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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Results 6-10 of 87
... convex set in E. ( a ) Prove cl C C C + € B for any real € > 0 . ( b ) For sets D and F in E with D open , prove D + F is open . ** ( c ) For x in int C and 0 < ≤ 1 , prove λx + ( 1 - X ) el C C C. Deduce int C + ( 1 − λ ) cl C C int ...
... convex set C in E , denoted ri C , is its interior relative to its affine hull . In other words , a point x lies in ri C if there is a real 80 with ( x + 8B ) naff C C C. ( a ) Find convex sets C1 C C2 with ri C1 Z ri C2 . n 1 ( b ) ...
... sets of perturbed log barriers ) ( a ) For 8 in R ++ , prove the function t Є R ++ → St - log t has compact level sets ... set of matrices { X1 , X2 , ...... . , Xm ) has a simultaneous ordered spectral de- composition . ** ( Singular ...
... convex cone which arises frequently in optimization is the normal cone to a convex set C at a point ≈ Є C , written Nc ( x ) . This is the convex cone of normal vectors , vectors d in E such that ( d , x − x ) ≤ 0 for all points x in ...
... set C of interest , so the normal cone Nc ( x ) is not simply { 0 } . The next result shows that when ƒ is convex the first order condition above is sufficient for x to be a global minimizer of ƒ on C. Proposition 2.1.2 ( First order ...
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 |