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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Results 6-10 of 55
... matrix X in S " satisfies ( X2 ) 1/2 X. ( b ) Find matrices X Y in S2 such that X2 Y2 . ( c ) For matrices X ≥ Y in S2 , prove X1 / 2 the relationship Y1 / 2 . ( Hint : Consider ( ( X1 / 2 + Y1 / 2 ) x , ( X1 / 2 – Y1 / 2 ) x ) = ( ( X ...
... matrix X in S " . 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 = x1 Zy for some doubly stochastic matrix Z. ( b ) Use ...
... matrices { X1 , X2 , ... , Xm } has a simultaneous ordered spectral de- composition . ** 17. ( Singular values and von Neumann's lemma ) Let M " denote the vector space of nxn real matrices . For a matrix A in M " we define the singular ...
... matrix B in M " , use part ( a ) and Fan's inequality ( 1.2.2 ) to prove tr ( ATB ) ≤ o ( A ) To ( B ) . ( c ) If A lies in S , prove λ ( A ) = σ ( A ) . ( d ) By considering matrices of the form A + al and B + BI , deduce Fan's ...
... matrix A in S " , define a function g ( x ) = xa Ax / || x || 2 for nonzero x in R " . Prove g has a minimizer . ( c ) Calculate Vg ( x ) for nonzero x . ( d ) Deduce that minimizers of g must be eigenvectors , and calculate the minimum ...
Contents
7 | |
15 | |
Fenchel Duality | 33 |
Convex Analysis | 65 |
Special Cases | 97 |
Nonsmooth Optimization | 123 |
Fixed Points | 183 |
Infinite Versus Finite Dimensions | 209 |
List of Results and Notation | 221 |
Bibliography | 241 |
Other editions - View all
Convex Analysis and Nonlinear Optimization: Theory and Examples Jonathan M. Borwein,Adrian S. Lewis No preview available - 2000 |