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Many papers formulate their (NP-hard) problem as ILP or MILP and then show Cplex/Gurobi are unable to solve instances of a certain size. Then they introduce a meta-heuristic solution approach to solve these large instances.

My question is: have there been any attempts to build a meta-heuristic solver that can be directly applied to any MIP-formulation? If not, why not?

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I believe there are long answers to this question, but a very short one is: yes. Here is an example that originated not from the use case you describe, but I believe that they can directly apply their local search to a MIP: LocalSolver.

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