# Niching-Driven Divide-and-Conquer Hill Exploration
## 4 Experiments
### Effect of NDDCHE in Complex Cases  
Although there exists an exponential difference in the size of the BoA between the optima in the problem used in the above experiments, some other factors may also increase the difficulty of the problem. As shown in Figure 5, the distribution of the optima shows a fractal structure in the previous test problem, which is chaotic in many cases. Additionally, the fitness landscape in the BoA of each optimum is symmetric and isotropic around the optimum in the test problem, which might be asymmetric and ill-conditioned in practical cases. To investigate the effect of NDDCHE in those complex cases, we use the free peaks [27] to generate eight test problems F1 to F8. