Lagrangian equations or direct minimization algorithm for constrained optimization?

[sodemus]

Hello everybody!
I guess my question is mainly concerned with numerical algorithms...
Given a problem of the form
min w = f(x)
subject to
g1(x)=0
:
:
gn(x)=0
where x is a m x 1 vector, n < m.

From a numerical standpoint, how can I know whether it is preferably to solve it by setting up the Lagrangian and solve the resulting system of m + n non linear equations with appropriate algorithms OR to implement an appropriate algorithm to solve the minimization problem directly? As far as my particular problem goes, let's say n = 2 and m = 25.

Any help is more than appreciated!

 

[fresh_42]

If you had knowledge about the constraints, e.g. linearity, you could chose another approach. Without any further information, Lagrange multipliers should be fine.