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 73
... Proposition 1.1.3 ( Weierstrass ) Suppose that the set D C E is non- empty and closed , and that all the level sets of the continuous function f : D → R are bounded . Then f has a global minimizer . Just as for sets , convexity of ...
Theory and Examples Jonathan Borwein, Adrian S. Lewis. → R Proposition 1.1.5 For a convex set C C E , a convex function ƒ : C has bounded level sets if and only if it satisfies the growth condition ( 1.1.4 ) . The proof is outlined in ...
... Proposition 1.1.3 ( Weierstrass ) . = 9. ( Composing convex functions ) Suppose that the set CCE is convex and that the functions f1 , f2 , ... , fn : C → R are convex , and define a function f : CR " with components fi . Suppose ...
... Proposition 1.1.5 . The relative interior Some arguments about finite - dimensional convex sets C simplify and lose no generality if we assume C contains 0 and spans E. The following exer- cises outline this idea . 11 . 12 ...
... Proposition 1.2.4 ( Hardy - Littlewood - Pólya ) Any vectors x and y in R satisfy the inequality xTy≤ [ x ] [ y ] . У We describe a proof of Fan's theorem in the exercises , using the above proposition and the following classical ...
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 |