Update: This bug has been fixed in the just released Mathematica 11.1.
A good workaround was suggested by ilianilian.
For a workaround, please try Method -> "InteriorPoint".
This workaround worked for me. In particular, I can now run the first FindMinimum
command for the code listing in my question as many times as I want, and the second FindMinimum
command does not cause a crash either. Even better, the solution found with this option seems to be more accurate. I also did not notice a slow down when I applied this workaround to the original application.