Convex Analysis and Monotone Operator Theory in Hilbert Spaces

✍ Scribed by Heinz H. Bauschke, Patrick L. Combettes (auth.)

• Publisher: Springer-Verlag New York
• Year: 2011
• Tongue: English
• Leaves: 486
• Series: CMS Books in Mathematics
• Edition: 1
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Front Matter....Pages i-xvi
Background....Pages 1-25
Hilbert Spaces....Pages 27-42
Convex Sets....Pages 43-58
Convexity and Nonexpansiveness....Pages 59-74
Fejér Monotonicity and Fixed Point Iterations....Pages 75-86
Convex Cones and Generalized Interiors....Pages 87-106
Support Functions and Polar Sets....Pages 107-112
Convex Functions....Pages 113-127
Lower Semicontinuous Convex Functions....Pages 129-141
Convex Functions: Variants....Pages 143-153
Convex Variational Problems....Pages 155-165
Infimal Convolution....Pages 167-180
Conjugation....Pages 181-195
Further Conjugation Results....Pages 197-206
Fenchel–Rockafellar Duality....Pages 207-222
Subdifferentiability....Pages 223-240
Differentiability of Convex Functions....Pages 241-259
Further Differentiability Results....Pages 261-274
Duality in Convex Optimization....Pages 275-292
Monotone Operators....Pages 293-309
Finer Properties of Monotone Operators....Pages 311-321
Stronger Notions of Monotonicity....Pages 323-331
Resolvents of Monotone Operators....Pages 333-350
Sums of Monotone Operators....Pages 351-362
Zeros of Sums of Monotone Operators....Pages 363-379
Fermat’s Rule in Convex Optimization....Pages 381-397
Proximal Minimization....Pages 399-413
Projection Operators....Pages 415-430
Best Approximation Algorithms....Pages 431-440
Back Matter....Pages 441-468

✦ Subjects

Calculus of Variations and Optimal Control; Optimization; Algorithms; Visualization