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 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 elementary real analysis and [ 1 ] ...
... Prove the set D - C is closed and convex . ( b ) Deduce that if in addition D and C are disjoint then there ex- ists ... function f ( x ) 8. Prove Proposition 1.1.3 ( Weierstrass ) . = 9. ( Composing convex functions ) Suppose that the ...
... 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 that ( 1.1.4 ) fails . Deduce the existence of a sequence ( x ) in C with f ( x ) ≤ || xm ...
... Prove the function is norm - preserving . ( c ) Explain why Fan's inequality is a refinement of the Cauchy- Schwarz inequality . 8. Prove the inequality tr Z + tr Z - 1 > 2n for all matrices Z in S with equality if and only if Z = I. ++ ...
... prove the function t Є R ++ → St - log t has compact level sets . ( b ) For c in R2 + , prove the function x = R → c1x has compact level sets . ++ n → c1x - log Xi ( c ) For C in S ++ , prove the function X € S3 ++ i = 1 → ( C , 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 |