Optimization, block designs and no free lunch theorems

Authors:
Evan J. Griffiths;Pekka Orponen
Affiliations:
Laboratory for Theoretical Computer Science, Helsinki University of Technology, Finland;Laboratory for Theoretical Computer Science, Helsinki University of Technology, Finland
Venue:
Information Processing Letters
Year:
2005

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Abstract

We study the precise conditions under which all optimization strategies for a given family of finite functions yield the same expected maximization 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 n- variate Boolean functions that satisfies the "No Free Lunch" condition with respect to searches of length Ω(n1/2/log1/2 n). Modifying the construction, we also obtain nontrivial "No Free Lunch" families of functions with large ranges.