I want to find the minimum to the following problem:

$$ min_{X} ||f(X)||_F (1) $$ where $X$ is a rectangular matrix, and $f$ is a function of it, involving other matrices, the norm is Frobenius norm.

I want to know if the solution of problem (1) is equivalent to the following problem: $$ min_{X} ||f(X) A||_F (2) $$ where $A$ is a rectangular matrix.

Are the two problems equivalent? if not, are there some properties on $A$ so that they become equivalent?

Thank you in advance.