Qualitatively, in my experience in the solving of large scale MILPs, it is common that binary variables corresponding to "edge possibility" components are frequently chosen. Intuitively, these seem associated with various "critical values" within the problem domain.
This seems to resonate with the approach of Lagrange Multipliers which identifies such critical values. I have therefore wondered whether a Lagrangian-Multiplier analysis could add significant value in the solving of MILPs. A quick online search seems to confirm that Lagrange Multipliers do indeed have a role to play.
Does anyone have practical hands-on experience about the usefulness of Lagrange-Multipliers in the solving of real-world large-scale MILP problems? If so, what techniques have you found to be the most useful?