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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... Calculate the adjoint map A * . 12. * ( Fan's inequality ) For vectors x and y in R " and a matrix U in On , define απ ( Diag x , UT ( Diag y ) U ) . ( a ) Prove a = xT Zy for some doubly stochastic matrix Z. ( b ) Use Birkhoff's ...
... Calculate Vg ( x ) for nonzero x . ( d ) Deduce that minimizers of g must be eigenvectors , and calculate the minimum value . ( e ) Find an alternative proof of part ( d ) by using a spectral decom- position of A. ( Another approach to ...
... calculate the nearest point in the subspace { x ЄE | ( a , x ) = 0 } to the point y E. ← ( e ) ( Projection on R2 and S3 ) Prove the nearest point in R to a vector y in R is y + , where y = max { yi , 0 } for each i . For a matrix U in ...
... a given root ) Consider the Eu- clidean space of complex polynomials of degree no more than n , with inner product n j = 0 Xj zĎ n n Luz ) = Σ9 ; j = 0 j = 0 · Given a polynomial p in this space , calculate the 2.1 Optimality Conditions 21.
... 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 |