I know of a technique, but I don't know of its name and I don't have any real literature on the technique.
On the wikipedia page for Lagrange multipliers a method is provided to convert a Lagrangian function to another function which has minima where the Lagrangian has any kind of stationary point.
For function $f(\mathbf{x})$ subject to $g(\mathbf{x})=0$, the Lagrangian is given by
$L(\mathbf{x},\lambda) = f(\mathbf{x}) + \lambda g(\mathbf{x})$
When this isn't solvable analytically, we could try to use some numerical optimization techniques, but most of those techniques are made to find minima or maxima and not saddle points. We therefore define
$L^2(\mathbf{x},\lambda) = \left( \frac{\partial L(\mathbf{x},\lambda)}{\partial \mathbf{x}} \right)^2 + \left( \frac{\partial L(\mathbf{x},\lambda)}{\partial \lambda} \right)^2 $.
This function clearly has minima where $L(\mathbf{x},\lambda)$ has any kind of stationary point. We can therefore safely use most kinds of numerical optimization techniques in order to find them. We then know that for the found values of $\mathbf{x}$ and $\lambda$, $f(x)$ is minimal or maximal and that the constraints are met.
What is the name of this technique? Is there any literature on this technique?