What conditions need to be hold for a nonconvex optimization problem to have a unique solution?

Specifically, I have the following minimization problem that I'd like to know whether it has a unique solution or not:

$ \min\_{\bf A, B} \| {\bf Y - XAB} \|\_2^2 + \lambda\_1 \|{\bf A}\|\_1 + \lambda\_2 \|{\bf B}\|\_1$

where ${\bf Y}\_{n \times q}$ and ${\bf X}\_{n \times p}$ are fixed, and ${\bf A}\_{p \times r}$ and ${\bf B}\_{r,q}$ are the variables ($r \leq \min(p,q)$).

Also, I know that this minimzation problem is convex if either ${\bf A}$ or ${\bf B}$ are fixed (and so it has a unique solution in that case).

Is there a certain relaxation that transforms this into a convex problem?