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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... nonzero element a of E satisfying ( a , y ) > b ≥ ( a , x ) for all points x in C. Sets in E of the form { x | ( a , x ) = b } and { x | ( a , x ) ≤ b } ( for a nonzero element a of E and real b ) are called hyperplanes and closed ...
... nonzero element a in E with infrED ( a , x ) > supec ( a , y ) . Interpret geometrically . ( c ) Show part ( b ) fails for the closed convex sets in R2 , D = { X | X1 > 0 , X1X2 ≥ 1 } , 6 . ** C = { x | x2 = 0 } . ( Recession cones ) ...
... nonzero . ) These two cones have some important differences ; in particular , R is a polyhedron , whereas the cone of positive semidefinite matrices S is not , even for n = 2. The cones R2 and S2 are important largely because of the ...
... nonzero matrices X and Y in S3 such that RX R + Y and ( X + Y ) / 2 S3 + . 4. ( A nonlattice ordering ) Suppose the matrix Z in S2 satisfies WY [ 1 0 0 0 8 ] and W > 0 0 0 1 ↔ W > Z. ( a ) By considering diagonal W , prove Z = [ 1 a a ...
... nonzero y in N ( A ) then ≈ is a local minimizer . We are already beginning to see the broad interplay between analytic , geometric and topological ideas in optimization theory . A good illustration is the separation result of Section ...
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 |