Customer Reviews

Linear Optimization and Extensions: Problems and Solutions (Universitext)

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This book is certainly a very good reference for theoretical topics of linear programming. It covers the Simplex method and the Ellipsoid algorithms. It also covers the geometry of linear programming (polyhedra and polytopes, etc). It certainly covers more topics than most other linear programming texts. As expected, a book writen for theoretical topics is certainly not easy to read, especially for people with no training in doing rigorous mathematical proofs. Also, not many examples or illustrations are given in this book, and this might be a problem for some readers.

For nearly 30 years, Padberg has been a leader in computational integer programming and in combinatorial optimization theory.

In practice, Padberg has helped to design and implement "branch-and-cut" methods for finding exact optimal solutions to large traveling salesman problems, and this approach is a method of choice for finding approximately optimal solutions to tough industrial problems. The book provides the mathematical and computational background for understanding branch-and-cut; the established mathematical texts by Nemhauser and Wolsey and by Schrijver are less detailed and more condensed, and omit numerical issues. The treatment of modern simplex algorithms for linear programming---updating LU factorizations and using column- and constraint-generation and -purging---is excellent, and a large bibliography contains recent references. Besides industrial and Berlin-airlift scheduling problems, the book contains TSP examples of circuit-board wiring, U.S. state capitals, and Odysseus!

Three more highlights: The double description algorithm receives a complete description, and this is useful for combinatorial geometers. The discussion of integer-arithmetic and complexity theory is very readable, and these technical topics are slighted by interior-point books (besides Wright's quickie), despite their importance in integer programming and combinatorial optimization. The discussion of interior-point algorithms emphasizes projective geometry, a beautiful theory that has inspired so much of optimization theory---besides Karmarkar's interior-point algorithm, Dantzig's simplex algorithm, Fenchel duality, Davidon's conic algorithm for nonlinear optimization, etc.).

The book is not a comprehensive survey of linear programming,
and lacks a treatment of Nesterov's theory of self-concordant barrier-functions. Also, no treatment is given of pivoting algorithms besides Dantzig's (e.g., Terlaky's criss-cross method, Todd's oriented matroid algorithm).