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. |
From inside the book
... bounded if there is a real k satisfying kB D , and it is compact if it is closed and bounded . The following result is a central tool in real analysis . Theorem 1.1.2 ( Bolzano - Weierstrass ) Bounded sequences in E have convergent ...
... bounded . Then f has a global minimizer . Just as for sets , convexity of functions will be crucial for us . convex set CCE , we say that the function ƒ : C → R is convex if ƒ ( Xx + ( 1 − X ) y ) ≤ \ ƒ ( x ) + ( 1 − X ) ƒ ( y ) ...
... 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 ] for lin- ear ...
... 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 || / m → + ∞ . For a fixed point ≈ in C , derive ...
... develop in Section 7.1 . Proposition 2.1.7 If the function f : E → R is differentiable and bounded below then there are points where f has small derivative . Proof . Fix any real € > 0. The function 2.1 Optimality Conditions 17.
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 |