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
... cone polarity ) , we constantly emphasize the power of abstract models and notation . Good reference works on finite - dimensional convex analysis already ex- ist . Rockafellar's classic Convex Analysis [ 167 ] has been indispensable ...
... cone . ( Notice we require that cones contain the origin . ) Examples are the positive orthant = R = { x R " | each x ; ≥ 0 } , and the cone of vectors with nonincreasing components R2 = 1 Chapter 1 Background 1.1 Euclidean Spaces.
... convex sets originate with Minkowski [ 142 ] . The theory of the relative interior ( Exercises 11 , 12 , and 13 ) is devel- oped extensively in [ 167 ] ( which is also a good reference for the recession cone , Exercise 6 ) . 1. Prove ...
... convex sets in R2 , D = { X | X1 > 0 , X1X2 ≥ 1 } , 6 . ** C = { x | x2 = 0 } . ( Recession cones ) Consider a nonempty closed convex set CC E. We define the recession cone of C by 0 * ( C ) = { d € E│C + R + d C C } . ( a ) Prove 0+ ...
... cones have some important differences ; in particular , R is a polyhedron , whereas the cone of positive semidefinite matrices S is not , even for n = 2. The cones R2 and S2 are important largely because of the orderings they induce ...
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 |