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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... lemma ) Suppose C is a convex set in E. ( a ) Prove cl C C C + € B for any real € > 0 . ( b ) For sets D and F in E with D open , prove D + F is open . ** ( c ) For x in int C and 0 < ≤ 1 , prove λx + ( 1 - X ) el C C C. Deduce int C + ...
... lemma ) Let M ” denote the vector space of n × n real matrices . For a matrix A in M " we define the singular values of A by σ ; ( A ) √ ( ATA ) for i = 1 , 2 , ... , n , and hence define a map σ : Mn → R. ( Notice zero may be a ...
... By considering matrices of the form A + al and B + ẞI , deduce Fan's inequality from von Neumann's lemma ( part ( b ) ) . Chapter 2 Inequality Constraints 2.1 Optimality Conditions Early in multivariate 14 1. Background.
... lemma ( which we derive at the end of this section ) . Our first approach , however , relies on a different theorem of the alternative . Theorem 2.2.1 ( Gordan ) For any elements ao , a1 , , ... , am of E , exactly one of the following ...
... lemma , one of the cornerstones of many approaches to optimality conditions . The proof uses the idea of the projection onto a linear subspace Y of E. Notice first that Y becomes a Euclidean space by equipping it with the same inner ...
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 |