Linear Programming Computation

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.

這本專著代表了線性規劃（LP）領域的一個歷史性突破，自1947年George Dantzig首次發現單純形法以來。它既思考深入又資訊豐富，著重於反映和推動LP領域的最新成就。這個新版本分為兩卷。第一卷涵蓋了LP的基礎知識，包括可行區域的幾何形狀、單純形法及其實現、對偶性和對偶單純形法、原始對偶單純形法、敏感度分析和參數化LP、廣義單純形法、分解法、內點法和整數LP方法。第二卷主要介紹了作者自己的貢獻，例如高效的原始/對偶樞紐規則、原始/對偶Phase-I方法、簡化/減少單純形法、廣義簡化單純形法、原始/對偶不完全基方法、原始/對偶面方法、一個新的分解原則等等。

這個版本進行了許多重要的改進。第一卷包括了新的結果，例如混合兩階段單純形算法、雙重消除、用於減少成本的新價格方案、雙層LP模型和最佳解集的截取。特別是，整數LP方法的章節以新的ILP求解器「controlled-cutting/branch」方法的目標削減為基礎進行了重寫，同時還引入了一個有吸引力的controlled-branch方法的實現。

在第二卷中，單純形可行點算法被重寫，並從關鍵內點法章節中移除，形成了一個獨立的章節，標題為「單純形內點法」，因為它代表了從傳統單純形算法轉化而來的高效內點算法類別。原始章節的標題則改為「面內點法」，因為剩下的算法代表了另一類從正常內點算法轉化而來的高效內點算法。在不利用稀疏性的情況下，原始的原始/對偶面方法使用了Cholesky分解。為了處理稀疏計算，第二卷增加了兩個討論LU分解的新章節。最令人興奮的改進來自對簡化單純形法的重新發現。在第一版中，它的原型推導在一個同樣的章節中呈現，然後在另一個章節中轉換為所謂的「改進」版本。幸運的是，作者最近找到了一個非常簡潔的新推導，因此他現在可以在一個單獨的章節中介紹這種獨特的新單純形法。令人興奮的是，簡化單純形法有望成為有史以來最好的LP求解器。

這個版本著重於計算，包含了許多新穎的想法、理論和方法，並有實際的數值結果支持。內容清晰簡潔，從簡單到深入地呈現。特別是，有更多的例子用來演示算法。這本書是LP領域中一部罕見的著作，是本科生、研究生、教師、從業人員和LP及相關領域的研究人員必不可少的工具。

Ping-Qi Pan is Professor and Doctoral Supervisor of Mathematics Department at Southeast University, Nanjing, China. He was Visiting Scholar at University of Washington (1986-1987), and Visiting Scientist at Cornell University (1987-1988). His research interest focuses on mathematical programming and operations research, especially large-scale linear optimization. He was standing council member of Mathematical Programming Society of China, and standing council member of Operation Research Society of China. Professor Pan has received the honorary title of Outstanding Scientific-Technical Worker of Jiangsu Province of China. He was nominated as Top 100 scientist of 2012 by the International Biographical Centre, Cambridge, England. He won the LIFETIME ACHIEVEMENT AWARD by Who's Who in the World in 2017.

潘平奇是中國南京東南大學數學系的教授和博士生導師。他曾在華盛頓大學（1986-1987年）擔任訪問學者，並在康奈爾大學（1987-1988年）擔任訪問科學家。他的研究興趣集中在數學規劃和運籌學，尤其是大規模線性優化。他曾擔任中國數學規劃學會常務理事和中國運籌學會常務理事。潘教授獲得了中國江蘇省優秀科技工作者的榮譽稱號。他被國際傳記中心（位於英國劍橋）提名為2012年的百大科學家。他在2017年獲得了《世界名人錄》終身成就獎。