I have the following problem:
$ \min_{A} || A - X||_{F}\\s.t. \theta = (A^TA)^{-1}A^Ty$
where $A$ and $X$ are $n\space x \space p$ matrices and $y$ is a $n\space x \space 1$ vector.
$\theta$ is the estimates of the population parameters in linear regression with OLS $(\hat{\beta})$.
I am thinking of using the Lagrangian. However, although it is fairly easy to minimize the Frobenius norm without the constraint, I am not sure how to integrate the constraint into the Lagrangian as it is a equation involving matrices and vectors.