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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Theory and Examples Jonathan Borwein, Adrian S. Lewis. Chapter 1 Background 1.1 Euclidean Spaces We begin by reviewing some ... Examples are the positive orthant = R = { x R " | each x ; ≥ 0 } , and the cone of vectors with nonincreasing ...
... example , the interior of R2 is R + = { x Reach x¿ > 0 } . = = We say the point x in E is the limit of the sequence ... examples of closed sets . Easy exercises show that D is open exactly when its complement Dc is closed , and that ...
... example , that for any real a the level set { x = D | f ( x ) ≤ a } is closed providing D is closed . ← Given another Euclidean space Y , we call a map A : E → Y linear if any points x and z in E and any reals A and μ satisfy A ( Ax ...
Theory and Examples Jonathan Borwein, Adrian S. Lewis. ― = ----- ( -∞ , ∞ ] denotes the interval RU { + } . We try ... example is the stronger condition f ( x ) lim inf || x || → ∞ || x || > 0 , ( 1.1.4 ) where we define f ( x ) lim ...
Theory and Examples Jonathan Borwein, Adrian S. Lewis. 5. * ( Strong separation ) Suppose that the set C C E is closed and convex , and that the set DCE is compact and convex . ( a ) Prove the set D - C is closed and convex . ( b ) ...
Contents
1 | |
Chapter 2 Inequality Constraints | 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 |