Convex Analysis and Nonlinear Optimization: Theory and Examples
Springer Science & Business Media, Nov 30, 2005 - Mathematics - 310 pages
Optimization 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.
More generally, the idea of convexity is central to the transition from classical analysis to various branches of modern analysis: from linear to nonlinear ...
These exercises fall into three categories, marked with zero, one, or two asterisks, respectively, as follows: examples that illustrate the ideas in the ...
Chapter 1 Background 1.1 Euclidean Spaces We begin by reviewing some of the fundamental algebraic, geometric and analytic ideas we use throughout the book.
The smallest cone containing a given set D C E is clearly R+D. The fundamental geometric idea of this book is convexity. A set C in E is convex if the line ...
Just as for sets, geometric and topological ideas also intermingle for the functions we study. Given a set D in E, we call a function R continuous (on D) if ...
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Chapter 2 Inequality Constraints
Chapter 3 Fenchel Duality
Chapter 4 Convex Analysis
Chapter 5 Special Cases
Chapter 6 Nonsmooth Optimization
Chapter 7 KarushKuhnTucker Theory
Chapter 8 Fixed Points