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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... function f : D → R is a point ≈ in D at which ƒ attains its infimum inf ƒ = inf f ( D ) = inf { ƒ ( x ) | x Є D } . D f In this case we refer to ĩ as an optimal solution of the optimization problem infp f . For a positive real 8 and a ...
... convex set C C E , a convex function ƒ : C has bounded level sets if and only if it satisfies the growth condition ( 1.1.4 ) . The proof is outlined in Exercise 10 . Exercises and Commentary Good general references are [ 177 ] for ...
... function with bounded level sets which does not satisfy the growth condition ( 1.1.4 ) . ( b ) Prove that any function satisfying ( 1.1.4 ) has bounded level sets . ( c ) Suppose the convex function f : C → R has bounded level sets but ...
... function X € S2 + → log det X. An exercise shows this function is differentiable on S with derivative X - 1 . ++ ++ H A convex cone which arises frequently in optimization is the normal cone to a convex set C at a point ≈ Є C ...
... function subject to constraints . In that case local minimizers ĩ may not lie in the interior of the set C of interest , so the normal cone Nc ( x ) is not simply { 0 } . The next result shows that when ƒ is convex ... convex and that the ...
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 |