Linear, convex and integer programming, and the solvers that plan supply chains, schedules and networks.
Optimising a linear function over a polyhedron: solved fast in practice by simplex and interior-point methods, with polynomial-time guarantees and one famous gap, strong polynomiality.
Open-source LP and mixed-integer solvers (HiGHS, SCIP), the MIPLIB instance library with its list of open instances, and VIPR certificates that check optimality claims in exact arithmetic.