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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... entries x . This map embeds R " as a subspace of Sn and the cone R2 as a subcone of S2 . The determinant of a square matrix Z is written det Z. + We write On for the group of n × n 1.2 Symmetric Matrices 6 1.2 Symmetric Matrices.
... subspace . ( d ) C a closed halfspace : { x | ( a , x ) ≤ b } where 0 a € E and bЄ R. ( e ) C = { x ЄR " | x≥ 0 for all j € J } ( for J C { 1,2 , ... , n } ) . xj 3. ( Self - dual cones ) Prove each of the following cones K satisfy ...
... subspace { x ЄE | ( a , x ) = 0 } to the point y E. ← ( e ) ( Projection on R2 and S3 ) Prove the nearest point in R to a vector y in R is y + , where y = max { yi , 0 } for each i . For a matrix U in On and a vector y in R " , prove ...
... subspace LC S " of matrices with ( i , j ) th entry of zero for all ( i , j ) in △ satisfies Ln S + 0. By considering the problem ( for CEST ) ++ inf { ( C , X ) – log det X | X Є Lns " } , — ++ use Section 1.2 , Exercise 14 and ...
... subspace Y of E. Notice first that Y becomes a Euclidean space by equipping it with the same inner product . The projection of a point x in E onto Y , written Pyx , is simply the nearest point to x in Y. This is well - defined ( see ...
Contents
1 | |
15 | |
Chapter 3 Fenchel Duality | 33 |
Chapter 4 Convex Analysis | 65 |
Chapter 5 Special Cases | 97 |
Chapter 6 Nonsmooth Optimization | 123 |
Chapter 7 KarushKuhnTucker Theory | 153 |
Chapter 8 Fixed Points | 179 |
Chapter 9 More N onsmooth Structure | 213 |
Infinite Versus Finite Dimensions | 239 |
Chapter 11 List of Results and Notation | 253 |
Bibliography | 275 |
Index | 289 |
Other editions - View all
Convex Analysis and Nonlinear Optimization: Theory and Examples Jonathan M. Borwein,Adrian S. Lewis No preview available - 2000 |