IF $\mathbf{X=AS}$ where $\mathbf{X} \in R_+^{n \times m}$, $\mathbf{A} \in R_+^{n \times r}$ are known variable and $\mathbf{S} \in R_+^{r \times m}$ is unknown variable, How to solve the below problem,
$\underset{\mathbf{S}}{\text{maximize }} \|\mathbf{S}\|_F^2$
subject to $\mathbf{X=AS}$, $\mathbf{S}\succeq 0$, $\|\mathbf{S}\|_1=1$
I will appreciate your help as I am not from mathematics background. I have tried many formulation to suit the available algorithms, but I could not manage to end up with any. At Least tips and suggestion with guideline to solve similar problem will help greatly.
Thanks for reading.