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An Introduction to Nonsmooth Analysis

  • 1st Edition - November 26, 2013
  • Latest edition
  • Author: Juan Ferrera
  • Language: English

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 i… Read more

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Description

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.

Key features

  • 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

Readership

graduate students in mathematics, handbook for a graduate course or a reference book in an undergraduate course of advanced analysis with the aim of introduce the nonsmooth analysis as a complement to differential calculus, showing how smooth tools can be employed in the lack of differentiability

Table of contents

DedicationPrefaceAcknowledgment1. Basic Concepts and ResultsAbstract1.1 Upper and Lower Limits1.2 Semicontinuity1.3 Differentiability1.4 Two Important Theorems1.5 Problems2. Convex FunctionsAbstract2.1 Convex Sets and Convex Functions2.2 Continuity of Convex Functions2.3 Separation Results2.4 Convexity and Differentiability2.5 Problems3. The Subdifferential of a Convex FunctionAbstract3.1 Subdifferential Properties3.2 Two Examples3.3 Problems4. The Subdifferential: General CaseAbstract4.1 Definition and Basic Properties4.2 Geometrical Meaning of the Subdifferential4.3 Density of Subdifferentiability Points4.4 Proximal Subdifferential4.5 Problems5. CalculusAbstract5.1 Sum Rule5.2 Constrained Minima5.3 Chain Rule5.4 Regular Functions: Elementary Properties5.5 Mean Value Results5.6 Decreasing Functions5.7 Problems6. Lipschitz Functions and the Generalized GradientAbstract6.1 Lipschitz Regular Functions6.2 The Generalized Gradient6.3 Generalized Jacobian6.4 Graphical Derivative6.5 Problems7. ApplicationsAbstract7.1 Flow Invariant Sets7.2 Viscosity Solutions7.3 Solving Equations7.4 ProblemsBibliographyIndex

Review quotes

"...starting from the very beginning, adopting a slow, easy to follow linear development and reaching to a self-contained theory...oriented towards undergraduate students, as a first quick introduction to the topic."—MathSciNet

"...devoted to presenting the theory of the subdifferential of lower semicontinuous functions which is a generalization of the subdifferential of convex functions...a good reference for researchers in optimization and applied mathematics."—Zentralblatt MATH, Sep-14

Product details

  • Edition: 1
  • Latest edition
  • Published: November 26, 2013
  • Language: English

About the author

JF

Juan Ferrera

Affiliations and expertise
Universidad Complutense de Madrid, Spain

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