Convex analysis and monotone operator theory in Hilbert spaces / Heinz H. Bauschke, Patrick L. Combettes

Auteur: Bauschke, Heinz H. (1964-) - AuteurCo-auteur: Combettes, Patrick Louis - AuteurType de document: Monographie Collection: CMS books in mathematics Langue: anglaisPays: Etats UnisÉditeur: New York : Springer, cop. 2011Description: 1 vol. (XVI-468 p.) : ill. ; 25 cm ISBN: 9781441994660 ; rel. Note: 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. The choice of the Hilbert space setting enables the authors to cover many interesting applications, while avoiding the technical difficulties related to a general Banach space framework. The monograph is divided into 29 chapters, entitled as follows: Background (Chapter 1), Hilbert Spaces (Chapter 2), Convex Sets (Chapter 3), Convexity and Nonexpansiveness (Chapter 4), Fejér Monotonicity and Fixed Point Iterations (Chapter 5), Convex Cones and Generalized Interiors (Chapter 6), Support Functions and Polar Sets (Chapter 7), Convex Functions (Chapter 8), Lower Semicontinuous Convex Functions (Chapter 9), Convex Functions: Variants (Chapter 10), Convex Variational Problems (Chapter 11), Infimal Convolution (Chapter 12), Conjugation (Chapter 13), Further Conjugation Results (Chapter 14), Fenchel-Rockafellar Duality (Chapter 15), Subdifferentiability (Chapter 16), Differentiability of Convex Functions (Chapter 17), Further Differentiability Results (Chapter 18), Duality in Convex Optimization (Chapter 19), Monotone Operators (Chapter 20), Finer Properties of Monotone Operators (Chapter 21), Stronger Notions of Monotonicity (Chapter 22), Resolvents of Monotone Operators (Chapter 23), Sums of Monotone Operators (Chapter 24), Zeros of Sums of Monotone Operators (Chapter 25), Fermat’s Rule in Convex Optimization (Chapter 26), Proximal Minimization (Chapter 27), Projection Operators (Chapter 28), and Best Approximation Algorithms (Chapter 29). 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. (zbMath)Bibliographie: Bibliogr. p. 449-459. Index. Sujets MSC: 47H05 Operator theory -- Nonlinear operators and their properties -- Monotone operators and generalizations
47H09 Operator theory -- Nonlinear operators and their properties -- Contraction-type mappings, nonexpansive mappings, A-proper mappings, etc
52A41 Convex and discrete geometry -- General convexity -- Convex functions and convex programs
46C05 Functional analysis -- Inner product spaces and their generalizations, Hilbert spaces -- Hilbert and pre-Hilbert spaces: geometry and topology
90C25 Operations research, mathematical programming -- Mathematical programming -- Convex programming
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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. The choice of the Hilbert space setting enables the authors to cover many interesting applications, while avoiding the technical difficulties related to a general Banach space framework.

The monograph is divided into 29 chapters, entitled as follows: Background (Chapter 1), Hilbert Spaces (Chapter 2), Convex Sets (Chapter 3), Convexity and Nonexpansiveness (Chapter 4), Fejér Monotonicity and Fixed Point Iterations (Chapter 5), Convex Cones and Generalized Interiors (Chapter 6), Support Functions and Polar Sets (Chapter 7), Convex Functions (Chapter 8), Lower Semicontinuous Convex Functions (Chapter 9), Convex Functions: Variants (Chapter 10), Convex Variational Problems (Chapter 11), Infimal Convolution (Chapter 12), Conjugation (Chapter 13), Further Conjugation Results (Chapter 14), Fenchel-Rockafellar Duality (Chapter 15), Subdifferentiability (Chapter 16), Differentiability of Convex Functions (Chapter 17), Further Differentiability Results (Chapter 18), Duality in Convex Optimization (Chapter 19), Monotone Operators (Chapter 20), Finer Properties of Monotone Operators (Chapter 21), Stronger Notions of Monotonicity (Chapter 22), Resolvents of Monotone Operators (Chapter 23), Sums of Monotone Operators (Chapter 24), Zeros of Sums of Monotone Operators (Chapter 25), Fermat’s Rule in Convex Optimization (Chapter 26), Proximal Minimization (Chapter 27), Projection Operators (Chapter 28), and Best Approximation Algorithms (Chapter 29).

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. (zbMath)

Bibliogr. p. 449-459. Index

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