Linear-programming

In my work I needed to find the shortest path from a single point to a set of points. This book really helped me to find the suitable method: the Dijkstra algorithm. I began reading Chapter 9, which is "Shortest Paths and Discrete Dynamic Programming". The material is presented clearly and with relevant and adequate variety of examples. I haven't read the other chapters since they are not required for my work at this moment and I don't have ample time to make a full review; however, I can say this: My many years in research in several fields have often put me in a position of transfering mathematical algorithms in one field to another or to search for an effecient one. I frequently get a limited time period to do literature search and I usually page-read many books. This is one of the rare books which are easy to read and comprehend. I thank and congratulate the author for doing a wonderful service.

This is an excellent book for those who need to use the power of operations research methods (esp the newer algorithms, interior point methods etc.) but dont have the time to chew on pages of theory. Must congratulate the author on a job well done. Is it possible to bring out a cheaper paper back edition ? That would benefit the student and research community immensely.

If you are taking a graduate or an undergraduate course in OR, this book is a must! I have not seen ANY book able to present OR with such simple, direct examples and WITHOUT sacrificing theory.
This is the best written textbook I have ever read. When I compare it with the hundereds of dollars I spend on badly written books, even as a PG (poor graduate) student I would gladly pay twice of what this book is priced at.

Beside the great ideas provided in this book for problem solving, it provides a deep wisdom of for piecing some of the puzzles of our life. I recommend it.

I first ordered this book thinking it was George Polya 's book "How to solve it", then I realized it wasn't and I bought it anyway since I thought it might turn out as a "must read" book, just like Polys'a book.

One one hand it was a dissapointment, because the books are not written in the same manner and don't attact similar problelsm.

But then, this book makes you look into problems, and realize that usually we people are usually good in solving problems of the sort we learned how to (well... duh!), but surprisingly, we have a hard time solving even trivial problems if they are not placed in the context we got used to seeing them.

This book comes and tries to make things better in this department, showing you some general methods for solving problems, and also showing problems and suggested solutions along with a long discussion.

You should be able, once you've read the book and put your mind to it, to be better in understanding problems, understanding which tool to use for solving them and finally, understanding the tools enough to be able to actually solve the problem.

I enjoyed the overview of methods, and there are many such methods throughout the book (perhaps a complementary book for learning which "machine learning" methods are available these days and what sorts of problems they are useful for solving would be Tom Mitchell's "Machine Learning" book).

I wasn't sorry for buying this book. I'm happy I was fortunate enough to bump into it.

I am a computer scientist, but have gotten impatient over the years with the needless formalization that occurs in algorithmic texts. This is a delightful breath of fresh air in terms of balancing erudition with attempts to be "user friendly". If you want the latest and greatest twist to a well known technique, this book won't provide it. But it does a great job of competently and lucidly explaining the value proposition behind each optimization method and how to gradually upgrade from applying it naively to the more intricately optimized applications. Well done!

This is a very good book for people who are involved in evaluating different alternatives and each alternative has its own strengths and weaknesses. It is also a very good reference for students and researchers who are studying/working in the area of decision analysis. It is especially useful for people who are interested in trying to find out which MCDM method is the best and who have used or intend to use the AHP method in their work or research. It has extensive and intensive coverage on the AHP method for ranking alternatives, other pairwise comparison methods for extracting weights, their shortcomings and improvement, and their comparisons with other popular MCDM methods. The discussions on the ranking abnormalities of the AHP method and its revised versions are very informative. The comparisons are supported using numerous numerical examples, the conclusions are convincing, and the contents easy to understand.

I read this book with great interest. I am a Faculty Member at the New York University. This book is very clear to the point. I liked very much the in-depth analyses of many real-life MCDM issues and also the many numerical examples. The computational results are very thorough and revealing. The first book of its kind. An Excellent piece of work!!!

Multi-Criteria Decision Making (MCDM) has been one of the fastest growing problem areas during at least the last two decades. Thus, due to the importance of this field, many MCDM methods have been proposed. Ironically, often these methods contradict each other when they are posed to solve the same decision problem. This, in turn, has increased the confusion in MCDM theory and practice.

As Professor H.-J. Zimmermann from Aachen, Germany, remarked in the foreword for this book, this is exactly where this book has its focus and why it is that important: Rather than suggesting another MCDM method without any convincing justification, this book concentrates on the best known and most frequently used methods, compares them extensively and makes the reader aware of quite a number of "abnormalities" of some of the methods of which users are often not conscious. The book also considers very critically the most touchy points in solving real MCDM problems, namely, the quantification of qualitative data, how to derive relative weights of importance from ratio and difference comparisons, and also how to perform a sensitivity analysis with many MCDM methods.

