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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... convex cone with interior S + 2. Explain why S2 is not a polyhedron . ' ++ • + 3. ( S3 is not strictly convex ) Find nonzero matrices X and Y in S3 such that RX R + Y and ( X + Y ) / 2 S3 + . 4. ( A nonlattice ordering ) Suppose the ...
... convex cone which arises frequently in optimization is the normal cone to a convex set C at a point ≈ Є C , written Nc ( x ) . This is the convex cone of normal vectors , vectors d in E such that ( d , x − x ) ≤ 0 for all points x in ...
... cone is a closed convex cone . 2. ( Examples of normal cones ) For the following sets CC E , check C is convex and compute the normal cone Nc ( x ) for points ≈ in C : ( a ) C a closed interval in R. ( b ) CB , the unit ball . ( c ) C ...
... cone to the set { x ЄE | Ax = b } at any point in it is A * Y . Hence deduce Corollary 2.1.3 ( First order conditions for linear constraints ) . 5. Prove that the differentiable function x2 + x2 ( 1 − x1 ) 3 has a unique critical point ...
... cone C = m R } { Σ | μia2 0 ≤ μ1 , M2 , . . . , μm Ɛ R 1 ... , ( 2.2.11 ) can be separated from C by a hyperplane . If a solves system ( 2.2.9 ) then C is contained in the closed halfspace { a | ( a , x ) ≤ 0 } , whereas c is ...
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 |