Overview
- 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
Part of the book series: Springer Texts in Statistics (STS)
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Table of contents (11 chapters)
Keywords
About this book
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 convexity serves as bridge between linear and nonlinear programming and makes it possible to give a modern exposition of linear programming based on the interior point method rather than the simplex method.
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.
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 differentstudy 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
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)
Authors and Affiliations
Bibliographic Information
Book Title: Optimization
Authors: Kenneth Lange
Series Title: Springer Texts in Statistics
DOI: https://doi.org/10.1007/978-1-4757-4182-7
Publisher: Springer New York, NY
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eBook Packages: Springer Book Archive
Copyright Information: Springer-Verlag New York 2004
eBook ISBN: 978-1-4757-4182-7Published: 09 March 2013
Series ISSN: 1431-875X
Series E-ISSN: 2197-4136
Edition Number: 1
Number of Pages: XIII, 255
Topics: Optimization, Statistical Theory and Methods, Operations Research/Decision Theory