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
... set D C E is clearly R + D . The fundamental geometric idea of this book is convexity . A set C in . E is convex ... convex sets are convex . Given any set DCE , the ... C C E is closed and convex , and that the point y does not lie in C. Then ...
... set D C E is non- empty and closed , and that all the level sets of the continuous function f : DR are bounded . Then ƒ has a global minimizer . Just as for sets , convexity of functions will be crucial for us . Given a convex set CCE ...
... convex sets is con- vex . Deduce that the convex hull of a set D C E is well - defined as the intersection of all convex sets containing D. 2. ( a ) Prove that if the set C C E is convex and if " ... " xm € С , 0 ≤ ... " 1 , λ2 ...
... set CCE is 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 ...
... set C of interest , so the normal cone Nċ ( x ) is not simply { 0 } . The next result shows that when ƒ is convex ... CCE is convex and that the function f : C → R is convex . Then for any points x and x in C , the directional derivative f ...
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 |