高等代数在线性规划问题求解中的应用

摘要

高等代数课程是数学类本科阶段最基本的核心课程之一，不仅是初等代数的延拓，也是后续课程——运筹学，尤其是线性规划部分的基础。运用高等代数知识求解了线性规划问题：利用线性规划问题标准型的线性约束条件是线性方程组的特点，从线性方程组的角度分析了可行解与最优解；利用线性规划问题标准型的矩阵形式，根据矩阵的知识推出基可行解与目标函数值的表达式，以及最优解的判别准则，并讨论了矩阵初等变换与单纯形法的联系。同时通过一个典型实例说明了以上各分析的正确性。%　　Advanced Algebra is one of the basic core courses for all undergraduates in the department of mathematics. It is not only the extension of elementary mathematics, but also the foundation of Operations Research, especially of the Linear Programming. The knowledge of advanced algebra is used to solve the linear programming problem. Because the linear constraints of the standard format of the linear programming problem are the linear equations, the feasible solutions and the optimal solution are deduced from the point of view of the linear equations. According to the matrix form of the standard format of the linear programming problem, the expressions of the basic feasible solutions and the corresponding objective function value can be obtained, in addition to the criteria for the optimal solution. Moreover, the close relationship between the elementary transformation of the matrix and the simplex method is analyzed by the knowledge of the matrix. The correction of the above analysis can be illustrated by one typical example.