I tried to find something before but it seems all the answers do not include my kind of problem.
Basically I want to minimize
$\vec{a}^TM\vec{a}+\vec{a}^T\vec{b}$, $M$ is symmetric, positive definite
subject to
$\vec{a}^T\vec{c}=\alpha$
and
$\vec{a}^TZ_i\vec{a}=0$ where $Z_i, i=1, 2, 3$ are general matrices, without assumption on them. The only thing you could assume is that they could be symmetrised in this expression.
I tried with the standard approach of Lagrange multipliers,
$L = \vec{a}^TM\vec{a}+\vec{a}^T\vec{b}-\sum_i \lambda_i (\vec{a}^TZ_i\vec{a})+\lambda (\alpha-\vec{a}^T\vec{c}) $
Taking the gradient of this and looking for stationary points
$2M\vec{a}-2\sum_i \lambda_i Z_i\vec{a} -\lambda \vec{c} -\vec{b}= 0$
Then you have also the constraints. Now, the presence of the $\sum_i \lambda_i Z_i\vec{a}$ makes quite difficult to me to solve the problem, and I am not sure if there is a trickier way.
Also, looking for the Hessian of the problem $M-\sum_i \lambda_i Z_i$. So I guess I should look at all the configuration that make this positive definite? Is this a doable thing analytically speaking?
I am not looking for a solution, although I would appreciate some hint or some resources that I could look into them.
Thanks!