This book provides a unique perspective into the core of MCDM methods and practice. Many theoretical and empirical analyses are presented and are complementary to each other. This allows the reader to gain a deep theoretical and practical insight into the topics covered in this book. In addition, the author offers at the end of each chapter and at the end of the book suggestions for further research. More information on this book can be found in the personal web site of the author which is located at the Louisiana State University in the U.S.A.

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.

I owe Dr Luenberger a million thanks for writing this book. As his student, I think he is the master of putting complex issues in simple words. Your faithful student..Jayanth Krishnan

When I decided to change my career path from B-school to mathematics, I know that only with taking calculus and linear algebra courses is definitely not enough for me to get into a decent math graduate program. I spent an afternoon in a local bookstore to find a book for functional analysis and Hilbert space which is comprehensible for me at that time. I found Luenberger. I was obsessed with its clarity and simplicity without sacrificing too much rigor. Especially for those finance student who want to learn some advanced math for quant finance but may not have enough background to deal with, Luenberger's book is a really good starting point!

Combinaorial Optimization is one of those rare books that is an instant classic. The authors weave a readable fabric of intuition and theory that is unmatched in this exciting discipline. The choice of topics covered begins with two fundamental optimization problems, namely, the minimum spanning tree and shortest path problems. Next, maximum flow and minimum cost flow problems are discussed, followed by matching problems, polyhedral issues arising in combinatorial optimization, and the famous traveling salesman problem. The text concludes with chapters on matroids and NP-Completeness. The exposition on these topics is very well written and the proofs are rigorous. There is a terrific blend of theory, algorithms and applications without overwhelming the reader with computational details. The authors also do a good job of developing an accurate historical perspective of the material, most of which evolved during the time period 1955 to 1995. The book is suitable for an upper-level undergraduate, or a graduate course. The exercises are very well thought out and are at an appropriate level. I have taught undergraduate courses in combinatorial optimization for over 10 years and have always struggled to find an appropriate text. My problem has now been solved.

This book was thoroughly written by great-minded Masters. It is well-organized in their topics and presentation. However, the book details is unbalnced, some chapters are overwhelm the data, and some others are insufficient. By the way, I graded this book a Very Good one. Worth Reading !!

A good introduction to Combinatorial optimisation and integer programming.

Especially recommended are the chapters on minimum weight matching and the TSP.

I lost my original copy during my last move. Therefore, I was overjoyed that an inexpensive paperback version had been printed. A must for the numerical analyst's library.

This is a good intermediate text on numerical analysis. The development of the underlying real variable theory is much more rigorous than the closely related and more recent text "Numerical Recipes in C". Also, there is more attention paid to function theoretic considerations such as notions of continuity and compactness. This is basically an introductory numerical functional analysis textbook. There are numerous good examples sprinkled throughout the text. To get the most out of this book, you need a working knowledge of advanced calculus, real analysis and linear algebra.

This is the republication of the 2nd edition published by McGraw-Hill, 1978, with minor corrections. This Dover edition also includes 50 pages of Hints and Answers to Problems, which is very helpful. It is one of the 14 reference books listed in the Numerical Recipe in C: The Art of Scientific Computing, and the authors of the Recipe book says, of the 14 books, "These are the books that we like to have within easy reach." A. Ralston, of SUNY Buffalo, also co-wrote a book, Discrete Algorithmic Mathematics(DAM), which is easy and fun to read. But I am puzzled by the words - "Well-known and highly regarded even by those who have never used it." - on the back cover of the A K Peters edition of DAM. What do they mean?

Nelder and Wedderburn wrote the seminal paper on generalized linear models in the 1970s. Since then John Nelder has pioneered the research and software development of the methods. This is the first of several excellent texts on generalized linear models. It illustrates how through the use of a link function many classical statistical models can be unified into one general form of model. This unification is helpful both theoretically and computationally. Various applications are presented in a clear manner.

The first edition is already a well-known text and reference, this expanded version is even better. Very comprehensive and very helpful.

This is an important book. It is a mature, deep introduction to generalized linear models.

General linear models extend multiple linear models to include cases in which the distribution of the dependent variable is part of the exponential family and the expected value of the dependent variable is a function of the linear predictor. Besides the normal (Gaussian) distribution, the binomial distribution, the Poisson distribution and the Gamma distribution, are just some of the exponential family members most frequently encountered in the scientific literature. Using appropriate functions to join the dependent variable to the linear predictor many classic models of applied statistics are included in the broad frame of generalized linear models: "logistic regression", log-linear models, Cox's proportional hazards models are just some of them.

Further extensions to the "base" family of generalized linear models, such as those based on the use of quasi-likelihood functions, and models in which both the expected value and the dispersion are function of a linear predictor, are well presented in the book.

Examples, and exercises, introduce many non-banal, useful, designs.

