Synopses & Reviews
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.
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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.
Review
From the reviews: "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. ... 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." (Sergiu Aizicovici, Zentralblatt MATH, Vol. 1218, 2011) "This timely, well-written, informative and readable book is a largely self-contained exposition of the main results ... in Hilbert spaces. ... The high level of the presentation, the careful and detailed discussion of many applications and algorithms, and last, but not least, the inclusion of more than four hundred exercises, make the book accessible and of great value to students, practitioners and researchers ... ." (Simeon Reich, Mathematical Reviews, Issue 2012 h)
Review
From the reviews:
"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. ... 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." (Sergiu Aizicovici, Zentralblatt MATH, Vol. 1218, 2011)
"This timely, well-written, informative and readable book is a largely self-contained exposition of the main results ... in Hilbert spaces. ... The high level of the presentation, the careful and detailed discussion of many applications and algorithms, and last, but not least, the inclusion of more than four hundred exercises, make the book accessible and of great value to students, practitioners and researchers ... ." (Simeon Reich, Mathematical Reviews, Issue 2012 h)
Synopsis
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.
Synopsis
This book examines results of convex analysis and optimization in Hilbert space, presenting a concise exposition of related theory that allows for algorithms to construct solutions to problems in optimization, equilibrium theory, monotone inclusions and more.
Table of Contents
Background * Convex sets * Finer properties of convex sets * Convex cones * Conjugation * Monotone operators * Subdifferentiability * Convex optimization * Lipschitz operators * Resolvents * Prox operators * Projections onto convex sets * Subgradient projections * Tau-class * Fejer monotonicity * Regularization * Finding common fixed points * Averaging techniques * Finding fixed points of compositions * Best approximation problems * Variational problems