Lagrange multiplier
![Figure 1: Find x and y to maximize f(x, y) subject to a constraint (shown in red) g(x, y) = c.](/Images/godic/202501/23/LagrangeMultipliers3D0418.png")
![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).](/Images/godic/202501/23/LagrangeMultipliers2D.svg0418.png")
![Figure 3: A paraboloid constrained along two intersecting lines.](/Images/godic/202501/23/As_wiki_lgm_parab.svg0418.png")
![Figure 4: Contour map of Figure 3.](/Images/godic/202501/23/As_wiki_lgm_levelsets.svg0419.png")
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