Journal of Mathematical Modelling and Algorithms 3: 313–322, 2004. © 2004 Kluwer Academic Publishers. Printed in the Netherlands. 313 A No-Free-Lunch Theorem for Non-Uniform Distributions of Target Functions CHRISTIAN IGEL and MARC TOUSSAINT Chair of Theoretical Biology, Institut für Neuroinformatik, Ruhr-Universität Bochum, 44780 Bochum, Germany. e-mail: {christian.igel,marc.toussaint}@neuroinformatik.rub.de Abstract. The sharpened No-Free-Lunch-theorem (NFL-theorem) states that, regardless of the per- formance measure, the performance of all optimization algorithms averaged uniformly over any finite set F of functions is equal if and only if F is closed under permutation (c.u.p.). In this paper, we first summarize some consequences of this theorem, which have been proven recently: The number of subsets c.u.p. can be neglected compared to the total number of possible subsets. In particular, problem classes relevant in practice are not likely to be c.u.p. The average number of evaluations needed to find a desirable (e.g., optimal) solution can be calculated independent of the optimization algorithm in certain scenarios. Second, as the main result, the NFL-theorem is extended. Neces- sary and sufficient conditions for NFL-results to hold are given for arbitrary distributions of target functions. This yields the most general NFL-theorem for optimization presented so far. Mathematics Subject Classifications (2000): 90C27, 68T20. Key words: evolutionary computation, No-Free-Lunch theorem. 1. Introduction Search heuristics such as evolutionary algorithms, grid search, simulated anneal- ing, and tabu search are general in the sense that they can be applied to any target function f : X → Y, where X denotes the search space and Y is a set of totally ordered cost-values. Much research is spent on developing search heuristics that are superior to others when the target functions belong to a certain class of problems. But under which conditions can one search method be better than another? The No- Free-Lunch-theorem for optimization (NFL-theorem) roughly speaking states that all non-repeating search algorithms have the same mean performance when aver- aged uniformly over all possible objective functions f : X → Y [14, 9, 2, 15, 8, 1]. Of course, in practice an algorithm need not perform well on all possible functions, but only on a subset that arises from the real-world problems at hand. Recently, a sharpened version of the NFL-theorem has been proven that states that NFL-results hold (i.e., the mean performance of all search algorithms is equal) for any subset F of the set of all possible functions if and only if F is closed under permutation (c.u.p.) and each target function in F is equally likely [11].