Lagrange乘数法是求条件极值的重要方法。教材中仅仅针对目标函数为二元函数,约束条件为一个二元方程时,给出了Lagrange乘数法的基本思想与详细的做法,但对于自变量多余两个、约束条件多余一个的情形的Lagrange乘数法只是简单提及,没有给出详尽的推导过程。本文分别从横向和纵向两个维度,通过层层递进的方式,按照五种情形,给出了Lagrange乘数法的一般推广,并进行了详细的理论推导以及给出了Lagrange乘数法的几何意义。研究结果不论对于一线的科研工作者还是初学者都有一定的启发与借鉴意义。Lagrange multiplier method is an important method for finding conditional extremum. The textbook only provides the basic idea and detailed method of Lagrange multiplier method when the objective function is a binary function and the constraint condition is a binary equation. However, for cases where there are more than two independent variables and more than one constraint, the Lagrange multiplier method is only briefly mentioned without providing a detailed derivation process. This article provides a general extension of the Lagrange multiplier method from both horizontal and vertical dimensions, using a progressive approach in five different scenarios. Moreover, detailed theoretical derivation and geometric significance of the Lagrange multiplier method are also presented. The research results of this article have certain inspirations and reference significance for both frontline researchers and beginners.
