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 70
... vector x in R ^ , we denote by [ x ] the vector with the same components permuted into nonincreasing order . We leave the proof of this result as an exercise . Proposition 1.2.4 ( Hardy - Littlewood - Pólya ) Any vectors x and y in Rn ...
... vectors , prove x1 + x2 + ... + xnn . Deduce the inequality for any vector z in R n · n n Zi ≥ ( II n Zi 1 1 1 / n → Rn 11. For a fixed column vector s in R " , define a linear map A : Sn by setting AX = Xs for any matrix X in S ...
... vector space of nxn real matrices . For a matrix A in M " we define the singular values of A by σ¿ ( A ) √ ( ATA ) for i = 1 , 2 , ... , n , and hence define a map σ : M " → R " . ( Notice zero may be a singular value . ) = ( a ) ...
... vectors , vectors d in E such that ( d , x - x ) ≤ 0 for all points x in C. Proposition 2.1.1 ( First order necessary condition ) Suppose that C is a convex set in E and that the point x is a local minimizer of the function f : CR ...
... vectors y in N ( A ) . Conversely , if ya ▽ 2ƒ ( x ) y > 0 for all nonzero y in N ( A ) then I is a local minimizer . We are already beginning to see the broad interplay between analytic , geometric and topological ideas in ...
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 |