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!