Local Optima Networks of the Black Box Optimisation Benchmark Functions

Compressed monotonic local optima networks (CMLONs) provide a powerful means to visualise the global structure of the fitness landscapes of optimisation problems as network graphs. Historically, they have been developed for discrete optimisation problems, but they have more recently been extended to continuous problems. However, experience is limited because they have only been applied to a few analytical problems and even fewer real-world problems. This work aims to address that gap by providing a systematic and comprehensive catalogue of CMLONs for the well-known black box optimisation benchmark functions. These are a set of continuous analytical functions with diverse properties. CMLONs are calculated for each of the twenty-four benchmark functions in three, five, eight, twelve, and twenty dimensions. Network metrics are also calculated for each of the CMLONs and dimensionality reduction used to classify and compare the functions. It was found that the CMLONs have diverse representations that are related to both the functions' properties and their dimensionality. Network metrics were important for multimodal functions in higher dimensional problems, where the CMLONs were too dense to visualise meaningfully. These results provide an extensive catalogue of CMLONs for a variety of continuous functions and provide a useful reference for real-world optimisation problems.