There are some minor drawbacks. Some more advanced topics might have been introduced more smoothly (i.e. conditional likelihood). Some other topics are better understood when you are already familiar with the specific object of study (i.e. Cox's proportional hazards models as a generalized linear model). The book does not provide software examples, nor is it related with any specific statistical package. However, the maturity of the reader to whom the book is addressed should be so high that translating the majority of the examples presented in the book in the "language" of a familiar statistical package should not be a problem.

Nemhauser is a great professor and this book is text book for IP. But this book requires that you are familiar with Linear programming. So read this book only after you had LP. Bertsimas book is really good for Lp. Or even Bazaara's book is a goo start.

This book starts with the mathematical basics behind linear programming and develops on these introducing new techniques like Bender's decomposition, various cuts, etc. The way the mathematics is dealt is flawless but I thought the methods required more examples for better understanding. But ofcourse the book had to be concise....

I have no opinion on the combinatorial optimisation part.

I first used this book as a text for a graduate course in Integer Programming. At first it seemed to be a very poorly organized book, but as I read it and grew familiar with the subject, I realized that this is a hybrid textbook-handbook. Though the title says "Integer and Combinatorial", the authors go a step ahead and present topics in advanced linear programming, computational complexity, polyhedral theory in a fashion appropriate to the learning of this subject. The exercises are challenging and it has a very good list of references (only up to 1988). Our professor had to supplement the text with recent papers to cover the latest advances. My only complaint is that model formulation could have been dealt with in more detail. You might want to use HP Williams' "Model Building in Mathematical Programming" to look up good Math Prog models.
Though this is an excellent book in all respects, I would recommend Papadimitriou's older book on combinatorial optimization for a good discussion of P, NP problems and decision / optimization problems.

Don't let the title fools you. This book goes beyond Integer and Combintarial optimization. While there are many books talk about NP and proof, this book makes you "understand" it! Still, I agree with others that this book is a reference tool for Integer and Combinatorial optimization. I'm so glad this book published in paperback so it will be more affordable for others.... still expensive though :(

The book offers an objective treatment of linear programming, in small self contained chapters. I consider this title the best introdutory text on LP, just because it is extremely well written. The major drawback of this book is the small and easy number of exercises proposed at the end of the chapters. The text is not an updated book on the subject, but I really recomend it.

I cut my teeth on this text in George Nemhauser's class. The book is clear and concise and does an excellent job explaining this topic to beginners. I've not come across a better introductory text yet. I still have this book in my reference library.

If you want an introduction to LP, this is the text for you.

A masterpiece on Linear Programming. Although it does not contain Interior Point Methods developed subsequently, it's always the first book I refer to, whenever I have any questions on Linear Programming. Strongly Recommended.

Some books are good for mathematicians, some books are good for managers. This book is different. Williams did a good job to combine both mathematic and application perfective in a single book. Even you have only high school background, this book is readable. For senior researchers or grad students or strong math background person, this book is still enjoyable to recall your fundamental of math modeling. The references are not quite updated, however. Also, this book should added some current optimization tools. Even though the title is model building, not solving, it won't be harmful to have the metaheuristics (only introduction) or KKT.

If there is anything that I would hold against my favorite Operations Research books - it would be the lack of emphasis on model and structure. Williams' book fills in that gap and is an essential companion to every Math Prog book. It is not a cookbook where one can look up a particular problem and the possible ways to model it. Instead, it takes a systematic and very sensible approach to modeling.
The three chapters on Integer Programming Models are amazingly easy to understand and were a real help during a graduate course in the subject. The huge number of practical examples in Parts 2, 3 and 4 of the book is the real value of the book. I would be hard-pressed for space to describe the range of problems that are modeled in Part 2... Part 3 covers a good deal of discussion on these formulations and Part 4 follows it up with solutions. Though solutions are not discussed in detail, they are a great help for someone who has worked hard through the problems and needs a verification of the solutions.
Another useful section in the book is a chapter on the interpretation of Linear Programming solutions. For a person without a Math Prog background (say, a manager), this kind of material is very useful. In fact, it once served as a good refresher for me in a hurry... and an excellent one at that.
The only sore point is a very limited discussion on nonlinear models.

This is one of the only books I have ever encountered that focuses on the practical aspects of model formulation. This is a frequently overlooked aspect of optimization, but models that are well formulated will often result in superior performance. It is particularly strong in the formulation of mixed-integer problems, with a variety of tips for linearizing variable products and for incorporation of logical constructs. It also shows how to model SOS1 and SOS2 variable types. One other area that I found to be particularly useful was a section covering convexity analysis. This was the only book that I've read that did a good job of explaining the concepts and ramifications of problem convexity. Finally, the examples in the book cover a wide range of practical problems. Most are fairly simple, but do a good job of illustrating important techniques.

I highly recommend this book for linear and mixed-integer modelers. However, if you don't use these types of solvers in your work, the book is less likely to be valuable.