The examples I see using Lagrange multipliers usually express a constraint as something like
$$x^2 + y^2 \ge C$$
but then solve the problem as if the constraint were
$$x^2 + y^2 = C$$
which works for many applications. But what if I'm not certain $L$ will be minimized on the constraint surface? Can Lagrange multipliers still be used?
The application is support vector machines where we minimize the 2-norm of a vector $\vec w$ subject to one constraint of the form
$$\vec w \cdot \vec x_i + b \ge 1$$
for many different vectors $\vec x_i$.
In this case, not all of the vectors will satisfy (and, indeed not all vectors can satisfy)
$$\vec w \cdot \vec x_i + b = 1$$
in the optimal solution.
If I create a Lagrangian like this:
$$L = \vec w \cdot \vec w - \lambda_i(1-\vec w \cdot \vec x_i + b)$$
then what's to stop $\lambda_i$ from going to $-\infty$ for those vectors for which
$$\vec w \cdot \vec x_i + b > 1 ?$$
Are Lagrange multipliers the right tool here?