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 1-5 of 84
... 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 [ 1 ] ...
... Prove the set D – C is closed and convex . ( b ) Deduce that if in addition D and C are disjoint then there ex- ists ... ( f ) Consider another nonempty closed convex set DCE such that 0+ ( C ) no + ( D ) is a linear subspace . Prove C - D ...
... f : CR has bounded level sets but that ( 1.1.4 ) fails . Deduce the existence of a sequence ( x ) in C with f ( x ) ... Prove cl CCC + B for any real € > 0 . ( b ) For sets D and F in E with D open , prove D + F is open . ( c ) For x in ...
... prove aff D = x + span ( D - x ) , and deduce the linear subspace span ( D − x ) is independent of x . - ( The ... F is another Euclidean space and the map A : E → F is linear , prove ri AC Ari C. 1.2 Symmetric Matrices Throughout most ...
... f : R → R has a critical point x . If x is a local minimizer then the ... prove . Theorem 2.1.6 ( Basic separation ) Suppose that the set CCE is ... f ( x ) = || x − y || 2 / 2 . Now by the Weierstrass proposition ( 1.1.3 ) there exists ...
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 |