Looking for a method to solve the following problem (not sure if it can be written as a convex optimization problem, and solved by a SDP solver):
$\mathop {\min }\limits_X {\left\| {A - {X^T}BX} \right\|_F} \\ Subject \; to: Some \; Convex \; Constraints $
Where $A \in \mathbb{S}^n$, $B \in \mathbb{S}^m$ and $X \in \mathbb{R}^{m \times n}$.
No, it is not convex as it essentially is a non-convex quartic polynomial (if you square the expression to get rid of the square-root, trivial monotonic manipulation). As a simple example, consider the scalar case with $a=1$ and $b=1$ which will lead to $(1-x^2)^2$ which has two distinct globally optimal solutions.
No simple way to solve it.