Kenneth Lange

Optimization (eBook, PDF)

• Format: PDF

Produktbeschreibung

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 and can serve as a bridge to more advanced treatises on nonlinear and convex programming. The emphasis on statistical applications will be especially appealing to graduate students of statistics and biostatistics. The intended audience also includes graduate students in applied mathematics, computational biology, computer science, economics, and physics as well as upper division undergraduate majors in mathematics who want to see rigorous mathematics combined with real applications.

Chapter 1 reviews classical methods for the exact solution of optimization problems. Chapters 2 and 3 summarize relevant concepts from mathematical analysis. Chapter 4 presents the Karush-Kuhn-Tucker conditions for optimal points in constrained nonlinear programming. Chapter 5 discusses convexity and its implications in optimization. Chapters 6 and 7 introduce the MM and the EM algorithms widely used in statistics. Chapters 8 and 9 discuss Newton's method and its offshoots, quasi-Newton algorithms and the method of conjugate gradients. Chapter 10 summarizes convergence results, and Chapter 11 briefly surveys convex programming, duality, and Dykstra's algorithm.

Kenneth Lange is the Rosenfeld Professor of Computational Genetics in the Departments of Biomathematics and Human Genetics at the UCLA School of Medicine. He is also Interim Chair of the Department of Human Genetics. At various times during his career, he has held appointments at the University of New Hampshire, MIT, Harvard, the University of Michigan, and the University of Helsinki. Whileat the University of Michigan, he was the Pharmacia & Upjohn Foundation Professor of Biostatistics. His research interests include human genetics, population modeling, biomedical imaging, computational statistics, and applied stochastic processes. Springer-Verlag previously published his books Mathematical and Statistical Methods for Genetic Analysis, Second Edition, Numerical Analysis for Statisticians, and Applied Probability.

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Produktdetails

• Verlag: Springer New York
• Seitenzahl: 255
• Erscheinungstermin: 9. März 2013
• Englisch
• ISBN-13: 9781475741827
• Artikelnr.: 44045559

Autorenporträt

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.

Inhaltsangabe

Elementary Optimization.- The Seven C's of Analysis.- The Gauge Integral.- Differentiation.- Karush-Kuhn-Tucker Theory.- Convexity.- Block Relaxation.- The MM Algorithm.- The EM Algorithm.- Newton's Method and Scoring.- Conjugate Gradient and Quasi-Newton.- Analysis of Convergence.- Penalty and Barrier Methods.- Convex Calculus.- Feasibility and Duality.- Convex Minimization Algorithms.- The Calculus of Variations.- Appendix: Mathematical Notes.- References.- Index.