OneMax in Black-Box Models with Several Restrictions

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 of a problem. Our testbed are so-called OneMax functions which require to minimize the Hamming distance to an unknown target string z∈{0,1}n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z \in \{0,1\}^n$$\end{document}. We analyze in particular the combined memory-restricted ranking-based black-box complexity of OneMax for different memory sizes. While its isolated (1+1) memory-restricted as well as its ranking-based black-box complexity for bit strings of length n is only of order n/logn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n/\log n$$\end{document}, the combined model does not allow for algorithms being faster than linear in n, as can easily be seen by standard information-theoretic considerations. Our main result is a matching upper bound. That is, we show that the (1+1) memory-restricted ranking-based black-box complexity of OneMax is linear. We also analyze its black-box complexity for memory sizes other than (1+1). Moreover, we show that these results apply to the (Monte Carlo) complexity of OneMax in the recently introduced elitist model [Doerr/Lengler GECCO 2015] that combines the above-mentioned memory-restrictions and ranking-based decisions with an enforced usage of truncation selection. Finally, we also provide improved lower bounds for the complexity of OneMax in the regarded models.