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
Results 1-5 of 82
... Theorem 160 7.3 Metric Regularity and the Limiting Subdifferential 7.4 Second Order Conditions 166 172 8 Fixed Points 179 8.1 The Brouwer Fixed Point Theorem . 179 8.2 Selection and the Kakutani - Fan Fixed Point Theorem 190 8.3 ...
... Theorem 1.2.1 ( Fan ) Any matrices X and Y in Sn satisfy the inequality tr ( XY ) ≤ \ ( X ) TM A ( Y ) . ( 1.2.2 ) Equality holds if and only if X and Y have a simultaneous ordered spectral decomposition : there is a matrix U in On ...
... theorems of the alternative " , and , in particular , the Farkas lemma ( which we derive at the end of this section ) . Our first approach , however , relies on a different theorem of the alternative . Theorem 2.2.1 ( Gordan ) For any ...
... theorem . We now proceed by using Gordan's theorem to derive the Farkas lemma , one of the cornerstones of many approaches to optimality conditions . The proof uses the idea of the projection onto a linear subspace Y of E. Notice first ...
... theorem appeared in [ 84 ] , and the Farkas lemma appeared in [ 75 ] . The standard modern approach to theorems of the alternative ( Exercises 7 and 8 , for example ) is via linear programming duality ( see , for example , [ 53 ] ...
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 |