# Black-Box Complexity of the Binary Value Function

## TL;DR

This paper refines the understanding of the black-box complexity of the BinVal function, providing tighter bounds and establishing its position among unimodal functions in evolutionary computation.

## Contribution

It offers more precise upper and lower bounds for the unbiased black-box complexity of BinVal and proves its status as the easiest unimodal pseudo-Boolean function for unbiased algorithms.

## Key findings

- Upper bound of log_2 n + 2.42141558 - o(1) for BinVal complexity
- Lower bound of log_2 n + 1.1186406 - o(1) for BinVal complexity
- BinVal is the easiest among unimodal pseudo-Boolean functions for unbiased algorithms

## Abstract

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.