OptimizationFinitedimensional optimization problems occur throughout the mathematical sciences. The majority of these problems cannot be solved analytically. This introduction to optimization attempts to strike a balance between presentation of mathematical theory and development of numerical algorithms. Building on students’ skills in calculus and linear algebra, the text provides a rigorous exposition without undue abstraction. Its stress on statistical applications will be especially appealing to graduate students of statistics and biostatistics. The intended audience also includes students in applied mathematics, computational biology, computer science, economics, and physics who want to see rigorous mathematics combined with real applications. In this second edition the emphasis remains on finitedimensional optimization. New material has been added on the MM algorithm, block descent and ascent, and the calculus of variations. Convex calculus is now treated in much greater depth. Advanced topics such as the Fenchel conjugate, subdifferentials, duality, feasibility, alternating projections, projected gradient methods, exact penalty methods, and Bregman iteration will equip students with the essentials for understanding modern data mining techniques in high dimensions.

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Contents
1  
2 The Seven Cs of Analysis  23 
3 The Gauge Integral  53 
4 Differentiation  75 
5 KarushKuhnTucker Theory  106 
6 Convexity  137 
7 Block Relaxation  171 
8 The MM Algorithm  184 
12 Analysis of Convergence  291 
13 Penalty and Barrier Methods  313 
14 Convex Calculus  340 
15 Feasibility and Duality  383 
16 Convex Minimization Algorithms  415 
17 The Calculus of Variations  445 
Appendix Mathematical Notes  473 
References  499 