Optimization

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Springer Science & Business Media, Mar 19, 2013 - Mathematics - 529 pages

Finite-dimensional 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 finite-dimensional 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 Elementary Optimization
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

9 The EM Algorithm
221
10 Newtons Method and Scoring
245
11 Conjugate Gradient and QuasiNewton
273

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About the author (2013)

Kenneth Lange is the Rosenfeld Professor of Computational Genetics at UCLA. He is also Chair of the Department of Human Genetics and Professor of Biomathematics and Statistics. At various times during his career, he has held appointments at the University of New Hampshire, MIT, Harvard, the University of Michigan, the University of Helsinki, and Stanford. He is a fellow of the American Statistical Association, the Institute of Mathematical Statistics, and the American Institute for Medical and Biomedical Engineering. His research interests include human genetics, population modeling, biomedical imaging, computational statistics, and applied stochastic processes. Springer previously published his books Mathematical and Statistical Methods for Genetic Analysis, Numerical Analysis for Statisticians, and Applied Probability, all in second editions.