Client is in a situation where when a process restarts or is shut down and comes back up again it is detected and reported as a new process. Therefore all problems/alerts on the restart/shutdown stay open forever until manually closed.
As far as I know there is no way to stop those alerts from firing.
But, is there a way to programatically (maybe via API or REST call?) close all open problems of this type?
You should customize your process group configuration, so that processes are identified as the same process group after each start. That's the idea of process group. See https://www.dynatrace.com/support/help/infrastructure/processes/can-i-customize-how-process-groups-a....
Closing alerts manually or automatically should be considered as an anti-pattern.