When a vacancy occurs in a single member district, it is often filled through a special/by election. In a PR system, this might break proportionality unless all the seats in a district or cluster go vacant at once. Giving some official the power to fill vacancies also may not maintain proportionality, and worse, this power is ripe for abuse (see: Rod Blagojevich).
One way of filling a seat could be to rerun the election on the original ballots with the departing representatives disqualified and the other original winners assured election (this has been done with non-list STV.) For an optimization method, this would be easy: limit the possible winner sets to those that contain the remaining reps. For methods such as RRV, where the order in which the previous candidates were elected has no effect on a given round, the remaining reps could be declared already elected, and the election would be rerun from there. For methods such as Sequential Monroe where the order of election matters, the two best options seem to be:
- Declare the order of election of the remaining reps in the rerun to be the same as that of the original election. So in the early rounds of the rerun, declare the winner to be the remaining rep who was seated earliest in the original election, (ignoring who the method actually says to seat, but using its reweight step.) Repeat until all of the remaining reps have been seated. Then use the method normally to fill the remaining seats.
- Rerun the election normally, but disqualify all candidates except the remaining reps until all remaining reps have been seated. The candidates seated after the remaining reps fill the vacancies.