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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... element a of E ) then we say f is ( Gâteaux ) differentiable at I , with ( Gâteaux ) derivative Vf ( x ) = a . If f is differentiable at every point in C then we simply say ƒ is differentiable ( on C ) . An example we use quite ...
... is differentiable at x then ▽ ƒ ( x ) Є A * Y . ( b ) Conversely , if Vƒ ( ĩ ) € A * Y and f is convex then ĩ is a global min- imizer for ( 2.1.4 ) . The element y Є Y satisfying Vƒ ( x ) 16 2. Inequality Constraints.
... element a of E such that ( a , y ) > b≥ ( a , x ) for all points x in C. Proof . We may assume C is nonempty , and define a function ƒ : E → R by f ( x ) = || x − y || 2 / 2 . Now by the Weierstrass proposition ( 1.1.3 ) there exists ...
... element a of E , calculate the nearest point in the subspace { x ЄE | ( a , x ) = 0 } to the point y Є E. ( e ) ... elements a of E. ) - ++ 10. ( a ) Prove the function f : S + → R defined by ƒ ( X ) = tr X - 1 is differentiable on S2 + ...
... elements ao , a1 , ... , am of E , exactly one of the following systems has a solution : m Σλία i = 0 m = 0 , Σ ; = 1 , 0 ≤ No0 , No1 , ... , Am E R i = 0 ( a ' , x ) < 0 for i = 0,1 , ... , m , x Є E. ( 2.2.2 ) ( 2.2.3 ) Geometrically ...
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 |