Is there a way to write the trace norm of the product of two matrices function of the separated trace norm of those two matrices. Or maybe a good approximation for it function of the same matrices. I want this because i am trying to optimize an objective function that has the trace of the product of two matrices as a regulizer. Having this approximation, this would allow me to use the proximal gradient method as an optimization method to solve the problem.
$$\min_A f(A) +\|AX\|$$
where $\|.\|$ is the trace norm. I want to minimize it with respect to $A$. $f$ is differentiable and convex.