Is Maxima/Minima of Lagrange function same as Maxima/Minima of function under consideration?

Question

Given:

To find critical points of function f(x) subject to constraints:
g(x) = 0

We create a Lagrange function: L(x, λ) = f(x) - λg(x)

Now, are the critical points and max/min of Lagrange function same as critical points of f(x)?

If yes, why?

Answer 1

This is a good question that has a lot of depth to the answer, but the essence is no, this is not true in general for even what you're asking, which is a pretty specific case of the KKT theorem. Essentially this answer gets more complicated when dealing with the KKT theorem, which is a more generalized form of Lagrange multipliers. But the method of Lagrange multipliers is only a necessary condition, meaning that not all the points are guaranteed to be optimal. In fact, part of the exercise of working with Lagrange multipliers in problems is checking the solutions from what you get from the Lagrangian and seeing which is the optimal solution. However, in the case you're specifying, we do have that the Lagrangian will find any point that is optimal in our original problem.

Edit: for further reading, page 7 of this lecture may help solidify some of this.