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In their GECCO'12 paper, Doerr and Doerr proved that the k-ary unbiased black-box complexity of OneMax on n bits is O(n/k) for 2 ≤ k ≤ O(log n). We propose an alternative strategy for achieving this unbiased black-box complexity when 3 ≤ k ≤ log 2 n. While it is based on the same idea of block-wise optimization, it uses k-ary unbiased operators in a different way.For each block of size 2 k−1 − 1 we set up, in O(k) queries, a virtual coordinate system, which enables us to use an arbitrary unrestricted algorithm to optimize this block. This is possible because this coordinate system introduces a bijection between unrestricted queries and a subset of k-ary unbiased operators. We note that this technique does not depend on OneMax being solved and can be used in more general contexts.This together constitutes an algorithm which is conceptually simpler than the one by Doerr and Doerr, and at the same time achieves better constant factors in the asymptotic notation. Our algorithm works in (2 + o(1)) · n/(k − 1), where o(1) relates to k. Our experimental evaluation of this algorithm shows its efficiency already for 3 ≤ k ≤ 6. arXiv:1804.05443v2 [cs.NE] 10 Jul 2018 heuristics. Often, complexities of rather simple problems are studied, such as the famous OneMax problem, defined on bit strings of length n as follows:The notion of unbiased black-box complexity was introduced in [16] for pseudo-Boolean problems (see also the journal version [17]) partially as a response to unrealistically low black-box complexities of various NP-hard problems [13]. Since evolutionary algorithms and other randomized search heuristics are designed as general-purpose solvers, they shall not prefer one instance of a problem over another one. This means that unbiased black-box algorithms are a better model of randomized search heuristics, since the unbiased model does not allow certain ways of being "too fast". Unfortunately, the ways were found to perform most of the work without making queries in the unbiased model too [5,8]. In fact, it was shown that, with a proper notion of unbiasedness for the given type of individuals, the unbiased black-box complexity coincides to the unrestricted one [18]. Several alternative restricted models of black-box algorithms were subsequently introduced as a reaction, namely ranking-based algorithms [11], limited-memory algorithms [10], and elitist algorithms [12].One of possible restrictions to the unbiased black-box search model is the use of unbiased operators with restricted arity. The original paper [16] studied mostly unary unbiased black-box complexity, e.g. the class of algorithms allowing only unbiased operators taking one individual and producing another one, or mutationbased algorithms. This model appeared to be quite restrictive, e.g. the unary unbiased black-box complexity of OneMax was proven to be Θ(n log n) [6,16]. Together with the rather old question, whether crossover is useful in evolutionary algorithms (which was previously positively, but only in artificial settings [15]), this inspired a nu...

The binary value function, or BinVal, has appeared in several studies in theory of evolutionary computation as one of the extreme examples of linear pseudo-Boolean functions. Its unbiased black-box complexity was previously shown to be at most ⌈log 2 n⌉ + 2, where n is the problem size.We augment it with an upper bound of log 2 n + 2.42141558 − o(1), which is more precise for many values of n. We also present a lower bound of log 2 n+1.1186406−o(1). Additionally, we prove that BinVal is an easiest function among all unimodal pseudo-Boolean functions at least for unbiased algorithms.

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