Why Benchmarking is an (Asymptotically) Optimal Approach to Numerical Methods: A Geombinatoric Proof

In numerical mathematics, one of the most frequently used ways of gauging the quality of different numerical methods is benchmarking. Specifically, once we have methods that work well on some (but not all) problems from a given problem class, we find the problem that is the toughest for the existing methods. This problem becomes a benchmark for gauging how well different methods solve problems that previous methods could not. Once we have a method that works well in solving this benchmark problem, we repeat the process again -- by selecting, as a new benchmark, a problem that is the toughest to solve by the new methods, and by looking for a new method that works the best on this new benchmark. At first glance, this idea sounds like a heuristic, but its success in numerical mathematics indicates that this heuristic is either optimal or at least close to optimality. In this paper, we use the geombinatoric approach to prove that benchmarking is indeed asymptotically optimal