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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... 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 } is closed providing D is closed . Given another Euclidean space ...
... continuous function f : DR are bounded . Then ƒ has a global minimizer . Just as for sets , convexity of functions will be crucial for us . Given a convex set CCE , we say that the function f : C → R is convex if ƒ ( λx + ( 1 − λ ) y ) ...
... continuously differentiable near x . If ≈ is a local minimizer then y1V2ƒ ( x ) y ≥ 0 for all 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 ...
... continuous , satisfying > 0 in R and nonzero x in R " . Prove f f ( x ) = f ( x ) for all has a minimizer . ( b ) Given a matrix A in S " , define a function g ( x ) = xa Ax / || x || 2 for nonzero x in R " . Prove g has a minimizer ...
... continuous . ( d ) Given a nonzero element a of E , calculate the nearest point in the subspace { x ЄE | ( a , x ) = 0 } to the point y Є E. ( e ) ( Projection on R and S2 ) Prove the nearest point in R to a vector y in R " is y + ...
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 |