Hi I would like to optimize the following system: $$\min_{q,\|q\|=1} \sum_i^n |q^T M_i q|$$

More details:

- the size of the unknown vector q is $4\times 1$,
- M_i is a matrix of size $4\times 4$. It is symmetric but
*not*positive definite, - $n$ is about $200$ in my case.

Anyone knows how to solve this problem?

It might be related to the following problem: $$\max_{q,\|q\|=1} q^T M q$$ The solution is simply the eigenvector of $M$ associated to the highest eigenvalue.

But in our system, the absolute function and the sum make the problem more complicated.

I also tried reformulating the system as: $$\min_{q,t} \sum_i^n t_i \text{ subject to }|q^T M_i q|\leq t_i\text{ for all }i=1,\dots,n\text{ and }\|q\|=1$$ but it did not really help.

It is not a standard quadratically constrained quadratic program (QCQP) because the matrices are not positive definite.