How to deal with non-existent derivatives in Lagrangian?

I am stucked at a detail in a constrained optimization problem:

Question

Assume that the objective function is continuous on its domain $D$, but at some points $Z \subseteq D$ it is not differentiable. Further assume that the constraints implicitly force some or all feasible and optimal solutions to take values in $Z$. Doesn't this contradict somehow the Lagrangian approach? How to deal with this?

Example

Think of the following optimization problem (with the continuous extension $0 \cdot \log 0 := 0$) on $n$ variables $x_1, \ldots, x_n$:

minimize $\sum_i x_i \cdot \log x_i$

subject to $A~x = b$ and all $x_i \geq 0$

for some given matrix $A \in \{0,1\}^{n \times n}$ and some positive vector $b$. It has a unique optimum (if any) because the objective is strictly convex with convex constraints.

The Lagrangian $\Lambda(x, \mathbf{\lambda}) = \sum_i~ x_i \cdot \log x_i - \lambda^T ( A~x - b)$ provides the derivatives

$\nabla_{x_i}\Lambda(x,\alpha) = \log x_i + 1 - \sum_k \lambda_k a_{ki}$

and

$\nabla_{\lambda_i}\Lambda(X,\lambda) = (\sum_k a_{ik} x_k) - b_i$.

Probably one should not simply continue calculations and just ignore the fact that the derivatives are not defined for $x_i=0$? Indeed the entries in $A$ may be such that every feasible solution $f$ forces some of the $x_i$'s to $0$, that is, it determines some index set $I(f) \subseteq \{1, \ldots, n\}$ with $x_i = 0$ for all $i \in I(f)$. For any such $x_i$ the derivative is $\nabla_{x_i}\Lambda(x,\alpha) = -\infty$. Hence for each feasible solution separately, "in retrospective" we could set all the $x_i$'s from $I(f)$ to zero already from the beginning and then define $\nabla_{x_i}\Lambda(x,\alpha) := 0$ for them). But this seems to be a painful argument and somehow contradicts the clear formulation of the Lagrangian as a function in the free variables $x_i$. So what is the right approach here?

Some rough ideas: Can one for example argue somehow by the continuity of the derivatives? Or should one study all the optimization problems on every index subspace $S \subseteq \{1, \ldots, n\}$ separately constrained to $x_i > 0$ on $S$, in order to then find the optimum among all of them? Or can one extend the Lagrangian approach to a weaker notion of derivatives? Or are there related techniques that directly work on the KKT conditions without explicitly taking the derivatives?