Convex Analysis and Nonlinear Optimization

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

Résumé
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.

Contenu

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.