In this paper, a double-pivot simplex method is proposed. Two upper bounds of iteration numbers are derived. Applying one of the bounds to some special linear programming (LP) problems, such as LP with a totally unimodular matrix and Markov decision problem with a fixed discount rate, indicates that the double-pivot simplex method solves these problems in a strongly polynomial time. Applying the other bound to a variant of Klee–Minty cube shows that this bound is actually attainable. Numerical test on three variants of Klee–Minty cubes is performed for the problems with sizes as big as 200 constraints and 400 variables. The test result shows that the proposed algorithm performs extremely good for all three variants. Dantzig’s simplex method cannot handle the Klee–Minty cube problems with 200 constraints because it needs about \(2^{200} \approx 10^{60}\) iterations. Numerical test is also performed for randomly generated problems for both the proposed and Dantzig’s simplex methods. This test shows that the proposed method is promising for large-size problems.

本文提出了一种双轴单纯形法。得出两个迭代次数的上限。将边界之一应用于某些特殊的线性规划（LP）问题，例如具有完全单模矩阵的LP和具有固定贴现率的Markov决策问题，表明双轴单纯形法可在强多项式时间内解决这些问题。将另一个边界应用于Klee-Minty多维数据集的变体，表明该边界实际上是可以达到的。对Klee-Minty多维数据集的三个变体进行了数值测试，以解决尺寸最大为200个约束和400个变量的问题。测试结果表明，所提出的算法对于这三个变体都表现出极好的效果。Dantzig的单纯形方法无法处理具有200个约束的Klee-Minty立方体问题，因为它需要大约\（2 ^ {200} \ approx 10 ^ {60} \）次迭代。对于提议的和Dantzig的单纯形法，还对随机产生的问题进行了数值测试。该测试表明，所提出的方法在解决大型问题方面很有希望。