Let $A$ be a $m\times n$ matrix and $u\in\mathbb R^m$. Denote by $0\preccurlyeq x$ for some vector $x$ in $\mathbb R^m$ the relation such that $\forall i$, $0\leq x_i$. I am trying to solve the following minimization problem : \begin{align*} \min_{\substack{v\in\mathbb R^n:\\0\preccurlyeq Av}} \left\| v-A^T u \right\|_2^2 \end{align*}
I am wondering if there is a nice closed form solution to this problem or if there are some good algorithm to solve it. If it can be of any help, we can assume $0\preccurlyeq u$.
Since this is convex, the KKT conditions are sufficient, here they are ($\mu$ are the lagrange multipliers) : \begin{cases} 2v-2A^Tu-A^T\mu=0\\ 0\preccurlyeq Av\\ 0\preccurlyeq \mu\\ \mu^T A v = 0 \end{cases}
I tried playing around with those without much success.