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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... Vector Spaces [ 90 ] , ease of ex- tension beyond finite dimensions substantially motivates our choice of ap- proach . Where possible , we have chosen a proof technique permitting those readers familiar with functional analysis to ...
... 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 standard inner product ) , but a more abstract , coordinate ...
Theory and Examples Jonathan Borwein, Adrian S. Lewis. and the cone of vectors with nonincreasing components R2 = { x € R ” | X1 ≥ X2 > ··· > Xn } · The smallest cone containing a given set DCE is clearly R + D . The fundamental ...
... vector u in E , prove u € 0+ ( C ) if and only if there is a sequence ( x " ) in C satisfying || x " || → ∞ and ... vectors a1 , a2 , max ; ( a2 , x ) is convex on E. am in E , prove the function f ( x ) 8. Prove Proposition 1.1.3 ...
... vector space S " into a Euclidean space by defining the inner product ( X , Y ) = tr ( XY ) for X , Y € S " . ... Any matrix X in S " has n real eigenvalues ( counted by multiplicity ) , which we write in nonincreasing order λ1 ( 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 |