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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... functions we study . Given a set D in E , we call a function f : D continuous ( on D ) if f ( x2 ) f ( x ) for any sequence → x in D. In this case it easy to check , for example , that for any real a the level set { x € D│f ( x ) ≤ a } ...
... real 8 and a function g : ( 0,6 ) → R , we define lim inf g ( t ) = lim inf g t ↓ 0 ( 0 , t ) t10 lim sup g ( t ) ... function f : DR are bounded . Then ƒ has a global minimizer . Just as for sets , convexity of functions will be crucial ...
... function f : C → R has bounded level sets if and only if it satisfies the growth condition ( 1.1.4 ) . The proof is outlined in Exercise 10 . Exercises and Commentary Good general references are [ 156 ] for elementary real analysis and ...
... function with bounded level sets which does not satisfy the growth condition ... real € > 0 . ( b ) For sets D and F in E with D open , prove D + F is open ... real \ . The affine hull of a set D in E , denoted aff D , is the smallest ...
... real eigenvalues ( counted by multiplicity ) , which we write in nonincreasing order A1 ( X ) > λ2 ( X ) ≥ ... .. ≥ λn ( X ) . In this way we define a function \ : Sn → R " . We also define a linear map Diag R " → S " , where for a ...
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 |