Consider finding $$\underset{X}{\operatorname{argmin}} \left\|S-AX\right\|^2$$ where $S, A, X$ are real-valued matrices and $\|\cdot\|$ denotes the Frobenius norm.
Is it there a straightforward (i.e. easy-to-implement) algorithm to solve this problem when there is an additional constraint on the elements of $X$, namely that $X_{i,j} \in [-1,1]$? Is this a quadratic programming problem?
If you disregard the additional constraint, it's a linear problem (or actually a least-squares problem). You just need to rewrite it using vectorization and Kronecker products. However you could probably linearize the constraint $X_{i,j} \in [-1,1]$ by adding $\lambda\|{\bf x}\|_F$ and then adjust $\lambda$ which could be a diagonal matrix so you have an individual $\lambda$ for each ${X_{i,j}}$ and solving a couple of linear optimizations, updating the lambdas in between each run.