I am interested in deriving first-order optimality conditions for
\begin{equation} \min_{x\in\mathbb{R}^{n}}f(x)\\ \text{s.t. }x\in\mathcal{K} \end{equation}
where $f$ is a smooth function and $\mathcal{K}$ is a cone, for concreteness let's take \begin{equation} \mathcal{K}=\lbrace x\in\mathbb{R}^{n}\text{ s.t. }x_{1}\geq\|x_{2:n}\|_{2}\rbrace \end{equation} where \begin{equation} x=\begin{bmatrix} x_{1}\\ x_{2:n} \end{bmatrix}, \,x_{1}\in\mathbb{R},\,x_{2:n}\in\mathbb{R}^{n-1} \end{equation} the optimization problem is then equivalently reformulated as \begin{equation} \min_{x\in\mathbb{R}^{n}}f(x)\\ \text{s.t. }x_{1}\geq \|x_{2:n}\|_{2} \end{equation}
If the constraint $x_{1}-\|x_{2:n}\|_{2}\geq 0$ and the objective $f(x)$ were both smooth, then I would form a Lagrangian \begin{equation} \mathcal{L}(x,z)=f(x)-z\cdot(x_{1}-\|x_{2:n}\|_{2}) \end{equation} and then require that \begin{equation} \nabla_{x}\mathcal{L}=0\\ z\geq 0,\,x_{1}-\|x_{2:n}\|_{2}\geq 0,\,z\cdot(x_{1}-\|x_{2:n}\|_{2})=0 \end{equation} however $\|\cdot\|_{2}$ is not a continuously differentable function and so $\nabla_{x}\mathcal{L}$ isn't defined everywhere.
I was able to recover optimality conditions in the literature by replacing $\nabla_{x}\|x\|_{2}$ by the subderivative but am not sure why this works. Any pointers or references on deriving optimality conditions for this problem would be greatly appreciated.
