Bauschke H.H., Combettes P.L. Convex Analysis and Monotone Operator Theory in Hilbert Spaces

2nd Edition. — Springer, 2017. — 620 p. — (CMS Books in Mathematics). — ISBN: 9783319483108.Taking a unique comprehensive approach, the theory is developed from the ground up, with the rich connections and interactions between the areas as the central focus, and it is illustrated by a large number of examples. The Hilbert space setting of the material offers a wide range of applications while avoiding the technical difficulties of general Banach spaces.
The authors have also drawn upon recent advances and modern tools to simplify the proofs of key results making the book more accessible to a broader range of scholars and users. Combining a strong emphasis on applications with exceptionally lucid writing and an abundance of exercises, this text is of great value to a large audience including pure and applied mathematicians as well as researchers in engineering, data science, machine learning, physics, decision sciences, economics, and inverse problems.
The second edition of Convex Analysis and Monotone Operator Theory in Hilbert Spaces greatly expands on the first edition, containing over 140 pages of new material, over 270 new results, and more than 100 new exercises. It features a new chapter on proximity operators including two sections on proximity operators of matrix functions, in addition to several new sections distributed throughout the original chapters. Many existing results have been improved, and the list of references has been updated.Background
Hilbert Spaces
Convex Sets
Convexity and Notions of Nonexpansiveness
Fejér Monotonicity and Fixed Point Iterations
Convex Cones and Generalized Interiors
Support Functions and Polar Sets
Convex Functions
Lower Semicontinuous Convex Functions
Convex Functions: Variants
Convex Minimization Problems
Infimal Convolution
Conjugation
Further Conjugation Results
Fenchel–Rockafellar Duality
Subdifferentiability of Convex Functions
Differentiability of Convex Functions
Further Differentiability Results
Duality in Convex Optimization
Monotone Operators
Finer Properties of Monotone Operators
Stronger Notions of Monotonicity
Resolvents of Monotone Operators
Proximity Operators
Sums of Monotone Operators
Zeros of Sums of Monotone Operators
Fermat’s Rule in Convex Optimization
Proximal Minimization
Projection Operators
Best Approximation Algorithms