Is there any stricy way of eliminating Lagrangian Multipliers or it depends on the specific equations after derivative equations?

Question

I am pretty new for the concept of langragian and might be a naive question. After computing the first derivative equation of lagrange version of the function, it is required to get away multipliers but it is a little hasting for me at the moment to see how to do that. Do I need to come up with new method for this elimination for each different problem or is there any intuitive general way to get rid of this multipliers to find the optimal value of objective variable?

Answer 1

If there would be a simple "general way to get rid of these multipliers" this way would have been condensed into a theorem, and you would have heard about it.

Whether one should get rid of the multipliers right away depends on the problem at hand.

In many situations where there is just one multiplier $\lambda$ it is possible to express the actually interesting unknowns $x_i$ (resp., $x$, $y$, $z$, etc.) in terms of $\lambda$. The given condition $F(x_1,x_2,\ldots, x_n)=0$ then turns into an equation for $\lambda$ that hopefully can be solved, and finally one obtains the values of the corresponding $x_i$.