Engineering
Lagrangean Relaxation for Capacity-Constrained Facility Location
Quick fact
With Lagrangean relaxation, you can often solve a capacitated facility location problem with 100 facilities and 1000 customers in under a second — something that would take minutes or hours with a generic integer-programming solver.
Why this is interesting
You need to decide where to build warehouses so that you can serve all customers cheaply — but each warehouse can only handle so much demand. How do you solve that almost instantly, even for hundreds of cities?