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Convex Analysis and Nonlinear Optimization de Jonathan Borwein

Convex Analysis and Nonlinear Optimization: CMS Books in Mathematics

Autor Jonathan Borwein, Adrian S. Lewis
en Limba Engleză Paperback – dec 2010

Observăm în literatura academică dedicată matematicii aplicate o lacună între tratatele teoretice rigide și manualele de calcul algoritmic, spațiu pe care Convex Analysis and Nonlinear Optimization îl ocupă cu o eleganță rară. Considerăm că forța acestui volum, aflat la a doua ediție, rezidă în capacitatea de a unifica teoria optimizării computaționale prin limbajul analizei convexe. Față de ediția princeps, autorii Jonathan Borwein și Adrian S. Lewis au introdus secțiuni noi despre optimizarea semismooth și au rafinat demonstrațiile pentru a oferi un text complet autonom.

Subliniem structura logică a lucrării, care progresează natural de la fundamentele inegalităților și dualitatea Fenchel către aplicații complexe în optimizarea nonsmooth și teoria Karush-Kuhn-Tucker. Această abordare acoperă aceeași arie tematică precum volumul clasic Convex Analysis de Ralph Tyrell Rockafellar, dar cu o orientare mai pronunțată spre nevoile pedagogice ale studenților de masterat, integrând exerciții opționale care transformă teoria pură în instrumente de lucru.

În contextul operei sale vaste, Jonathan Borwein face aici trecerea de la explorarea istorică sau experimentală, prezentă în lucrări precum Pi: A Source Book sau The Computer as Crucible, către rigoarea analizei matematice fundamentale. Descoperim aici un echilibru între precizia matematică și accesibilitate, volumul fiind esențial pentru înțelegerea mecanismelor din spatele algoritmilor moderni folosiți în inginerie și economie.

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Specificații

ISBN-13: 9781441921277
ISBN-10: 1441921273
Pagini: 324
Ilustrații: XII, 310 p.
Dimensiuni: 155 x 235 x 18 mm
Greutate: 0.49 kg
Ediția:Second Edition 2006
Editura: Springer
Colecția CMS Books in Mathematics
Seria CMS Books in Mathematics

Locul publicării:New York, NY, United States

Public țintă

Graduate

De ce să citești această carte

Recomandăm această carte studenților și cercetătorilor care doresc o bază solidă în teoria optimizării. Spre deosebire de manualele pur algoritmice, acest volum oferă fundamentul matematic necesar pentru a înțelege „de ce” funcționează tehnicile de calcul. Cititorul câștigă o perspectivă unitară asupra analizei convexe, sprijinită de numeroase exerciții și demonstrații noi, esențiale pentru parcursul academic în matematică sau cercetare operațională.

Despre autor

Jonathan Borwein a fost un matematician de renume mondial, profesor laureat la Universitatea din Newcastle și director al centrului CARMA. Recunoscut ca cercetător de elită de către ISI și laureat al premiului Chauvenet, Borwein a publicat extensiv în diverse ramuri ale matematicii, de la analiză funcțională la matematică experimentală. Expertiza sa în utilizarea tehnologiei pentru explorarea conceptelor matematice se reflectă în claritatea și structura modernă a lucrării Convex Analysis and Nonlinear Optimization, facilitând accesul studenților la idei matematice complexe.

Descriere scurtă

Optimization is a rich and thriving mathematical discipline. The theory underlying current computational optimization techniques grows ever more sophisticated. The powerful and elegant language of convex analysis unifies much of this theory. The aim of this book is to provide a concise, accessible account of convex analysis and its applications and extensions, for a broad audience. It can serve as a teaching text, at roughly the level of first year graduate students. While the main body of the text is self-contained, each section concludes with an often extensive set of optional exercises. The new edition adds material on semismooth optimization, as well as several new proofs that will make this book even more self-contained.

Cuprins

Background.- Inequality Constraints.- Fenchel Duality.- Convex Analysis.- Special Cases.- Nonsmooth Optimization.- Karush—Kuhn—Tucker Theory.- Fixed Points.- More Nonsmooth Structure.- Postscript: Infinite Versus Finite Dimensions.- List of Results and Notation.

Recenzii

From the reviews:
MATHEMATICAL REVIEWS
"The present book gives a concise treatment of the area, aiming to show the relevance in particular of new developments in nonsmooth analysis to optimization theory…The book is of a manageable size and as such should appeal to the student. Further, the proofs are generally short and snappy, revealing the power of the abstract structural approach and fruitful interplay of geometrical and topological ideas. However, considerable ground is covered and, as a graduate text should, it develops the subject up to the frontiers of current research, giving an idea of areas for further exploration…This text will give impetus to the teaching of analysis because it makes evident its significant applications in optimization. But it will also bring added attraction to the study of optimization because it reveals so much of its abstract structural base."
"The book is divided into 11 chapters and provides a comprehensive presentation of the main features of convex analysis and nonlinear optimisation. Each result is sustained by a set of theorems, propositions and corollaries and includes rigorous proofs and clarifying discussions. They are complemented by a series of theoretical exercises. … This book is warmly recommended for an advanced course in analysis for mathematicians or as a first graduate course for students involved with optimization theory." (Carlos Narciso Bouza Herrera, Zentralblatt MATH, Vol. 1116 (18), 2007)

Textul de pe ultima copertă

A cornerstone of modern optimization and analysis, convexity pervades applications ranging through engineering and computation to finance.
This concise introduction to convex analysis and its extensions aims at first year graduate students, and includes many guided exercises. The corrected Second Edition adds a chapter emphasizing concrete models. New topics include monotone operator theory, Rademacher's theorem, proximal normal geometry, Chebyshev sets, and amenability. The final material on "partial smoothness" won a 2005 SIAM Outstanding Paper Prize.
Jonathan M. Borwein, FRSC is Canada Research Chair in Collaborative Technology at Dalhousie University. A Fellow of the AAAS and a foreign member of the Bulgarian Academy of Science, he received his Doctorate from Oxford in 1974 as a Rhodes Scholar and has worked at Waterloo, Carnegie Mellon and Simon Fraser Universities. Recognition for his extensive publications in optimization, analysis and computational mathematics includes the 1993 Chauvenet prize.
 
Adrian S. Lewis is a Professor in the School of Operations Research and Industrial Engineering at Cornell. Following his 1987 Doctorate from Cambridge, he has worked at Waterloo and Simon Fraser Universities. He received the 1995 Aisenstadt Prize, from the University of Montreal, and the 2003 Lagrange Prize for Continuous Optimization, from SIAM and the Mathematical Programming Society.
 
About the First Edition:
 
"...a very rewarding book, and I highly recommend it... "
- M.J. Todd, in the International Journal of Robust and Nonlinear Control
 
"...a beautifully written book... highly recommended..."
- L. Qi, in the Australian Mathematical Society Gazette
 
"This book represents a tour de force for introducing so many topics of present interest in such a small space and with such clarity and elegance."
- J.-P. Penot,in Canadian Mathematical Society Notes
 
"There is a fascinating interweaving of theory and applications..."
- J.R. Giles, in Mathematical Reviews
"...an ideal introductory teaching text..."
- S. Cobzas, in Studia Universitatis Babes-Bolyai Mathematica

Caracteristici

Reviews the increasingly sophisticated state of computational optimization techniques Provides an accessible account of convex analysis and its applications and extensions New Edition adds material on semismooth optimization, as well as several new proofs The self-contained main body of the book is supplemented with optional exercises at the end of each section Includes supplementary material: sn.pub/extras