Black-Box Complexity of the Binary Value Function

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 $\lceil \log_2 n \rceil + 2$, 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.Comment: 24 pages, one figure. An extended two-page abstract of this work will appear in proceedings of the Genetic and Evolutionary Computation Conference, GECCO'1

A simple two-module problem to exemplify building-block assembly under crossover

Theoretically and empirically it is clear that a genetic algorithm with crossover will outperform a genetic algorithm without crossover in some fitness landscapes, and vice versa in other landscapes. Despite an extensive literature on the subject, and recent proofs of a principled distinction in the abilities of crossover and non-crossover algorithms for a particular theoretical landscape, building general intuitions about when and why crossover performs well when it does is a different matter. In particular, the proposal that crossover might enable the assembly of good building-blocks has been difficult to verify despite many attempts at idealized building-block landscapes. Here we show the first example of a two-module problem that shows a principled advantage for cross-over. This allows us to understand building-block assembly under crossover quite straightforwardly and build intuition about more general landscape classes favoring crossover or disfavoring it

Estimating the Number of Stable Configurations for the Generalized Thomson Problem

Given a natural number N, one may ask what configuration of N points on the two-sphere minimizes the discrete generalized Coulomb energy. If one applies a gradient-based numerical optimization to this problem, one encounters many configurations that are stable but not globally minimal. This led the authors of this manuscript to the question, how many stable configurations are there? In this manuscript we report methods for identifying and counting observed stable configurations, and estimating the actual number of stable configurations. These estimates indicate that for N approaching two hundred, there are at least tens of thousands of stable configurations.Comment: The final publication is available at Springer via http://dx.doi.org/10.1007/s10955-015-1245-