to nonsmooth analysis, serving not only to introduce the basic ideas, but also to illuminate the calculations and derivations in the applied sections dealing with the calculus of variations and optimal control. Written in a lively, engaging style and stocked with numerous figures and practice problems, this book offers an ideal introduction to this vigorous field of current research. It is suitable as a graduate text for a one-semester course in optimal control or as a manual for self-study. Each chapter closes with a list of references to ease the reader's transition from active learner to contributing researcher.

Nonsmooth Optimization

Nonsmooth Optimization
  • Author : Marko M. Mäkelä,Pekka Neittaanmäki
  • Publisher :Unknown
  • Release Date :1992-01-01
  • Total pages :254
  • ISBN : 9810207735
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Summary : Introduces various methods for nonsmooth optimization and applies these methods to solve discretized nonsmooth optimal control problems of systems governed by boundary value problems. Annotation copyrighted by Book News, Inc., Portland, OR

Nonlinear Analysis

Nonlinear Analysis
  • Author : Leszek Gasinski,Nikolaos S. Papageorgiou
  • Publisher :Unknown
  • Release Date :2005-07-27
  • Total pages :984
  • ISBN : 1584884843
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Summary : Nonlinear analysis is a broad, interdisciplinary field characterized by a remarkable mixture of analysis, topology, and applications. Its concepts and techniques provide the tools for developing more realistic and accurate models for a variety of phenomena encountered in fields ranging from engineering and chemistry to economics and biology. This volume focuses on topics in nonlinear analysis pertinent to the theory of boundary value problems and their application in areas such as control theory and the calculus of variations. It complements the many other books on nonlinear analysis by addressing topics previously discussed fully only in scattered research papers. These include recent results on critical point theory, nonlinear differential operators, and related regularity and comparison principles. The rich variety of topics, both theoretical and applied, make Nonlinear Analysis useful to anyone, whether graduate student or researcher, working in analysis or its applications in optimal control, theoretical mechanics, or dynamical systems. An appendix contains all of the background material needed, and a detailed bibliography forms a guide for further study.

Mathematics of Optimization: Smooth and Nonsmooth Case

Mathematics of Optimization: Smooth and Nonsmooth Case
  • Author : Giorgio Giorgi,A. Guerraggio,J. Thierfelder
  • Publisher :Unknown
  • Release Date :2004-03-10
  • Total pages :614
  • ISBN : 008053595X
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Summary : 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