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  • Book
  • © 2011

Convex Analysis and Monotone Operator Theory in Hilbert Spaces

  • Tight interplay between the key notions of convexity, monotonicity, and nonexpansiveness.

  • Accessible to a broad audience

  • Coverage of many applications of interest to practitioners in finite- and infinite-

  • dimensional spaces

  • More than 500 exercises are distributed throughout the book

  • Includes supplementary material: sn.pub/extras

Part of the book series: CMS Books in Mathematics (CMSBM)

This is a preview of subscription content, access via your institution.

Table of contents (29 chapters)

  1. Front Matter

    Pages i-xvi
  2. Background

    • Heinz H. Bauschke, Patrick L. Combettes
    Pages 1-25
  3. Hilbert Spaces

    • Heinz H. Bauschke, Patrick L. Combettes
    Pages 27-42
  4. Convex Sets

    • Heinz H. Bauschke, Patrick L. Combettes
    Pages 43-58
  5. Convexity and Nonexpansiveness

    • Heinz H. Bauschke, Patrick L. Combettes
    Pages 59-74
  6. Fejér Monotonicity and Fixed Point Iterations

    • Heinz H. Bauschke, Patrick L. Combettes
    Pages 75-86
  7. Convex Cones and Generalized Interiors

    • Heinz H. Bauschke, Patrick L. Combettes
    Pages 87-106
  8. Support Functions and Polar Sets

    • Heinz H. Bauschke, Patrick L. Combettes
    Pages 107-112
  9. Convex Functions

    • Heinz H. Bauschke, Patrick L. Combettes
    Pages 113-127
  10. Lower Semicontinuous Convex Functions

    • Heinz H. Bauschke, Patrick L. Combettes
    Pages 129-141
  11. Convex Functions: Variants

    • Heinz H. Bauschke, Patrick L. Combettes
    Pages 143-153
  12. Convex Variational Problems

    • Heinz H. Bauschke, Patrick L. Combettes
    Pages 155-165
  13. Infimal Convolution

    • Heinz H. Bauschke, Patrick L. Combettes
    Pages 167-180
  14. Conjugation

    • Heinz H. Bauschke, Patrick L. Combettes
    Pages 181-195
  15. Further Conjugation Results

    • Heinz H. Bauschke, Patrick L. Combettes
    Pages 197-206
  16. Fenchel–Rockafellar Duality

    • Heinz H. Bauschke, Patrick L. Combettes
    Pages 207-222
  17. Subdifferentiability

    • Heinz H. Bauschke, Patrick L. Combettes
    Pages 223-240
  18. Differentiability of Convex Functions

    • Heinz H. Bauschke, Patrick L. Combettes
    Pages 241-259
  19. Further Differentiability Results

    • Heinz H. Bauschke, Patrick L. Combettes
    Pages 261-274
  20. Duality in Convex Optimization

    • Heinz H. Bauschke, Patrick L. Combettes
    Pages 275-292

About this book

This book presents a largely self-contained account of the main results of convex analysis, monotone operator theory, and the theory of nonexpansive operators in the context of Hilbert spaces. Unlike existing literature, the novelty of this book, and indeed its central theme, is the tight interplay among the key notions of convexity, monotonicity, and nonexpansiveness. The presentation is accessible to a broad audience and attempts to reach out in particular to the applied sciences and engineering communities, where these tools have become indispensable.

 

Graduate students and researchers in pure and applied mathematics will benefit from this book. It is also directed to researchers in engineering, decision sciences, economics, and inverse problems, and can serve as a reference book.

Author Information:

Heinz H. Bauschke is a Professor of Mathematics at the University of British Columbia, Okanagan campus (UBCO) and currently a Canada Research Chair in Convex Analysis and Optimization. He was born in Frankfurt where he received his "Diplom-Mathematiker (mit Auszeichnung)" from Goethe Universität in 1990. He defended his Ph.D. thesis in Mathematics at Simon Fraser University in 1996 and was awarded the Governor General's Gold Medal for his graduate work. After a NSERC Postdoctoral Fellowship spent at the University of Waterloo, at the Pennsylvania State University, and at the University of California at Santa Barbara, Dr. Bauschke became College Professor at Okanagan University College in 1998. He joined the University of Guelph in 2001, and he returned to Kelowna in 2005, when Okanagan University College turned into UBCO.  In 2009, he became UBCO's first "Researcher of the Year".

Patrick L. Combettes received the Brevet d'Études du Premier Cycle from Académie de Versailles in 1977 and the Ph.D. degree from North Carolina State University in 1989. In 1990, he joined the City College and the Graduate Center of the City University of New York where he became a Full Professor in 1999. Since 1999, he has been with the Faculty of Mathematics of Université Pierre et Marie Curie -- Paris 6, laboratoire Jacques-Louis Lions, where he is presently a Professeur de Classe Exceptionnelle.

He was elected Fellow of the IEEE in 2005.

Reviews

From the reviews:

“This book is devoted to a review of basic results and applications of convex analysis, monotone operator theory, and the theory of nonexpansive mappings in Hilbert spaces. … Each chapter concludes with an exercise section. Bibliographical pointers, a summary of symbols and notation, an index, and a comprehensive reference list are also included. The book is suitable for graduate students and researchers in pure and applied mathematics, engineering and economics.” (Sergiu Aizicovici, Zentralblatt MATH, Vol. 1218, 2011)

“This timely, well-written, informative and readable book is a largely self-contained exposition of the main results … in Hilbert spaces. … The high level of the presentation, the careful and detailed discussion of many applications and algorithms, and last, but not least, the inclusion of more than four hundred exercises, make the book accessible and of great value to students, practitioners and researchers … .” (Simeon Reich, Mathematical Reviews, Issue 2012 h)

Authors and Affiliations

  • Okanagan Campus, Department of Mathematics and Statistic, University of British Columbia, Kelowna, Canada

    Heinz H. Bauschke

  • Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris, France

    Patrick L. Combettes

Bibliographic Information