Recent Results on No-Free-Lunch Theorems for Optimization

Details Recent Results on No-Free-Lunch Theorems for Optimization Recent Results on No-Free-Lunch Theorems for Optimization

2003-03-31

arxiv.org/abs/cs/0303032v1

Computer Science

Neural and Evolutionary Computing

10 pages, LaTeX, see http://www.neuroinformatik.rub.de/PROJECTS/SONN/

Scientific paper

The sharpened No-Free-Lunch-theorem (NFL-theorem) states that the performance of all optimization algorithms averaged over any finite set F of functions is equal if and only if F is closed under permutation (c.u.p.) and each target function in F is equally likely. In this paper, we first summarize some consequences of this theorem, which have been proven recently: The average number of evaluations needed to find a desirable (e.g., optimal) solution can be calculated; the number of subsets c.u.p. can be neglected compared to the overall number of possible subsets; and problem classes relevant in practice are not likely to be c.u.p. Second, as the main result, the NFL-theorem is extended. Necessary and sufficient conditions for NFL-results to hold are given for arbitrary, non-uniform distributions of target functions. This yields the most general NFL-theorem for optimization presented so far.

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