I want to solve the following problem: \begin{align} \min_{x\in\mathbb{R}^n} \quad&\sum_{i=1}^n x_i \log x_i \\ \text{s.t.}\quad & x \succeq 0 \\ \quad & \|Ax-b\|_2\leq c \end{align} Where $A$ is not full-ranked and not even square.
I tried to implement a projected proximal point algorithm but I'm having issues with the projection $P_C(x)$ of $x$ on the convex set $C=\{x\mid \|Ax-b\|_2\leq c\}$. Currently I solve the following equation by minimizing the $\|\cdot\|_2$ of the difference: \begin{gather} AP_C(x) = b + (Ax-b)\min\left(1,\frac{c}{\|Ax-b\|_2}\right) \end{gather} And then put to $0$ all negative coordinates of $P_C(x)$.