# When to use Lagrange-Multiplier to solve an optimization problem

I'm currently sturrling with the Lagrange-Multiplier. I do get it in theory but I kind of fail knowing when to use it.

Usually I get exercises like this:

Let $f:\mathbb R^n \to \mathbb R^m$ be some function and let A be some set.

I know, if I can parametrize A so that I get a function $g=0$ representing it's surface, can use lagrangian - right?

But what if I have $f: A \to R^m$ whereas $A\subset \mathbb R^n$, can I still use it or not?

Example: $A=\{(x,y,z)\in \mathbb R^3 | x^2+2y^2+3z^2 \leq 1\}$

$f: A\to\mathbb R, \quad (x,y,z)\mapsto x^2-y^2+1$

If I can find a parametrization for $A$ I could use Lagrange-Multipliers here, right? (for local points only)

• In fact, there is possible pitfall when the domain is a certain $A$: in this case, It does not prevent you to use Lagrange method but you have to check that the optimal value has not arise on the border of $A$. It is rather often the case in practise. – Jean Marie Jan 18 '17 at 11:00
• I added an example. – xotix Jan 18 '17 at 11:02
• Keep in mind, that optimization wirh Lagrange only works with $f : \mathbb{R}^n\rightarrow \mathbb{R}$ (aka having a single target value). You can not use this method, when mapping into the $\mathbb{R}^m$ with $m >1$ – Laray Jan 18 '17 at 11:06
• That's a good point, thanks. Starts to clear up. – xotix Jan 18 '17 at 11:17