For the version without the absolute value, a colleague and I published a few months ago an algorithm to solve that optimization problem and other related ones. I believe this is the fastest algorithm available as of today. It is not too simple as it relies on optimization on manifolds, however this is not a simple problem in general; the constraint on the eigenvalues is tricky to handle numerically. Some competitors are mentioned in the introduction and in the section with numerical experiments.

We have also published on https://github.com/fph/nearest-omega-stable some proof-of-concept Matlab code that you can use.

For the version without the absolute value, a colleague and I published a few months ago an algorithm to solve that optimization problem and other related ones (the problem you need is called "Schur stable" case in the paper). Some competitors are mentioned in the introduction and in the section with numerical experiments, but I believe ours is the fastest algorithm available as of today. It is not too simple as it relies on optimization on manifolds, however this is not a simple problem in general: the constraint on the eigenvalues is non-smooth and tricky to handle numerically.

We have also published on https://github.com/fph/nearest-omega-stable some proof-of-concept Matlab code that you can use.