Name: Convex Analysis and Monotone Operator Theory in Hilbert Spaces
ISBN: 9781441994677

Convex Analysis and Monotone Operator Theory in Hilbert Spaces

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.

What Hacker News says

cpp_frogJun 2021

While I can't give the exact prerequisites, I know that all of the things that appear in the paper relate to:

(1) Linear Algebra

(2) Optimization Theory (Convex Analysis, non-convex optimization) [0], [2]

(3) Probability Theory and Statistics (Measure Theory, Multivariate Statistics) [1], [3], [4], [5]

(4) Analysis, to a lesser extent. (2) and (3) are the most important.

I would give more references, but my background is too theoretical (and my field is Numerical Analysis of PDE). From the classes I took in college, three or four on each of (1-4), a person with a similar background can recognize the tools without much digging. Maybe some folks here can provide some insights into books that center on applications. So I'm trying not to diverge into too much theory (i.e. for measures, [4] instead of Folland). There also seems to be good use of Analysis techniques in the paper, see theorem 2.1.

I love that the paper references the Moore-Penrose pseudo-inverse, an object of study in both statistics and optimization for which I had to give a lecture for a course.

[0] https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf Convex Optimization, Boyd and Vandenberghe

[1] An Introduction to Multivariate Statistical Analysis, Anderson

[2] Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Bauschke-Combettes

[3] Theory of Multivariate Statistics, Bilodeau-Brenner

[4] The Elements of Integration and Lebesgue Measure, Bartle

[5] Probability: Theory and Examples, Durrett

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dplarsonOct 2018

For numerical optimization, a couple good textbooks are:

- "Practical Optimization" by P. E. Gill, W. Murray and M. H. Wright: a little old (1982), but provides a solid foundation

- "Convex Optimization" by S. Boyd and L. Vandenberghe: the standard for learning convex optimization (also available as a free PDF from the author's website)

- "Convex Analysis and Monotone Operator Theory in Hilbert Spaces" by H. H. Bauschke and P. L. Combettes: covers a more specialized area of numerical optimization, but the notation is beautiful (IMO) and it acts as a useful reference for recent research on, e.g., operator splitting methods

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