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 1-5 of 91
... shows that intersections of convex sets are convex . Given any set D C E , the linear span of D , denoted span ( D ) ... show that D is open exactly when its complement Dc is closed , and that arbitrary unions and finite intersections of ...
... are convex , B is closed , and C is bounded , prove АС В. ( Hint : Observe 2A + C = A + ( A + C ) C 2B + C . ) ( b ) Show this result can fail if B is not convex . 5. * ( Strong separation ) Suppose that the set 1.1 Euclidean Spaces ст 5.
... Show part ( b ) fails for the closed convex sets in R2 , D = { X | X1 > 0 , X1X2 ≥ 1 } , 6 . ** C = { x | x2 = 0 } . ( Recession cones ) Consider a nonempty closed convex set CC E. We define the recession cone of C by 0 * ( C ) = { d ...
... shows , for example , that the function A is norm - preserving : || X || = || X ( X ) || for all X in S " . For any X in S , the spectral decomposition also shows there is a unique matrix X1 / 2 in S whose square is X. The Cauchy ...
... 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 , written Nc ( x ) . This is the convex cone of ...
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 |