A topological niching covariance matrix adaptation for multimodal optimization

Multimodal optimization attempts to find multiple global and local optima of a function. Finding a set of optimal solutions is particularly important for practical problems. However, this kind of problem requires optimization techniques that demand a high computational cost and a large amount of parameters to be adjusted. These difficulties increase in high-dimensional space problems. In this work, we propose a niching method based on recent developments in the basins (optimal locations) identification to reduce costs and perform better in high-dimensional spaces. Using Nearest-Better Clustering (NBC) and Hill-Valley (or Detect Multimodal) methods, an exploratory initialization routine is employed to identify basins on functions with different levels of complexity. To maintain diversity over the generations, we define a bi-objective function, which is composed by the original fitness function and the distance to the nearest better neighbor, assisted by a reinitialization scheme. The proposed method is implemented using Evolutionary Strategy (ES) known as Covariance Matrix Adaptation (CMA). Unlike recent multimodal approaches using CMA-ES, we use its step size to control the influence of niche, thus avoiding extra efforts in parameterization. We apply a benchmark of 20 test functions, specially designed for multimodal optimization evaluation, and compare the performance with a state-of-the-art method. Finally we discuss the results and show that the proposed approach can reach better and stable results even in high-dimensional spaces.