关于求线性规划初始正则解的一个新方法的注记

A Note on a New Method for Achieving an Initial Regular Solution of a Linear Program

在线性规划问题的求解中,对基变量取负值的情形,文献[6]提出一种求初始正则解的新方法.该文对这种方法作了进一步讨论,指出它实质上是由原有单纯形法和对偶单纯形法两个阶段组成.第一阶段通过引入非负右手边向量构造辅助线性规划问题,然后用单纯形法求解这个辅助问题获得原问题的一个正则解(如果存在);第二阶段由此正则解出发,用对偶单纯形法求得原问题的最优解(如果存在).通过大规模例子对这种算法进行数值试验,结果表明它的计算效率非常低,因而对这种方法进行了改进.

Linear optimization has been widely used in the various areas of economics and management. In the case of filling the basic variables with negative values in solving a linear programming problem, Liang presented a new method for achieving an initial regular solution to the problem. This paper further discusses the essence of the method and finds out that the method consists of two phases. In phase 1, a nonnegative right-hand side vector is introduced to construct an auxiliary problem, which can generate a regular solution （if any） to the original problem by the primal simplex algorithm. Phase 2 applies the dual simplex algorithm to obtain the optimal solution （if any） starting from the regular solution. Furthermore, a numerical test on some large-scale instances from NETLIB is performed, and the results show that the method needs a great computational effort. Therefore, the paper presents an improvement of the method.