I have the following optimization problem
\begin{equation} \begin{aligned} \min_{X \in \Bbb R^{n \times c}} \quad & \operatorname{tr} \left( X^T A X \right) \\ \text { subject to } \quad & X^T X =I_c \\ \end{aligned} \end{equation}
It is known that the solution is eigenvectors corresponding to the $c$ smallest eigenvalues of $A$. The problem that I have is that $A$ depends on $X$. In this case, I don't know the solution.
Since the problem is a sub-problem of a bigger one, and needs to solve this equation multiple times, hence ($X(t)$, And $A(t)$ are evolving). I found that in a paper, it considers that $A(t)$ dependes not on $X(t)$, but rather on $X(t-1)$. And thus return to the first problem that is known how to solve. Is this possible? What can we do to justify this approximation?
Note that if it helps, $$A= B_1 + PB_2P +P^{2} X B_3 X^T P^{2}$$ where $B_i$ does not depend on $X$ and $P$ is a diagonal matrix depends on $X$ !! with
$$P =\operatorname{diag}\left( \left( x_i^T x_i \right)^{-0.5} \right)$$ where $x_i$ is the i-th row of $X$.
Note that $A$ is always symmetric.
