TITLE: Generalizing Lagrange multipliers to use the subdifferential?
QUESTION [6 upvotes]: Background:
This is a followup to this question:
Lagrange multipliers with non-smooth constraints
Lagrange multipliers can be used for constrained optimization problems of the form 
$\min_{\vec x} f(\vec x)$ 
s.t. $g(\vec x) = 0$.
Briefly, the method works by constructing the Lagrangian, $L(\vec x, \lambda) = f(\vec x) + \lambda g(\vec x)$, then finding points where $\forall i, \frac{\partial L}{\partial x_i} L = 0$ and $\frac{\partial L}{\partial \lambda} L = 0$.
As was kindly pointed out in this answer, the method fails when $g$ is non-differentiable (but continuous), because the partial derivatives may not exist at points of optimality.  For example, in the problem, minimize $x_1$ subject to $g(x_1,x_2) = x_1 - |x_2| = 0$. The minimum is at $(0,0)$, where $\frac{\partial g}{\partial x_2}$ does not exist.
Question:
It seems that there should be a natural generalization of the method that uses subgradients and the subdifferential.  Does the following work?  Is there a reference that describes this in more detail?
Proposal: construct the Lagrangian as usual, but instead of seeking a point where all partial derivatives are 0, seek a point where 0 is in each partial subdifferential.  So in the example above, the subdifferential with respect to $x_2$ when $x_2=0$ is the interval $[-\lambda, \lambda]$.  Thus, if we were given the solution $x_1=0,x_2=0,\lambda=-1$, we could verify it is a critical point by noting that $\frac{\partial L}{\partial \lambda} = x_1 - |x_2| = 0$, $\frac{\partial L}{\partial x_1} = 1 + \lambda = 0$, and 0 is in the subdifferential of $L$ w.r.t. $x_2$.
Is this argument correct?  My intuitive justification is that for any value $f'(x)$ in some variable's subdifferential at $x$, we should be able to construct a smooth function that has $f'(x)$ as its partial derivative at $x$, then solve the smoothed problem with a standard application of Lagrange multipliers.
(Aside: my goal is actually not to find a method to optimize the function.  I have a method for optimizing such functions, and I'm trying to develop some theoretical understanding of the solutions that it produces.  In particular, I'm trying to understand if proving that a solution satisfies the condition described in the Proposal section is meaningful.)

REPLY [3 votes]: Another issue here is that the subdifferential is classically useful for convex functions only. In your example, the function $x_2\mapsto F(x_1,x_2,\lambda)$ is concave whenever $\lambda>0$, and then its subdifferential is empty at $x_2=0$. And unless the function $g$ is linear, the constraint set won't be convex, so you won't be able to construct a convex Lagrangian-type function either.
You could consider generalisations of the subdifferential to nonconvex functions, (or, more accurately generalisations of the notion of stationarity) but none are perfect. The best known is Clarke's subdifferential [1], which is pretty simple, but doesn't enjoy full calculus. 
[1] F.H. Clarke, Optimization and Nonsmooth Analysis, Wiley: New York, 1983.