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 90
... Deduce that the convex hull of a set DCE 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 1 2 x1 , x2 , ... , xm Є C , 0 ≤ A1 , A2 , ... , λm € R , and = 1 ...
... Deduce that if in addition D and C are disjoint then there ex- ists a nonzero element a in E with infrED ( a , x ) > supec ( a , y ) . Interpret geometrically . ( c ) Show part ( b ) fails for the closed convex sets in R2 , D = { X | X1 > ...
... Deduce int C + ( 1 − λ ) cl C C int C. ( d ) Deduce int C is convex . ( e ) Deduce further that if int C is nonempty then cl ( int C ' ) = cl C. Is convexity necessary ? ( Affine sets ) A set L in E is affine if the entire line through ...
... deduce the linear subspace span ( D − x ) is independent of x . ( The relative interior ) ( We use Exercises 11 and ... Deduce ( 1 / ( n + 1 ) ) Σ x2 Є int C. Hence deduce that any nonempty convex set in E has nonempty relative interior ...
... Deduce the inequality n n Σ Zi ≥ ( ÏÏ ži ) 11 1 / n Zi 1 1 for any vector z in R2 . 11. For a fixed column vector s ... deduce the in- equality a ≤ [ x ] T [ y ] . ( c ) Deduce Fan's inequality ( 1.2.2 ) . 13. ( A lower bound ) Use ...
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 |