I have an MILP model to which I get an integer feasible solution as a result of a heuristic search. In this particular example, the initial solution turns out to be the optimal solution, which I prove through B&B after solving 3000 LPs (20 Minutes of solution time), which is frustrating.
I can use the value of the initial solution as a lower bound on the optimisation problem (maximization case).
Can I derive anything else other than a lower bound from that solution?
Many solvers have an option to control the "emphasis" (feasibility versus optimality) of the tree search. If you suspect that your initial solution is already optimal, set this option to emphasize optimality, which will make cut generation more aggressive and do some other similar things.
Another approach is to explicitly apply reduced-cost fixing to fix the values of variables, without loss of optimality. Many solvers do this automatically, but maybe not as aggressively as you could do yourself.
A natural idea that has come up many times before is to add an explicit objective cut of the form $\sum_j c_j x_j \ge \hat{z}$, where $\hat{z}$ is the objective value of your initial solution. The conventional wisdom is that doing this is usually a bad idea, in part because it often adds a dense constraint that causes numerical difficulty for the underlying LP solves. But it is easy to try.
You can also submit the solution to the MILP solver. The solver should then use the bound from it automatically. Further, it might use the solution values for various improvement heuristics (e.g. neighborhood searches).
