Perfect example is the simplex algorithm.
We do have a polynomial algorithm for linear programming yet simplex (with exponential worst case performance) is our tool of choice.
Mostly as a subproblem solver for mixed-integer programs, due to ease of warm-start. MIP itself is NP (B&B), so NP*NP=NP
Mostly as a subproblem solver for mixed-integer programs, due to ease of warm-start. MIP itself is NP (B&B), so NP*NP=NP