z3 returns the number of unsat clauses for a Max-SMT problem together with only one model. I wonder if I can list them all (e.g. using API).
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As mentioned in this question, you can use the generated model to add a constraint to prevent the generated model and run the solver again. In your case however, I believe you will also need to check the number of constraints that are broken, after generating a new model, to ensure that you don't break your optimality constraints.