Linear Programming

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.

Contenu

1 Introduction.- 1.1 The Problem.- 1.2 Prototype Problems.- 1.3 About this Book.- 1.4 Notes.- 2 Background.- 2.1 Real Analysis.- 2.2 Linear Algebra and Matrix Analysis.- 2.3 Numerical Linear Algebra.- 2.4 Convexity and Separation Theorems.- 2.5 Linear Equations and Inequalities.- 2.6 Convex Polyhedral Sets.- 2.7 Nonlinear System of Equations.- 2.8 Notes.- 3 Duality Theory and Optimality Conditions.- 3.1 The Dual Problem.- 3.2 Duality Theorems.- 3.3 Optimality and Complementary Slackness.- 3.4 Complementary Pair of Variables.- 3.5 Degeneracy and Uniqueness.- 3.6 Notes.- 4 Boundary Methods.- 4.1 Introduction.- 4.2 Primal Simplex Method.- 4.3 Bounded Variable Simplex Method.- 4.4 Dual Simplex Method.- 4.5 Primal - Dual Method.- 4.6 Notes.- 5 Interior Point Methods.- 5.1 Primal Affine Scaling Method.- 5.2 Degeneracy Resolution by Step-Size Control.- 5.3 Accelerated Affine Scaling Method.- 5.4 Primal Power Affine Scaling Method.- 5.5 Obtaining an Initial Interior Point.- 5.6 Bounded Variable Affine Scaling Method.- 5.7 Affine Scaling and Unrestricted Variables.- 5.8 Dual Affine Scaling Method.- 5.9 Primal-Dual Affine Scaling Method.- 5.10 Path Following or Homotopy Methods.- 5.11 Projective Transformation Method.- 5.12 Method and Unrestricted Variables.- 5.13 Notes.- 6 Implementation.- 6.1 Implementation of Boundary Methods.- 6.2 Implementation of Interior Point Methods.- 6.3 Notes.- A Tables.