Linear Programming Computation

The first volume addresses foundations of LP, including the geometry of feasible region, the simplex method and its implementation, duality and the dual simplex method, the primal-dual simplex method, sensitivity analysis and parametric LP, the generalized simplex method, the decomposition method, the interior-point method and integer LP method. Organized into two volumes. this book represents a real breakthrough in the field of linear programming (LP). The first volume addresses fundamentals, including geometry of feasible region, simplex method, implementation of simplex method, duality and dual simplex method, sensitivity analysis and parametric LP, generalized simplex method, decomposition method, interior-point method and integer LP method, as well as reflects the state of art by highlighting new results, such as efficient primal and dual pivot rules, primal and dual Phase-I methods. The second volume introduces contributions of the author himself, such as reduced and D-reduced-simplex methods, generalized reduced and dual reduced simplex methods, deficient-basis and dual deficient-basis-simplex methods, and face and dual face methods with Cholesky factorization, as well as with LU factorization. As a monograph, this book is a rare work in LP, containing many noval ideas and methods, supported by complete computational results. As revealed from the perspective of theory, the most recently achieved results, such as reduced and D-reduced simplex methods, as well as ILP solvers--- controlled-cut and controlled-branch methods, are very significant and promising, though there are no computational results available at this stage. With a focus on computation, the content of this book ranges from simple to profound, clear and fresh. In particular, all algorithms are accompanied by examples for demonstration whenever possible. As a milestone of LP, this book is an indispensable tool for undergraduate and graduate students, teachers, practitioners and researchers, in LP and related fields. This monograph represents a historic breakthrough in the field of linear programming (LP)since George Dantzig first discovered the simplex method in 1947. Being both thoughtful and informative, it focuses on reflecting and promoting the state of the art by highlighting new achievements in LP. This new edition is organized in two volumes. The first volume addresses foundations of LP, including the geometry of feasible region, the simplex method and its implementation, duality and the dual simplex method, the primal-dual simplex method, sensitivity analysis and parametric LP, the generalized simplex method, the decomposition method, the interior-point method and integer LP method. The second volume mainly introduces contributions of the author himself, such as efficient primal/dual pivot rules, primal/dual Phase-I methods, reduced/D-reduced simplex methods, the generalized reduced simplex method, primal/dual deficient-basis methods, primal/dual face methods, a new decomposition principle, etc. Many important improvements were made in this edition. The first volume includes new results, such as the mixed two-phase simplex algorithm, dual elimination, fresh pricing scheme for reduced cost, bilevel LP models and intercepting of optimal solution set. In particular, the chapter Integer LP Method was rewritten with great gains of the objective cutting for new ILP solvers {\it controlled-cutting/branch} methods, as well as with an attractive implementation of the controlled-branch method. In the second volume, the `simplex feasible-point algorithm' was rewritten, and removed from the chapter Pivotal Interior-Point Method to form an independent chapter with the new title `Simplex Interior-Point Method', as it represents a class of efficient interior-point algorithms transformed from traditional simplex algorithms. The title of the original chapter was then changed to `Facial Interior-Point Method', as the remaining algorithms represent another class of efficient interior-point algorithms transformed from normal interior-point algorithms. Without exploiting sparsity, the original primal/dual face methods were implemented using Cholesky factorization. In order to deal with sparse computation, two new chapters discussing LU factorization were added to the second volume. The most exciting improvement came from the rediscovery of the reduced simplex method. In the first edition, the derivation of its prototype was presented in a chapter with the same title, and then converted into the so-called `improved' version in another chapter. Fortunately, the author recently found a quite concise new derivation, so he can now introduce the distinctive fresh simplex method in a single chapter. It is exciting that the reduced simplex method can be expected to be the best LP solver ever. With a focus on computation, the current edition contains many novel ideas, theories and methods, supported by solid numerical results. Being clear and succinct, its content reveals in a fresh manner, from simple to profound. In particular, a larger number of examples were worked out to demonstrate algorithms. This book is a rare work in LP and an indispensable tool for undergraduate and graduate students, teachers, practitioners, and researchers in LP and related fields.