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Linear Programming: A Modern Integrated Analysis, Vol. 1by Romesh Saigal
Synopses & Reviews
In Linear Programming: A Modern Integrated Analysis, both boundary (simplex) and interior point methods are derived from the complementary slackness theorem and, unlike most books, the duality theorem is derived from Farkas's Lemma, which is proved as a convex separation theorem. The tedium of the simplex method is thus avoided. A new and inductive proof of Kantorovich's Theorem is offered, related to the convergence of Newton's method. Of the boundary methods, the book presents the (revised) primal and the dual simplex methods. An extensive discussion is given of the primal, dual and primal-dual affine scaling methods. In addition, the proof of the convergence under degeneracy, a bounded variable variant, and a super-linearly convergent variant of the primal affine scaling method are covered in one chapter. Polynomial barrier or path-following homotopy methods, and the projective transformation method are also covered in the interior point chapter. Besides the popular sparse Cholesky factorization and the conjugate gradient method, new methods are presented in a separate chapter on implementation. These methods use LQ factorization and iterative techniques.
Book News Annotation:
Presents a unified approach to the study of boundary (simplex) and interior point methods for linear programming. Derives both classes of methods from the complementary slackness theorem, with the duality theorem derived from Farkas' lemma, which is proved as a convex separation theorem. Offers a new and inductive proof of Kantorovich's theorem related to the convergence of Newton's method, and discusses the primal, the dual, and the primal-dual affine scaling methods; the polynomial barrier method; and the projective transformation method. Includes a chapter on background material for the study of boundary methods, and a chapter detailing new methods using LQ factorization and iterative techniques. Can be used as a text in a one- or two- semester advanced graduate course on linear programming.
Annotation c. Book News, Inc., Portland, OR (booknews.com)
Table of Contents
Preface. 1. Introduction. 2. Background. 3. Duality theory and optimality conditions. 4. Boundary methods. 5. Interior point methods. 6. Implementation. A: Tables. Bibliography. Index.
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