In this paper, a modified version of the Classical Lagrange Multiplier method is developed for convex quadratic optimization problems. The method, which is evolved from the first order derivative test for optimality o...In this paper, a modified version of the Classical Lagrange Multiplier method is developed for convex quadratic optimization problems. The method, which is evolved from the first order derivative test for optimality of the Lagrangian function with respect to the primary variables of the problem, decomposes the solution process into two independent ones, in which the primary variables are solved for independently, and then the secondary variables, which are the Lagrange multipliers, are solved for, afterward. This is an innovation that leads to solving independently two simpler systems of equations involving the primary variables only, on one hand, and the secondary ones on the other. Solutions obtained for small sized problems (as preliminary test of the method) demonstrate that the new method is generally effective in producing the required solutions.展开更多
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...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.展开更多
文摘In this paper, a modified version of the Classical Lagrange Multiplier method is developed for convex quadratic optimization problems. The method, which is evolved from the first order derivative test for optimality of the Lagrangian function with respect to the primary variables of the problem, decomposes the solution process into two independent ones, in which the primary variables are solved for independently, and then the secondary variables, which are the Lagrange multipliers, are solved for, afterward. This is an innovation that leads to solving independently two simpler systems of equations involving the primary variables only, on one hand, and the secondary ones on the other. Solutions obtained for small sized problems (as preliminary test of the method) demonstrate that the new method is generally effective in producing the required solutions.
文摘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.
