Two-Phase Real-Valued Multimodal Optimization with the Hill-Valley Evolutionary Algorithm

The aim of multimodal optimization (MMO) is to obtain all global optima of an optimization problem. In this chapter, we introduce a general framework for two-phase MMO evolutionary algorithms (EAs), in which different high-fitness regions (niches) are located in the first phase via clustering, and each of the located niches is separately optimized with a core search algorithm in the second phase. One such two-phase MMO EA is the Hill-Valley Evolutionary Algorithm (HillVall-EA). In HillVallEA, the remarkably simple hill-valley clustering method is used. The idea behind hill-valley clustering is that two solutions belong to the same niche (valley) when there is no hill in between them, which can be easily tested by performing additional function evaluations. We compare hill-valley clustering to two other recently introduced fitness-informed clustering methods: nearest-better clustering and hierarchical Gaussian mixture learning. We show how these clustering methods, as well as different core search algorithms, influence the resulting optimization performance of the two-phase MMO framework on the commonly used CEC 2013 niching benchmark suite. Our results show that HillVallEA, equipped with the core search algorithm Adapted Maximum-Likelihood Gaussian Model Univariate (AMu) as core search algorithm, outperforms all other MMO EAs, both within the limited benchmark budget, and in the long run. HillVallEA-AMu was the winner of the GECCO niching competition in 2018 and 2019, and is currently, to the best of our knowledge, the best performing algorithm on this benchmark suite.