Springer Texts in Statistics

Optimization

Authors: Lange, Kenneth

  • The content of courses on optimization theory varies tremendously. This book views linear programming as a special case of nonlinear programming. The real bridge between linear and nonlinear programming is convexity. The theoretical side and applications of convexity in the design of algorithms for problems with either a large number of parameters or linear restraints are addressed in this book

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About this Textbook

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. While at 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.

Reviews

From the reviews:

"...An excellent, imaginative, and authoritative text on the difficult topic of modeling the problems of multivariate outcomes with different scaling levels, different units of analysis, and different study designs simultaneously." Biometrics, March 2005

"...As a textbook, Optimization does provide a valuable introduction to an important branch of applicable mathematics." Technometrics, August 2005

"...I found Optimization to be an extremely engaging textbook....the text is ideal for graduate students or researchers beginning research on optimization problems in statistics. There is little doubt that someone who worked through the text as part of a reading course or specialized graduate seminar would benefit greatly from the author's perspective..." Journal of the American Statistical Association, December 2005

"This is a book on optimization theory that includes some of the background mathematics necessary to understand … . provides a good overview of graduate-level topics in optimization, including some of the supporting mathematics and some applications. … The book has some every nice exercise sets to illuminate and extend the material covered in the textbook, as well as an extensive bibliography. … a valuable introduction to an important branch of applicable mathematics." (Marvin H.J. Gruber, Technometrics, Vol. 47 (3), August, 2005)


Table of contents (11 chapters)

Buy this book

eBook $69.99
price for USA (gross)
  • ISBN 978-1-4757-4182-7
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
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Bibliographic Information

Bibliographic Information
Book Title
Optimization
Authors
Series Title
Springer Texts in Statistics
Copyright
2004
Publisher
Springer-Verlag New York
Copyright Holder
Springer-Verlag New York
eBook ISBN
978-1-4757-4182-7
DOI
10.1007/978-1-4757-4182-7
Series ISSN
1431-875X
Edition Number
1
Number of Pages
XIII, 255
Topics