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 55
... - dual cones ) Prove each of the following cones K satisfy the relationship NK ( 0 ) = −K . ( a ) R ( b ) ST + ( c ) { x ЄR " | x1 ≥ 0 , x2 ≥ x2 + x3 + + x2 } → Y ( where 4. ( Normals to affine sets 18 2. Inequality Constraints.
... constraints ) . 5. Prove that the differentiable function x2 + x2 ( 1 − x1 ) 3 has a unique critical point in R2 , which is a local minimizer , but has no global minimizer . Can this happen on R ? 6. ( The Rayleigh quotient ) ( a ) Let ...
... sets . ( b ) Deduce the existence of a solution to the following equations ( describing " conservation of current " ) : Σ Xi Xj Tij = 0 for i in Vo jijЄE Χα Ꮖ Ᏸ = = 0 1 . 12 . 13 . ( c ) Prove the power 20 2. Inequality Constraints.
... constraints ) to prove there exists a matrix X in LOS with C – X - 1 having ( i , j ) th entry of zero for all ( i , j ) not ԼՈՏ ++ in A. ** ( BFGS update , cf. [ 80 ] ) Given a matrix C in S2 + and vectors s and y in R " satisfying ...
... , calculate the nearest polynomial with a given complex root a , and prove the distance to this polyno- mial is ( 02 ) ( − 1/2 ) | p ( a ) | . 2.2 Theorems of the Alternative One well - trodden route 22 2. Inequality Constraints.
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 |