Convex Analysis and Nonlinear Optimization: Theory and ExamplesOptimization 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. |
From inside the book
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... 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 ...
... minimizers and maximizers of func- tions . Given a set △ CR , the infimum of △ ( written inf A ) is the greatest lower bound on A , and the supremum ( written sup A ) is the least upper bound . To ensure these are always defined , it ...
... minimizer of a function f : D → R is a point ≈ in D at which ƒ attains its infimum inf ƒ = inf f ( D ) = inf { ƒ ( x ) | x Є D } . D f In this case we refer to ĩ as an optimal solution of the optimization problem infp f . For a ...
... minimizer of ƒ on C if f ( x ) ≥ f ( x ) for all points x in C close to x . The directional derivative of a function ƒ at ≈ in a direction d € E is f ( x + td ) − f ( x ) f ' ( x ; d ) - lim t ↓ 0 t when this limit exists . When the ...
... minimizers must be critical points ( that is , ▽ ƒ ( x ) = 0 ) . This book is largely devoted to the study of first order necessary optimality conditions for a local minimizer of a function subject to constraints . In that case local ...
Contents
1 | |
15 | |
Chapter 3 Fenchel Duality | 33 |
Chapter 4 Convex Analysis | 65 |
Chapter 5 Special Cases | 97 |
Chapter 6 Nonsmooth Optimization | 123 |
Chapter 7 KarushKuhnTucker Theory | 153 |
Chapter 8 Fixed Points | 179 |
Chapter 9 More N onsmooth Structure | 213 |
Infinite Versus Finite Dimensions | 239 |
Chapter 11 List of Results and Notation | 253 |
Bibliography | 275 |
Index | 289 |
Other editions - View all
Convex Analysis and Nonlinear Optimization: Theory and Examples Jonathan M. Borwein,Adrian S. Lewis No preview available - 2000 |