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Operations Research Letters
Volume 35, Issue 4, July 2007, Pages 559-560
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doi:10.1016/j.orl.2006.09.002    How to Cite or Link Using DOI (Opens New Window)  
Copyright © 2006 Elsevier B.V. All rights reserved.

Book reviews Andrzej Ruszczyński, Nonlinear Optimization, Princeton University Press, Princeton, NJ, 2006, ISBN 0-691-11915-5, 464 pp., $ 59.50.

Franz Rendla
aUniversität Klagenfurt, Austria

Available online 27 October 2006.



Books on nonlinear optimization come in various flavours. Some focus on convex optimization, others concentrate on theoretical issues like optimality conditions or emphasize numerical methods. The present book provides a well balanced mixture of theory and methods, covering nearly 450 pages.

The theoretical part consists of three chapters. The initial chapter summarizes basics from convex analysis. The author develops mathematical tools (cones, differential and subdifferential calculus, conjugate duality) which are the basis for the subsequent parts of the book.

The chapter on optimality conditions considers necessary (and sufficient) conditions for optimality. After a brief discussion on the simplest case of unconstrained optimization, the author discusses the constrained case in detail. The chapter concludes with a short look at sensitivity analysis. The final theory chapter covers duality, based on the Lagrangian function. Conic duality and the augmented Lagrangian dual are also considered in this chapter.

The second half of the book is devoted to methods for finding local optima. The development is again from ‘easy’ to ‘difficult’, starting out with unconstrained optimization of differentiable functions. After treating the scalar case (line-search problem), the author treats the unconstrained case much like in the standard literature. The steepest descent method (with convergence analysis) is followed by Newton's method. Then comes the conjugate gradient method, quasi-Newton methods, trust-region based methods and finally a short description of methods which do not use analytically given gradients.

The chapter on constrained optimization of differential functions is treated in the traditional style, featuring feasible point methods and the penalty function approach, followed by dual methods (including the augmented Lagrangian approach). The chapter ends with a short description of the barrier method.

The last chapter considers nondifferentiable optimization. Various versions of subgradient based methods are analysed with respect to convergence properties, including the proximal point and the bundle method.

The book maintains a good balance of detail and generality in the sense that the author touches on many topics, but does not get side-tracked in endless technical detail on a specific topic. Each chapter closes with a collection of exercises, which seem to be meant mostly for students to get some hands-on experience on the presented material. The author also covers a lot of mathematical results, not directly related to optimization, in the form of easy corollaries of results developed in the book. Example 3.9 for instance provides an easy and simple generalization of the Toricelli point of a triangle to the problem of finding the point with minimal total distance from three closed convex sets in the plane. Similarly, Helly's theorem is derived in a simple way through convex optimization arguments.

Here are some distinguishing features of this book:

On a formal level, the book is well compiled. I could not spot any typographical error or misprint.

The thoughtful selection of the material in the theoretical part provides a good starting basis for the algorithmic development. Here it becomes apparent that the author has polished the manuscript in the course of his teaching it over a long period.

The algorithmic analysis of the various optimization methods is carried out rigorously, including careful convergence considerations. It is also worth mentioning that non-smooth optimization gets a thorough treatment in both parts of the book.

As a small drawback, the reader wishing to find out which method to use for a particular problem will not find an answer in the book. The author describes virtually all the standard techniques but does not provide any further information about practicality or robustness of the methods presented.

In summary, this book competes with the topmost league of books on optimization. The wide range of topics covered and the thorough theoretical treatment of algorithms make it not only a good prospective textbook, but even more a reference text (which I am happy to have on my shelf).



Operations Research Letters
Volume 35, Issue 4, July 2007, Pages 559-560
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