TY - JOUR A1 - Lengler, Johannes A1 - Opris, Andre A1 - Sudholt, Dirk T1 - Analysing Equilibrium States for Population Diversity JF - Algorithmica (ISSN: 1432-0541) N2 - Population diversity is crucial in evolutionary algorithms as it helps with global exploration and facilitates the use of crossover. Despite many runtime analyses showing advantages of population diversity, we have no clear picture of how diversity evolves over time. We study how the population diversity of (μ+1)algorithms, measured by the sum of pairwise Hamming distances, evolves in a fitness-neutral environment. We give an exact formula for the drift of population diversity and show that it is driven towards an equilibrium state. Moreover, we bound the expected time for getting close to the equilibrium state. We find that these dynamics, including the location of the equilibrium, are unaffected by surprisingly many algorithmic choices. All unbiased mutation operators with the same expected number of bit flips have the same effect on the expected diversity. Many crossover operators have no effect at all, including all binary unbiased, respectful operators. We review crossover operators from the literature and identify crossovers that are neutral towards the evolution of diversity and crossovers that are not. KW - Evolutionary algorithms KW - Runtime analysis KW - Diversity KW - Population dynamics Y1 - 2024 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:101:1-2406282100292.409184699704 SN - 0178-4617 SN - 1432-0541 VL - 86 IS - 7 SP - 2317 EP - 2351 PB - Springer US CY - New York ER - TY - JOUR A1 - Dang, Duc-Cuong A1 - Opris, Andre A1 - Sudholt, Dirk T1 - Crossover can guarantee exponential speed-ups in evolutionary multi-objective optimisation JF - Artificial Intelligence (Online ISSN: 1872-7921) N2 - Evolutionary algorithms are popular algorithms for multi-objective optimisation (also called Pareto optimisation) as they use a population to store trade-offs between different objectives. Despite their popularity, the theoretical foundation of multi-objective evolutionary optimisation (EMO) is still in its early development. Fundamental questions such as the benefits of the crossover operator are still not fully understood. We provide a theoretical analysis of the well-known EMO algorithms GSEMO and NSGA-II to showcase the possible advantages of crossover: we propose classes of “royal road” functions on which these algorithms cover the whole Pareto front in expected polynomial time if crossover is being used. But when disabling crossover, they require exponential time in expectation to cover the Pareto front. The latter even holds for a large class of black-box algorithms using any elitist selection and any unbiased mutation operator. Moreover, even the expected time to create a single Pareto-optimal search point is exponential. We provide two different function classes, one tailored for one-point crossover and another one tailored for uniform crossover, and we show that some immune-inspired hypermutations cannot avoid exponential optimisation times. Our work shows the first example of an exponential performance gap through the use of crossover for the widely used NSGA-II algorithm and contributes to a deeper understanding of its limitations and capabilities. KW - Evolutionary computation KW - Runtime analysis KW - Recombination KW - Multi-objective optimisation KW - Unbiased black-box algorithms KW - Hypermutation Y1 - 2024 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:739-opus4-18732 VL - 2025 IS - 330 PB - Elsevier CY - Amsterdam ER -