Theory and Algorithms for Linear Optimization: An Interior Point ApproachWiley Interscience Series in Discrete Mathematics, ISSN 0277-2698Wiley Series in Discrete Mathematics & Optimization

Linear Optimization (LO) is one of the most widely taught and fast developing techniques in mathematics, with applications in many areas of science, commerce and industry. The dramatically increased interest in the subject is due mainly to advances in computer technology and to the development of Interior Point Methods (IPM) for LO. This book provides a unified presentation of the field by way of an interior point approach to both the theory of LO and algorithms for LO (design, convergence, complexity and asymptotic behaviour). A common thread throughout the book is the role of strictly complementary solutions, which play a crucial role in the interior point approach and distinguishes the new approach from the classical Simplex-based approach. The approach to LO in this book is new in many aspects. In particular the IPM based development of duality theory is surprisingly elegant. The algorithmic parts of the book contain a complete discussion of many algorithmic variants, including predictor-corrector methods, partial updating, higher order methods and sensitivity and parametric analysis. The comprehensive and up-to-date coverage of the subject, together with the clarity of presentation, ensures that this book will be an invaluable resource for researchers and professionals who wish to develop their understanding of LO and IPMs. Numerous exercises are provided to help consolidate understanding of the material and more than 45 figures are included to illustrate the characteristics of the algorithms. A general understanding of linear algebra and calculus is assumed and the preliminary chapters provide a self-contained introduction for readers who are unfamiliar with LO methods. These chapters will also be of interest for readers who wish to take a fresh look at the topic. Contents Introduction Introduction: Theory and Complexity Duality Theory for Linear Optimization A Polynomial Algorithm for the Skew - Symmetric Model Solving the Canonical Problem The Logarithmic Barrier Approach Preliminaries The Dual Logarithmic Barrier Method The Primal-Dual Logarithmic Barrier Method Initialization The Target-following Approach Preliminaries The Primal-Dual Newton Method Applications The Dual Newton Method The Primal Newton Method Application to the Method of Centres Miscellaneous Topics Karmarkar?s Projective Method More Properties of the Central Path Partial Updating High-Order Methods Parametric and Sensitivity Analysis Implementing Interior Point Methods Appendices Bibliography Indexes