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
Results 6-10 of 84
... result can fail without the last assumption . ( f ) Consider another nonempty closed convex set DC E such that 0+ ( C ) n0 + ( D ) is a linear subspace . Prove CD is closed . 7. For any set of vectors a1 , a2 , max ; ( a2 , x ) is ...
... result in linear algebra states that matrices X and Y have a simultaneous ( unordered ) spectral decomposition if and only if they com- mute . Notice condition ( 1.2.3 ) is a stronger property . The special case of Fan's inequality ...
... result where C is an open set is the canonical intro- duction to the use of calculus in optimization : local minimizers must be critical points ( that is , ▽ ƒ ( x ) = 0 ) . This book is largely devoted to the study of first order ...
... result is called a Lagrange multiplier . This kind of construction recurs in many different forms in our development . In the absence of convexity , we need second order information to tell us more about minimizers . The following ...
... result concerns derivatives . Exercises and Commentary The optimality conditions in this section are very standard ( see for example [ 132 ] ) . The simple variational principle ( Proposition 2.1.7 ) was suggested by [ 95 ] . 1. Prove ...
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 |