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. |
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... function f : R → R has a critical point x . If x is a local minimizer then the Hessian ▽ 2ƒ ( x ) is positive ... convex , and that the point y does not lie in C. Then there exist a real b and a nonzero element a of E such that ( a , y ) > ...
... function x3 + x2 ( 1 − x1 ) 3 has a unique critical point in R2 , which is a local minimizer , but has no global minimizer . Can this happen on R ? 6. ( The Rayleigh quotient ) ( a ) Let the function ƒ : R " \ { 0 } → R be continuous ...
... function f : E → R is differentiable and satisfies the growth condition lim || ~ || → ∞ f ( x ) / || x || = + ∞ . Prove that the gradient map Vƒ has range E. ( Hint : Minimize the function f ( · ) – ( a ,. ) for elements a of E ...
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Contents
7 | |
15 | |
Fenchel Duality | 33 |
Convex Analysis | 65 |
Special Cases | 97 |
Nonsmooth Optimization | 123 |
Fixed Points | 183 |
Infinite Versus Finite Dimensions | 209 |
List of Results and Notation | 221 |
Bibliography | 241 |
Other editions - View all
Convex Analysis and Nonlinear Optimization: Theory and Examples Jonathan M. Borwein,Adrian S. Lewis No preview available - 2000 |