Focused no free lunch theorems

Proofs and empirical evidence are presented which show that a subset of algorithms can have identical performance over a subset of functions, even when the subset of functions is not closed under permutation. We refer to these as focused sets. In some cases focused sets correspond to the orbit of a permutation group; in other cases, the focused sets must be computed heuristically. In the smallest case, two algorithms can have identical performance over just two functions in a focused set. These results particularly exploit the case where search is limited to m steps, where m is significantly smaller than the size of the search space.