Does anybody know why in the outer approximation approach for MINLP it is not necessarily/needed to solve MILP to optimality? What is the rationale or explanation behind it?
The outer approximation MILP is a relaxation of the MINLP. Any lower bound on the MILP outer approximation is also a lower bound on the MINLP. Therefore a lower bound on the outer approximation MILP can be used as a lower bound for the MINLP, without requiring it be solved to optimality.