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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... exists , satisfies f ' ( x ; x - x ) ≥ 0. In particular , if f is differentiable at x , ≥0 . then the condition − ▽ ƒ ( x ) Є Nc ( ĩ ) holds . Proof . If some point x in C satisfies f 15 Inequality Constraints Optimality Conditions.
... Proof . A straightforward exercise using the convexity of ƒ shows the function tЄ ( 0,1 ] H f ( x + t ( x − x ) ) − f ( x ) t - is nondecreasing . The result then follows easily ( Exercise 7 ) . In particular , any critical point of a ...
... Proof . We may assume C is nonempty , and define a function ƒ : E → R by f ( x ) = || x − y || 2 / 2 . Now by the Weierstrass proposition ( 1.1.3 ) there exists a minimizer ≈ for ƒ on C , which by the First order necessary condition ...
... proof relies on consideration of a nondifferentiable func- tion , even though the result concerns derivatives . Exercises and Commentary The optimality conditions in this section are very standard ( see for example [ 119 ] ) . The ...
... proof of part ( d ) by using a spectral decom- position of A. ( Another approach to this problem is given in Section 7.2 , Exercise 6. ) 7. Suppose a convex function g : [ 0 , 1 ] → R satisfies g ( 0 ) = 0. Prove the function t Є ( 0 ...
Contents
7 | |
15 | |
Fenchel Duality | 33 |
Convex Analysis | 65 |
Special Cases | 97 |
Nonsmooth Optimization | 123 |
Fixed Points | 183 |
Infinite Versus Finite Dimensions | 209 |
List of Results and Notation | 221 |
Bibliography | 241 |
Other editions - View all
Convex Analysis and Nonlinear Optimization: Theory and Examples Jonathan M. Borwein,Adrian S. Lewis No preview available - 2000 |