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 1-5 of 42
... zero , one , or two asterisks , respectively , as follows : examples that illustrate the ideas in the text or easy expansions of sketched proofs ; important pieces of additional theory or more testing examples ; longer , harder examples ...
... zero or one , and with exactly one entry of one in each row and in each column ) . Theorem 1.2.5 ( Birkhoff ) Doubly stochastic matrices are convex com- binations of permutation matrices . We defer the proof to a later section ( Section ...
... For a matrix A in M " we define the singular values of A by σ ; ( A ) √ ( ATA ) for i = 1 , 2 , ... , n , and hence define a map σ : Mn → R. ( Notice zero may be a singular value . ) ( a ) Prove 入[ & ] - [ 1.2 Symmetric Matrices 13.
... zero for all ( i , j ) in △ satisfies Ln S + 0. By considering the problem ( for CEST ) ++ inf { ( C , X ) – log det X | X Є Lns " } , — ++ use Section 1.2 , Exercise 14 and Corollary 2.1.3 ( First order con- ditions for linear ...
... zero , satisfying Aoc = Στο λια . If λo > 0 the proof is complete , so suppose Ao = 0 and without loss of generality Am > 0 . λo m Define a subspace of E by Y = { y | ( am , y ) = 0 } , so by assumption the system ≤ ( a ' , y ) 0 for i ...
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 |