Consider the following optimisation problem
$$ (1) \quad \min_{x\geq 0} a^\top x+ \epsilon ||x||_2,\\ \quad \quad \quad \text{s.t. }B^\top x=c $$ with $\epsilon>0$.
Suppose I assume that linear independence constraint qualification (LICQ) holds. That is, for every $x\in \{x\in \mathbb{R}^: x\geq 0, B^\top x=c\}$, the binding constraints are linearly independent.
Consider the Lagrangian of (1): $$ L(x,\mu,v)=a^\top x+ \epsilon ||x||_2-\nu^\top x+\mu^\top(B^\top x-c) $$
Consider the sets of $x$ and $\mu$ satisfying the KKT conditions. Let me call these sets $\mathcal{X}$ and $\mathcal{M}$ respectively.
Question: could you help to show that $\mathcal{X}$ and $\mathcal{M}$ are singleton?
Thoughts: I believe that $\mathcal{X}$ is singleton because $\epsilon ||x||_2$ is strictly convex. Instead, I cannot show why $\mathcal{M}$ is singleton.