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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... map A : E → Y linear if any points x and z in E and any reals A and μ satisfy A ( Ax + uz ) = Ax + μAz . In fact any linear function from E to R has the form ( a , ) for some element a of E. Linear maps and affine functions ( linear ...
... map A : E → Y is linear , and N ( A ) n0 + ( C ) is a linear subspace , prove AC is closed . Show this result can fail without the last assumption . ( f ) Consider another nonempty closed convex set DC E such that 0+ ( C ) n0 + ( D ) ...
... linear subspace span ( D − x ) is independent of x . ( The relative interior ) ( We use Exercises 11 and 12. ) The ... map A : E → F is linear , prove ri AC Ɔ Ari C. 1.2 Symmetric Matrices Throughout most of this book our setting 8 1 ...
... linear map Diag : R " → S " , where for a vector x in R2 , Diag x is an n × n diagonal matrix with diagonal 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 ...
... linear map A : S2 → R ” by setting AX = Xs for any matrix X in S " . Calculate the adjoint map A * . 12. * ( Fan's inequality ) For vectors x and y in R " and a matrix U in On , define απ ( Diag x , UT ( Diag y ) U ) . ( a ) Prove a ...
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 |