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Optimization

Kenneth Lange
Publisher: 
Springer
Publication Date: 
2013
Number of Pages: 
529
Format: 
Hardcover
Edition: 
2
Series: 
Springer Texts in Statistics
Price: 
89.95
ISBN: 
9781461458371
Category: 
Textbook
[Reviewed by
Allen Stenger
, on
08/23/2014
]

This is an interesting and well-constructed text in optimization aimed at graduate students in all mathematical fields. It appears in a statistics series but is not especially statistical (except for Chapters 7 and 9), drawing examples from all fields. It has excellent problem sections at the end of each chapter, and generally takes a concrete approach to everything. For example, most work is done in \(\mathbb{R}^n\) rather than more general spaces, and the chapter on calculus of variations is closer to the original Euler-Lagrange theory than to the modern functional analysis approach.

The book assumes knowledge of calculus and of proofs, but no prior knowledge of optimization. It develops all the necessary mathematics in the early chapters, although it is one of those concise developments that are much easier to follow if you have seen the material before.

Roughly the first half of the book develops the mathematical background and exact methods, while the second half develops algorithms for approximate solutions. It does not deal explicitly with numerical analysis concerns. There is quite a lot on majorization as a way to simplify problems to obtain an approximate solution. The book develops the theory of the gauge integral as an easier alternative to the Lebesgue integral, while admitting that probability theory still requires the full theory of measure. It makes heavy use of convexity throughout.


Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis.

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.

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