Interior point methods can benefit greatly from a hot restart too. You simply initialize it with the previous optimal point, and you start from there. For inte

Interior point methods can benefit greatly from a hot restart too. You simply initialize it with the previous optimal point, and you start from there. For integer linear programming, sure, you should probably use simplex for solving the relaxation. I'm talking about continuous problems in engineering, statistics, machine learning, etcetera. Very few of those problems are linear, and you don't care at all about the corner case you described. That is a completely different world than combinatorial optimization. In continuous optimization if you have two or more inequality constraints that make the space lower dimensional that simply means that your problem statement is wrong, because the parameters are not exact, and with a small variation in the parameters your space will become empty.