OneMax in Black-Box Models with Several Restrictions

TL;DR

This paper investigates how combining multiple restrictions like memory and ranking constraints affects the black-box complexity of OneMax functions, showing that the complexity remains linear in problem size, which informs the limits of optimization algorithms.

Contribution

It provides the first analysis of combined restrictions on black-box complexity for OneMax, establishing tight linear lower bounds and extending results to elitist models.

Findings

Combined restrictions lead to linear complexity in n.

Isolated memory and ranking complexities are of order n/log n.

Results apply to Monte Carlo and elitist models.

Abstract

Black-box complexity studies lower bounds for the efficiency of general-purpose black-box optimization algorithms such as evolutionary algorithms and other search heuristics. Different models exist, each one being designed to analyze a different aspect of typical heuristics such as the memory size or the variation operators in use. While most of the previous works focus on one particular such aspect, we consider in this work how the combination of several algorithmic restrictions influence the black-box complexity. Our testbed are so-called OneMax functions, a classical set of test functions that is intimately related to classic coin-weighing problems and to the board game Mastermind. We analyze in particular the combined memory-restricted ranking-based black-box complexity of OneMax for different memory sizes. While its isolated memory-restricted as well as its ranking-based black-box complexity for bit strings of length $n$ is only of order $n/\log n$, the combined model does not allow for algorithms being faster than linear in $n$, as can be seen by standard information-theoretic considerations. We show that this linear bound is indeed asymptotically tight. Similar results are obtained for other memory- and offspring-sizes. Our results also apply to the (Monte Carlo) complexity of OneMax in the recently introduced elitist model, in which only the best-so-far solution can be kept in the memory. Finally, we also provide improved lower bounds for the complexity of OneMax in the regarded models. Our result enlivens the quest for natural evolutionary algorithms optimizing OneMax in $o(n \log n)$ iterations.