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Optimization by Vector Space Methods, by David G. Luenberger
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Engineers must make decisions regarding the distribution of expensive resources in a manner that will be economically beneficial. This problem can be realistically formulated and logically analyzed with optimization theory. This book shows engineers how to use optimization theory to solve complex problems. Unifies the large field of optimization with a few geometric principles. Covers functional analysis with a minimum of mathematics. Contains problems that relate to the applications in the book.
- Sales Rank: #353429 in Books
- Published on: 1997-01-23
- Original language: English
- Number of items: 1
- Dimensions: 9.00" h x .73" w x 6.00" l, 1.03 pounds
- Binding: Paperback
- 344 pages
From the Back Cover
Unifies the field of optimization with a few geometric principles.
The number of books that can legitimately be called classics in their fields is small indeed, but David Luenberger's Optimization by Vector Space Methods certainly qualifies. Not only does Luenberger clearly demonstrate that a large segment of the field of optimization can be effectively unified by a few geometric principles of linear vector space theory, but his methods have found applications quite removed from the engineering problems to which they were first applied. Nearly 30 years after its initial publication, this book is still among the most frequently cited sources in books and articles on financial optimization.
The book uses functional analysis --the study of linear vector spaces --to impose simple, intuitive interpretations on complex, infinite-dimensional problems. The early chapters offer an introduction to functional analysis, with applications to optimization. Topics addressed include linear space, Hilbert space, least-squares estimation, dual spaces, and linear operators and adjoints. Later chapters deal explicitly with optimization theory, discussing
* Optimization of functionals
* Global theory of constrained optimization
* Local theory of constrained optimization
* Iterative methods of optimization.
End-of-chapter problems constitute a major component of this book and come in two basic varieties. The first consists of miscellaneous mathematical problems and proofs that extend and supplement the theoretical material in the text; the second, optimization problems, illustrates further areas of application and helps the reader formulate and solve practical problems.
For professionals and graduate students in engineering, mathematics, operations research, economics, and business and finance, Optimization by Vector Space Methods is an indispensable source of problem-solving tools.
About the Author
DAVID G. LUENBERGER is a professor in the School of Engineering at Stanford University. He has published four textbooks and over 70 technical papers. Professor Luenberger is a Fellow of the Institute of Electrical and Electronics Engineers and recipient of the 1990 Bode Lecture Award. His current research is mainly in investment science, economics, and planning.
Most helpful customer reviews
52 of 54 people found the following review helpful.
Simply the perfect math book
By James Arvo
Optimization by Vector Space Methods, by David Luenberger, is one of the finest math texts I have ever read, and I've read hundreds. Many years ago this book sparked my interest in optimization and convinced me that the abstract mathematics I had been immersed in actually would be applicable to real problems. Since then, Luenberger's book has inspired several of my graduate students. I merely lent them my copy, and Luenberger did the rest; he drew them in by carefully laying the foundation for an elegant theory, with just the right mix of formalism and intuition, and opened their eyes to the beauty and practicality of abstract mathematics. Anyone with an interest in higher-level mathematics (beyond multi-variable calculus, say) would benefit from exposure to this finely-crafted book. I daresay, the rampant math anxiety that is so prevalent in the West would be substantially reduced if more authors would take such meticulous care in presenting their material.
The format of Luenberger's book is also extremely appealing in a way that I cannot quite put my finger on. The typography and illustrations are inherently crisp and inviting; they draw you in. There is nothing at all superfluous or gratuitous in this book. It is utterly to-the-point, methodical, and above all, clear. The techniques are developed starting from an elementary treatment of vector spaces, then proceeding on to Banach spaces and Hilbert spaces. Along the way, Luenberger introduces convexity, cones, basic topology, random variables, minimum-variance estimators, and least squares, among many other things. There is a recurring theme of duality, which can be used in a way analogous to the inner product of a Hilbert space. In particular, the familiar projection theorems of Hilbert spaces can be echoed in simpler normed linear spaces using duality, which Luenberger motivates and covers beautifully.
The book also covers some of the standard fare of functional analysis, such as the Han-Banach theorem, strong and weak convergence, and the Banach inverse theorem. However, Luenberger never wanders too far off into abstract nonsense; around every corner lay tantalizing application of these ideas to optimization. Luenberger first explores optimization of functionals then covers constrained optimization, which builds upon concepts such as positive cones and Lagrange multipliers. The optimization methods themselves have endless applications in fields such as computer vision, computer graphics, economics, and physics. Indeed, the list is effectively endless as optimization techniques pervade math and science.
I'm certain that the appeal of this book is helped immeasurably by the inherent beauty of the subject matter. Hilbert-space methods are lovely in themselves--they possess a structure that engages one's geometric intuition while at the same time admitting convenient algebraic properties. Once you are in the habit of phrasing problems in abstract settings such as Hilbert spaces, it forever changes how you look at things; you cannot help but look past the clutter to the essence of a problem (or, at least try very hard to do so). While this material is not nearly as abstract as, say, category theory, it nevertheless hits a high point in mathematics--a point more people ought to experience.
If you've had some exposure to optimization methods, or need to apply them in the context of computer vision, graphics, or finance, to mention just a few areas, then I urge you to take a look at Luenberger's fine book. It too hits a high point in clarity of mathematical writing. Combine beautiful theory with endless applications and lucid writing, and you have a winner of a book.
6 of 6 people found the following review helpful.
Elegant and astonishing
By Mark A. Peot
Professor Luenberger unites many areas of optimization using a few principles from functional analysis. The explanations are clear and the proofs are compact and elegant. This book is your tool for understanding the deep connection between linear programming, convex optimization, game theory, optimal control and series approximation (e.g. Fourier series).
Luenberger's book has over 4517 citations as of March 2012. In my opinion, the material in this book is essential for any graduate student or professional who intends to contribute to the literature in optimization or optimal control.
6 of 6 people found the following review helpful.
This is a true classic
By Mark A. Mendlovitz
This book is a timeless classic, filled with extraordinarily powerful mathematics and applicable to just about every serious subject area. Luenberger did a masterful job of writing a book that will "unravel the spaghetti" seen in most other books. The visual perspectives he provides to seemingly abstract ideas are the key.
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