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 55
... Matrices 2 Inequality Constraints Optimality Conditions 2.2 Theorems of the Alternative 9 15 15 23 2.3 Max - functions . 28 3 Fenchel Duality 33 3.1 Subgradients and Convex Functions 33 3.2 The Value Function 43 3.3 The Fenchel ...
... y ) in C. ( iii ) R + ( C – x ) is a linear subspace . ( e ) If F is another Euclidean space and the map A : E → F is linear , prove ri AC Ɔ Ari C. 1.2 Symmetric Matrices Throughout most of this book our setting 8 1. Background.
... matrices ( with analogous definitions for and > ) . By contrast , it is straightforward to see S is not ' ++ ' a lattice cone ( Exercise 4 ) . We denote the identity matrix by I. The trace of a square matrix Z is the sum of the diagonal ...
... matrices ( those matrices U satisfying UTU = I ) . Then any matrix X in S " has an ordered spectral decomposition X = UT ( Diag \ ( X ) ) U , for some matrix U in O " . This shows , for example , that the function A is norm - preserving ...
... matrix X in S " satisfies ( X2 ) 1/2 ≥ X. ( b ) Find matrices X Y in S2 such that X2 Y2 . ( c ) For matrices X Y in Sn , prove X1 / 2 Y1 / 2 . ( Hint : Consider the relationship ( ( X1 / 2 + Y1 / 2 ) x , ( X1 / 2 - Y1 / 2 ) x ) = ( ( 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 |