On the Runtime Analysis of the Clearing Diversity-Preserving Mechanism

Abstract

Clearing is a niching method inspired by the principle of assigning the available resources among a niche to a single individual. The clearing procedure supplies these resources only to the best individual of each niche: the winner. So far, its analysis has been focused on experimental approaches that have shown that clearing is a powerful diversity-preserving mechanism. Using rigorous runtime analysis to explain how and why it is a powerful method, we prove that a mutation-based evolutionary algorithm with a large enough population size, and a phenotypic distance function always succeeds in optimising all functions of unitation for small niches in polynomial time, while a genotypic distance function requires exponential time. Finally, we prove that with phenotypic and genotypic distances, clearing is able to find both optima for [Formula: see text] and several general classes of bimodal functions in polynomial expected time. We use empirical analysis to highlight some of the characteristics that makes it a useful mechanism and to support the theoretical results.

Parallel Black-Box Complexity With Tail Bounds

“…Well-known examples include the function TwoMax := max{ [32], which has been used as a challenging test bed in theoretical studies of diversity-preserving mechanisms [6,7,49]. The function H-Iff (Hierarchical If and only If) [58] consists of hierarchical building blocks that need to attain equal values in order to contribute to the fitness.…”

“…Hence the number of peaks is an upper bound on the number of global (and local) optima. The two function classes were named Jansen-Zarges function classes in Covantes Osuna and Sudholt [6], where they were used as benchmarks for the clearing diversity mechanism.…”

Empirical Analysis of Diversity-Preserving Mechanisms on Example Landscapes for Multimodal Optimisation

“…The peak i yielding the largest value In the case of two peaks p 1 and p 2 , if these peaks are complementary, that is, p 2 = p 1 , then JZ 1 and JZ 2 generalise the TwoMax function, with TwoMax being the special case of p 1 = 0 n , p 2 = 1 n , a 1 = a 2 = 1 and b 1 = b 2 = 0 [9]. This setting was studied for the (µ+1) EA with clearing in [3].…”

“…The goal is to observe which mechanisms are able to escape from local optima by tunnelling through the fitness valley that separates two peaks. We choose µ = 32 as in [3,Sect. 7.3] and also consider the same two initialisations: the standard uniform random initialisation and biased initialisation where the whole population is initialised with copies of one peak (0 n for TwoMax).…”

“…We would like to investigate how the diversity mechanisms behave if peaks have different Hamming distances. Following [3], we focus on two peaks and vary their Hamming distance between 1 and n by choosing p 1 = 0 n and p 2 ∈ {0 n−1 1, 0 n−2 1 2 , . .…”