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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... Euclidean space E , by which we mean a finite - dimensional vector space over the reals R , equipped with an inner product ( , ) . We would lose no generality if we considered only the space Rn of real ( column ) n - vectors ( with its ...
... Euclidean space Y , we call a map A : E → Y linear if any points x and z in E and any reals A and μ satisfy A ( Ax + uz ) = Ax + μAz . In fact any linear function from E to R has the form ( a , ) for some element a of E. Linear maps ...
... Euclidean space , the map A : E → Y is linear , and N ( A ) n0 + ( C ) is a linear subspace , prove AC is closed . Show this result can fail without the last assumption . ( f ) Consider another nonempty closed convex set DC E such that ...
... 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.
... space beyond its simple Euclidean structure . As an example , in this short section we describe a Euclidean space which " feels " very different from R " : the space Sn of nxn real symmetric matrices . The nonnegative orthant R is a ...
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 |