Convex Analysis and Monotone Operator Theory in Hilbert Spaces

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Springer Science & Business Media, Apr 19, 2011 - Mathematics - 468 pages

This book provides a largely self-contained account of the main results of convex analysis and optimization in Hilbert space. A concise exposition of related constructive fixed point theory is presented, that allows for a wide range of algorithms to construct solutions to problems in optimization, equilibrium theory, monotone inclusions, variational inequalities, best approximation theory, and convex feasibility. The book is accessible to a broad audience, and reaches out in particular to applied scientists and engineers, to whom these tools have become indispensable.

 

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Contents

Background
1
Hilbert Spaces
27
Convex Sets
43
Convexity and Nonexpansiveness
59
Fejér Monotonicity and Fixed Point Iterations
75
Convex Cones and Generalized Interiors
87
Support Functions and Polar Sets
107
Convex Functions
113
Further Differentiability Results
261
Duality in Convex Optimization
275
Monotone Operators
293
Finer Properties of Monotone Operators
310
Stronger Notions of Monotonicity
323
Resolvents of Monotone Operators
332
Sums of Monotone Operators
351
Zeros of Sums of Monotone Operators
363

Lower Semicontinuous Convex Functions
128
Convex Functions Variants
143
Convex Variational Problems
154
Infimal Convolution
167
Conjugation
181
Further Conjugation Results
196
FenchelRockafellar Duality
207
Subdifferentiability
223
Differentiability of Convex Functions
241
Fermats Rule in Convex Optimization
381
Proximal Minimization
398
Projection Operators
415
Best Approximation Algorithms
431
Bibliographical Pointers
441
Symbols and Notation
443
References
449
Index
461
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