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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... Semidefinite Programming . 109 5.4 Convex Process Duality 6 Nonsmooth Optimization 6.1 Generalized Derivatives 6.2 Regularity and Strict Differentiability 6.3 Tangent Cones 6.4 The Limiting Subdifferential . 114 123 123 130 137 145 7 ...
... semidefinite matrices , S. ( We call a matrix X in S " positive semidefinite if xa Xx ≥ 0 for all vectors x in R2 , and positive definite if the inequality is strict whenever x is nonzero . ) These two cones have some important ...
... semidefinite . Con- versely , if the Hessian is positive definite then I is a local minimizer . ( In fact for a to be a local minimizer it is sufficient for the Hessian to be positive semidefinite locally ; the function x Є R → x1 ...
... semidefinite matrix to UT Diag yU is UTDiag y + U . 9. * ( Coercivity ) Suppose that the function f : E → R is differentiable and satisfies the growth condition lim |||| ∞ f ( x ) / || x || *** + ∞ . Prove that the gradient map Vƒ ...
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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 |