I have the following problem. Let's say we have $x_{jk}$ it is an expression value of gene $j$ in a sample $k$. It is the average of expression levels across the cell types $s_{ij}$, weighted by respective proportions $a_{ki}$ ($i = 1 \cdots N$, $N$ is the disease type):
$$ x_{jk} = \sum_{i=1}^{N} a_{ki}s_{ij} $$
Generally this can be expressed as matrix form
$$ X = AS $$
What I want to do is to solve this equation
\begin{align} &\min_{A}\ || AS- X||^{2}\\ &\text{subject to } \left\{ \begin{array}{c l} \sum_{i} a_{ki} = 1\\ a_{ki} \ge0, \forall i \end{array}\right. \end{align}
Typically this is solved by using quadratic programming. The number of genes is large (e.g. ~30K).
I'm wandering if there is any good probabilistic model to deal such kind of problem?