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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... closed if D = cl D. Linear subspaces of E are important examples of closed sets . Easy exercises show that D is open exactly when its complement Dc is closed , and that arbitrary unions and finite intersections of open sets are open ...
... closed halfspace whereas y is not . Thus there is a “ dual " representation of C as the intersection of all closed halfspaces containing it . The set D is bounded if there is a real k satisfying kB D , and it is compact if it is closed ...
... 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 functions will be crucial for us . convex set CCE , we say that the function ƒ : C ...
... convex combinations of elements of D. 3. Prove that a convex set DCE has convex closure , and deduce that cl ( conv D ) is the smallest closed convex set containing D. 4. ( Radstrom cancellation ) Suppose sets A , B , C C E satisfy A + ...
... closed and convex , and that the set DCE is compact and convex . ( a ) Prove the set D - C is closed and convex . ( b ) Deduce that if in addition D and C are disjoint then there ex- ists a nonzero element a in E with infrED ( a , x ) ...
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 |