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.