I need to find the projection $x \in \mathbb{R}^{k}$ of a vector $z \in \mathbb{R}^{k}$ on the set defined by $Y \cdot x \geq 0$ where $Y$ is a (given but no specific property) matrix of size $m \cdot k$.
I first build the Lagrangian as follows :
$$L=\frac{1}{2} ||x-z||^2 -\sum_{i=1}^m \lambda_i Y(i,:)x$$ and set its gradient w.r.t. $x$ to $0$ which leads to $$x=z+\sum_{i=1}^m \lambda_i Y(i,:)^T$$
The inequality constraints add an additional KKT condition :
$$\lambda_i \cdot Y(i,:)x=0 \; \forall i=1,...,m$$
Replacing $x$ inside gives $$\lambda_i \cdot Y(i,:)[z+\sum_{i=1}^m \lambda_i Y(i,:)^T]=0 \; \forall i=1,...,m$$
It is where it becomes harder for me. I guess that either $\lambda_i$ or $Y(i,:)[z+\sum_{i=1}^m \lambda_i Y(i,:)^T]$ should be equal to $0$ but then I am stuck. Indeed, the second condition is a scalar product linking all the $\lambda_i$ and I don't know how to deal with it properly.
Any help appreciated