Nonsmooth Analysis: Universitext

Preliminaries.- The Conjugate of Convex Functionals.- Classical Derivatives.- The Subdifferential of Convex Functionals.- Optimality Conditions for Convex Problems.- Duality of Convex Problems.- Derivatives and Subdifferentials of Lipschitz Functionals.- Variational Principles.- Subdifferentials of Lower Semicontinuous Functionals.- Multifunctions.- Tangent and Normal Cones.- Optimality Conditions for Nonconvex Problems.- Extremal Principles and More Normals and Subdifferentials.

From the reviews:
"The main idea of the presented monograph is to deal with extremum problems connected with non-differentiable data. The text is divided into 13 chapters with an Appendix and 229 references. Each chapter ends with recommended references and exercises. … The text contains a big amount of latest results achieved in nonsmooth analysis together with applications in optimization. It can be recommended both to graduate students and the researchers in applied mathematics and optimization." (Igor Bock, Zentralblatt MATH, Vol. 1120 (22), 2007)
"The goal of this work by Schirotzek (Technische Universität, Dresden) is to present ‘subdifferentials for general lower semicontinuous functions’--technical, even to a research mathematician’s ear. … Schirotzek’s interesting case study reveals how research-level mathematics revisits and revises elementary ideas. … Summing Up: … Upper-division undergraduates through professionals." (D. V. Feldman, CHOICE, Vol. 45 (9), 2008)
"This book tells the story of the development of nonsmooth analysis … . may be used for a graduate course in nonsmooth analysis or as a basic reference for a short course. It provides a comprehensive overview … and a valuable source of information on forty years of development during which the term ‘nonsmooth analysis’ itself was coined by Clarke. It is even worth reading for those who took part in this development but whose works often lack the reasoned description offered by this book." (Gérard Lebourg, MathSciNet, September, 2008)

The book treats various concepts of generalized derivatives and subdifferentials in normed spaces, their geometric counterparts (tangent and normal cones) and their application to optimization problems. It starts with the subdifferential of convex analysis, passes to corresponding concepts for locally Lipschitz continuous functions and finally 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. The presentation is rigorous, with detailed proofs. Each chapter ends with bibliographic notes and exercises.

The reader is required to have a basic knowledge of functional analysis only. All further prerequisites are presented in the book The presentation strictly proceeds from simple to more difficult The presentation is rigorous, with detailed proofs. Each chapter ends with suggestions for further study and with exercises Includes supplementary material: sn.pub/extras