What you describe is a special case of Large Neighborhood Search.
Large Neighborhood search is an optimization method that consists in destroying a part of a solution and recreating it with a constructive approach. The constructive approach can be a greedy algorithm, a constraint programming model... or a MILP model.
When the constructive part is a MILP, it is often called "Kernel search", and the destruction part takes the form of unfixing some variables (possibly restraining their domain). Here is an example for an Inventory Routing Problem:
- Archetti C, Guastaroba G, Huerta‐Muñoz DL, Speranza MG (2021) A kernel search heuristic for the multivehicle inventory routing problem. International Transactions in Operational Research n/a: https://doi.org/10.1111/itor.12945
In this case, the neighborhood considered is larger than the neighborhood of a classical Local Search algorithm. Indeed, to explore a 2-opt neighborhood, it's faster to do a manual loop than to solve a MILP model.
This approach is attractive for problems with continuous variables (such as Inventory Routing Problems) since it's not straightforward to design a classical Local Search algorithm in these cases