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. |
From inside the book
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... finite intersections of open sets are open . The interior of D is just the largest open set contained in D , while cl D is the smallest closed set containing D. Finally , a subset G of D is open in D if there is an open set UCE with G ...
... finite intersection of closed halfspaces , and is therefore both closed and convex . The adjoint of the map A above is the linear map A * : Y → E defined by the property ( A * y , x ) = ( y , Ax ) for all points x in E and y in Y ...
... finite - dimensional convex sets C simplify and lose no generality if we assume C contains 0 and spans E. The following exer- cises outline this idea . 11 . ** ( Accessibility lemma ) Suppose C is a convex set in E. ( a ) Prove cl CCC + ...
... finite dimensions . But more concretely , identifying E with R2 may obscure properties of a space beyond its simple Euclidean structure . As an example , in this short section we describe a Euclidean space which " feels " very different ...
... finite , undirected , connected graph with vertex set V and edge set E. Suppose that a and ẞ in V are distinct vertices and that each edge ij in E has an associated “ resistance " rij > 0 in R. We consider the effect of applying a unit ...
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 |