Optimization, block designs and No Free Lunch theorems

We study the precise conditions under which all optimisation strategies for a given family of finite functions yield the same expected maximisation performance, when averaged over a uniform distribution of the functions. In the case of bounded-length searches in a family of Boolean functions, we provide tight connections between such “No Free Lunch” conditions and the structure of t-designs and t-wise balanced designs for arbitrary values t. As a corollary, we obtain a nontrivial family of nvariate Boolean functions that satisfies the “No Free Lunch“ condition with respect to searches of length Ω(n1/2/ log n). Modifying the construction, we also obtain nontrivial “No Free Lunch“ families of functions with large ranges.