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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... subspace containing D. It consists exactly of all linear combinations of elements of D. Analogously , the convex ... subspaces of E are important examples of closed sets . Easy exercises show that D is open exactly when its complement Dc ...
... subspace G of E , the orthogonal complement of G is the subspace - G1 = { y Є E | ( x , y ) = 0 for all x Є G } , so called because we can write E as a direct sum GG1 . ( In other words , any element of E can be written uniquely as the ...
... subspace , prove AC is closed . Show this 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 ...
... and only if it is a translate of a linear subspace . ( c ) Prove aff D is the set of all affine combinations of elements of D. ( d ) Prove cl DC aff D and deduce aff D = aff ( cl D ) . 13 . ** ( e ) For any point x 1.1 Euclidean Spaces ་ 7.
... subspace span ( D − x ) is independent of x . ( The relative interior ) ( We use Exercises 11 and 12. ) The relative interior of a convex set C in E , denoted ri C , is its interior relative to its affine hull . In other words , a ...
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 |