An Introduction to Nonsmooth Analysis

By

  • Juan Ferrera, Universidad Complutense de Madrid, Spain

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.
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Audience

This book is mainly directed to graduate students in mathematics. It may be used as 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.

 

Book information

  • Published: November 2013
  • Imprint: ACADEMIC PRESS
  • ISBN: 978-0-12-800731-0

Reviews

"...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



Table of Contents

Chapter 1. Basic concepts and results: Upper and lower limits. Semicontinuity. Differentiability. Two important Theorems.
Chapter 2. Convex Functions: Convex sets and convex functions. Continuity of convex functions. Separation Results. Convexity and Differentiability.
Chapter 3. The subdifferential of a Convex function: Subdifferential properties.  Examples.
Chapter 4. The subdifferential. General case: Definition and basic properties. Geometrical meaning of the subdifferential. Density of subdifferentiability points. Proximal subdifferential
Chapter 5. Calculus: Sum Rule. Constrained minima. Chain Rule. Regular functions: Elementary properties. Mean Value results. Decreasing Functions
Chapter 6. Lipschitz functions and the generalized gradient: Lipschitz regular functions. The generalized gradient. Generalized Jacobian. Graphical derivative
Chapter 7. Applications: Flow invariant sets. Viscosity solutions. Solving equations.