by Jonathan M. Borwein

Preface

Optimization is a rich and thriving mathematical discipline. Properties of
minimizers and maximizers of functions rely intimately on a wealth of techniques
from mathematical analysis, including tools from calculus and its
generalizations, topological notions, and more geometric ideas. The theory
underlying current computational optimization techniques grows ever
more sophisticated – duality-based algorithms, interior point methods, and
control-theoretic applications are typical examples. The powerful and elegant
language of convex analysis unifies much of this theory. Hence our aim of
writing a concise, accessible account of convex analysis and its applications
and extensions, for a broad audience.

For students of optimization and analysis, there is great benefit to blurring
the distinction between the two disciplines. Many important analytic
problems have illuminating optimization formulations and hence can be approached
through our main variational tools: subgradients and optimality
conditions, the many guises of duality, metric regularity and so forth. More
generally, the idea of convexity is central to the transition from classical
analysis to various branches of modern analysis: from linear to nonlinear
analysis, from smooth to nonsmooth, and from the study of functions to
multifunctions. Thus although we use certain optimization models repeatedly
to illustrate the main results (models such as linear and semidefinite
programming duality and cone polarity), we constantly emphasize the power
of abstract models and notation.

Good reference works on finite-dimensional convex analysis already exist.
Rockafellar’s classic Convex Analysis [149] has been indispensable and ubiquitous
since the 1970’s, and a more general sequel with Wets, Variational
Analysis [150], appeared recently. Hiriart-Urruty and Lemar´echal’s Convex
Analysis and Minimization Algorithms [86] is a comprehensive but gentler
introduction. Our goal is not to supplant these works, but on the contrary
to promote them, and thereby to motivate future researchers. This book
aims to make converts.

We try to be succinct rather than systematic, avoiding becoming bogged
down in technical details. Our style is relatively informal: for example, the
text of each section sets the context for many of the result statements. We
value the variety of independent, self-contained approaches over a single,
unified, sequential development. We hope to showcase a few memorable
principles rather than to develop the theory to its limits. We discuss no
algorithms. We point out a few important references as we go, but we make
no attempt at comprehensive historical surveys.

Infinite-dimensional optimization lies beyond our immediate scope. This
is for reasons of space and accessibility rather than history or application:
convex analysis developed historically from the calculus of variations, and
has important applications in optimal control, mathematical economics, and
other areas of infinite-dimensional optimization. However, rather like Halmos’s
Finite Dimensional Vector Spaces [81], ease of extension beyond finite
dimensions substantially motivates our choice of results and techniques.
Wherever possible, we have chosen a proof technique that permits those readers
familiar with functional analysis to discover for themselves how a result
extends. We would, in part, like this book to be an entr´ee for mathematicians
to a valuable and intrinsic part of modern analysis. The final chapter
illustrates some of the challenges arising in infinite dimensions.

This book can (and does) serve as a teaching text, at roughly the level
of first year graduate students. In principle we assume no knowledge of real
analysis, although in practice we expect a certain mathematical maturity.
While the main body of the text is self-contained, each section concludes with
an often extensive set of optional exercises. These exercises fall into three categories,
marked with zero, one or two asterisks respectively: examples which
illustrate the ideas in the text or easy expansions of sketched proofs; important
pieces of additional theory or more testing examples; longer, harder
examples or peripheral theory.

We are grateful to the Natural Sciences and Engineering Research Council
of Canada for their support during this project. Many people have helped
improve the presentation of this material. We would like to thank all of
them, but in particular Guillaume Haberer, Claude Lemar´echal, Olivier Ley,
Yves Lucet, Hristo Sendov, Mike Todd, Xianfu Wang, and especially Heinz
Bauschke.

[此贴子已经被作者于2008-9-20 17:15:34编辑过]