Let $x_0 \in \mathbb R^n$ and $A \in \mathbb R^{n \times n} $ is lower triangular matrix with all element being $1$:
$$ A= \begin{bmatrix} 1 & 0 & \cdots & 0 \\ 1 & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 1 & 1 & \cdots &1\\ \end{bmatrix} $$
Define set $C$ as: $S=\{ x \in \mathbb R^n: b_{min} \leq Ax \leq b_{max} \}$, where $b_{min}$ and $b_{max}$ are vectors that have identical element:
$$ b_{min}= \begin{bmatrix} x_{min} \\ x_{min} \\ \vdots \\ x_{min} \\ \end{bmatrix}, b_{max}= \begin{bmatrix} x_{max} \\ x_{max} \\ \vdots \\ x_{max} \\ \end{bmatrix} $$
I want to calculate the projection of $x_0$ on $S$. I'm wondering if there is any efficient way to do this, since $A$ clearly has a special structure. I've been trying to derive a closed form expression but no luck so far. Any idea would be highly appreciated!