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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... cone which arises frequently in optimization is the normal cone to a convex set C at a point ≈ Є C , written Nc ( x ) . This is the convex cone of normal vectors , vectors d in E such that ( d , x − x ) ≤ 0 for all points x in C ...
... normal cone Nc ( x ) is not simply { 0 } . The next result shows that when ƒ is convex the first order condition above is sufficient for x to be a global minimizer of ƒ on C. Proposition 2.1.2 ( First order sufficient condition ) ...
... normal cone is a closed convex cone . 2. ( Examples of normal cones ) For the following sets CC E , check C is convex and compute the normal cone Nc ( x ) for points ≈ in C : ( a ) C a closed interval in R. ( b ) CB , the unit ball ...
... normal cone to the set { x ЄE | Ax = b } at any point in it is A * Y . Hence deduce Corollary 2.1.3 ( First order conditions for linear constraints ) . 5. Prove that the differentiable function x2 + x2 ( 1 − x1 ) 3 has a unique ...
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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 |