This is related to two previous questions which I asked about the history of Lagrange Multipliers and intuition behind the gradient giving the direction of steepest ascent.
I am wondering if the constant $\lambda$ in the Lagrange equation $$\nabla f=\lambda \nabla g$$ has any significance.
For instance, can the sign of $\lambda$ tell us anything about the nature of the critical point? Does its magnitude have any significance?
Best Answer
Suppose you want to maximize $z=f(x,y)$ subject to the constraint $g(x,y)=c$. You've used the method of Lagrange multipliers to have found the maximum $M$ and along the way have computed the Lagrange multiplier $\lambda$. Then $\lambda={dM\over dc}$, i.e. $\lambda$ is the rate of change of the maximum value with respect to $c$.
Said another way, you can think of $\lambda$ as approximately the change in $M$ that results from a one unit change in $c$.
Elaborating further: Optimizing $f(x,y)$ subject to $g(x,y)=c$ via Lagrange multipliers leads to $\nabla f=\lambda g$. Let $L(x,y;\lambda):=f(x,y)+\lambda(c-g(x,y))$. Then the constrained optimization problem can be cast as $\nabla L=0$.
From this perspective, ${\partial L\over \partial c}=\lambda$, i.e. $\lambda$ is the rate of change of the quantity being optimized, $L$, with respect to the constraint value, $c$.