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单词 Lagrange multiplier method
释义

Lagrange multiplier method

中文百科

拉格朗日乘数 Lagrange multiplier

（重定向自Lagrange multiplier method）
图1：绿线标出的是约束g(x,y) = c的点的轨迹。蓝线是f的等高线。箭头表示斜率，和等高线的法线平行。
Figure 1: Find x and y to maximize f(x, y) subject to a constraint (shown in red) g(x, y) = c.
Figure 2: Contour map of Figure 1. The red line shows the constraint g(x, y) = c. The blue lines are contours of f(x, y). The point where the red line tangentially touches a blue contour is the solution. Since d1 > d2, the solution is a maximization of f(x, y).
Figure 3: A paraboloid constrained along two intersecting lines.

在数学中的最优化问题中，拉格朗日乘数法（以数学家约瑟夫·拉格朗日命名）是一种寻找多元函数在其变量受到一个或多个条件的约束时的极值的方法。这种方法可以将一个有n个变量与k个约束条件的最优化问题转换为一个解有n + k个变量的方程组的解的问题。这种方法中引入了一个或一组新的未知数，即拉格朗日乘数，又称拉格朗日乘子，或拉氏乘子，它们是在转换后的方程，即约束方程中作为梯度（gradient）的线性组合中各个矢量的系数。

比如：

要求f(x, y) \,在g(x, y) = c\, 时的最大值时，我们可以引入新变量拉格朗日乘数\lambda，这时我们只需要下列拉格朗日函数的极值：

英语百科

Lagrange multiplier 拉格朗日乘数

（重定向自Lagrange multiplier method）
Figure 1: Find x and y to maximize f(x, y) subject to a constraint (shown in red) g(x, y) = c.
Figure 2: Contour map of Figure 1. The red line shows the constraint g(x, y) = c. The blue lines are contours of f(x, y). The point where the red line tangentially touches a blue contour is the solution. Since d1 > d2, the solution is a maximization of f(x, y).
Figure 3: A paraboloid constrained along two intersecting lines.
Figure 4: Contour map of Figure 3.

In mathematical optimization, the method of Lagrange multipliers (named after Joseph Louis Lagrange) is a strategy for finding the local maxima and minima of a function subject to equality constraints.

For instance (see Figure 1), consider the optimization problem
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更新时间：2025/6/16 23:06:03