Convex Analysis and Monotone Operator Theory in Hilbert Spaces

Heinz H. Bauschke is a Full Professor of Mathematics at the Kelowna campus of the University of British Columbia, Canada.Patrick L. Combettes, IEEE Fellow, was on the faculty of the City University of New York and of Université Pierre et Marie Curie – Paris 6 before joining North Carolina State University as a Distinguished Professor of Mathematics in 2016.

• Background.- Hilbert Spaces.- Convex Sets.- Convexity and Notation of Nonexpansiveness.- Fejer 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.