Suppose we are trying to solve the following problem:

\begin{align} \text{minimize} & \hspace{8pt} f(x) \\ \text{subject to} & \hspace{8pt} g(x) = 0, \end{align} where $f$ and $g$ are both differentiable.

The method of Lagrange multipliers says that we should solve the following system of equations: \begin{align} \nabla f(x) &= \lambda \nabla g(x) \\ g(x) &= 0. \end{align}

If I form the Lagrangian, then I get this equation: $$L(x,\lambda) = f(x) - \lambda g(x).$$

If I take the gradient of the Lagrangian $L$ and set it equal to $0$, I find that I get the system of equations that I must solve for the method of Lagrange multipliers. The derivative with respect to $x$ gives me the first equation, and the derivative with respect to $\lambda$ gives me the second equation.

Why is this the case?

  • $\begingroup$ Are you asking if there is a connection between the derivatives of the Lagrangian function and the equations in the method of Lagrange multipliers? Definitely yes. $\endgroup$ – Matthew Leingang May 8 '18 at 2:18
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    $\begingroup$ I think you want $L(x, \lambda) = f(x) - \lambda g(x)$ in this formulation; with $L(x, \lambda) = f(x) + \lambda g(x)$ one gets $\nabla f = -\lambda \nabla g$, which works, but is not what is stated in the problem. Cheers! $\endgroup$ – Robert Lewis May 8 '18 at 2:32
  • $\begingroup$ @MatthewLeingang Excellent. Would you explain what the connection is and why? $\endgroup$ – NicNic8 May 8 '18 at 2:41
  • $\begingroup$ What is the problem here? It seems that you are fine with the statement "The method of Lagrange multipliers says that we should solve the following system of equations: \begin{align} \nabla f(x) &= \lambda \nabla g(x) \\ g(x) &= 0." \end{align} Now this statement is obviously equivalent with the quoted statement about the Lagrangian $L(x,\lambda)$. $\endgroup$ – Christian Blatter May 8 '18 at 19:21
  • $\begingroup$ @ChristianBlatter In introductory calculus books, they develop and prove the two equations you listed. I'd like a way to get to the equations from the Lagrangian, but I don't see an intuitive way to understand why the solution would be a saddle point of the Lagrangian. $\endgroup$ – NicNic8 May 9 '18 at 0:05

The connection comes from calculus.

Consider the problem:

\begin{align} \text{minimize} \quad & f(x) \\ x \in \quad & [a,b] \end{align}

If the point where $f$ attains its minimum is $x^*$, then from the extreme value theorem, $$\frac{\mathrm{d}f}{\mathrm{d}x}(x^*) = 0$$

Now add some equality constraints to this problem to get a general constrained problem.

\begin{align} \text{minimize} \quad & f(x) \\ \text{subject to} \quad & g(x) = 0 \\ x \in \quad & X \end{align}

The method of Lagrange multipliers forms a new function called the Lagrangian: $$\mathcal{L}(x, \lambda) = f(x) \ +\ \lambda g(x) $$

Under some assumptions, it can be proved that if $x^*$ is the point where the equality constrained problem above is minimised, then there is a unique $\lambda^*$ such that

$$\nabla \mathcal{L}(x^*, \lambda^*) = 0$$

This along with the original equality constraint simplifies to the system of equations in your question.

  • $\begingroup$ Why is a critical point of $\mathcal{L}$ the solution? Is the proof complicated? Is there some intuitive way to understand this? $\endgroup$ – NicNic8 May 8 '18 at 18:05
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    $\begingroup$ To put it very roughly, if $x^*$ is an optimal solution for $(3)$, then it is also feasible, which means that $g(x^*) =0$, i.e., the Lagrangian is just the calculus problem in disguise. Now the reason why we bother forming the Lagrangian at all is because the domain $X$ need not be as simple as in the case of the calculus problem or the constraints may be inequalities or non-linear and then, the theory of Lagrange multipliers gives us important information on the solutions. $\endgroup$ – Hikaru May 8 '18 at 18:37
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    $\begingroup$ For a proof refer to the book by Bertsekas $\endgroup$ – Hikaru May 8 '18 at 18:43

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