In literature,a separable program is formulated like this:
$$\min_{x_{1},...x_{n}}\sum_{i=1}^{n}f_{i}(x_{i})$$
where $f_{i}$ is a closed proper convex function.
My question is what does 'closed' mean? and is the following problem separable?
$$\min_{X \in R^{mn}}||X||_{*}+\lambda||\mathcal AX-b||_{2}^{2}$$
A convex function is called closed iff $f = \operatorname{cl} f$, where $\operatorname{cl} f$ is defined as follows: If $f(x) = -\infty$ for any $x$, then $(\operatorname{cl} f)(x) = -\infty$ for all $x$, otherwise $\operatorname{cl} f$ is the function whose epigraph is given by the $\overline{\operatorname{epi} f}$.
Since the $f$ in the question is proper,the function is closed iff the epigraph is closed iff the function is lower semicontinuous.
Without knowing what $\|\cdot\|_*$ is, it is difficult to give conditions under which the problem might be separable.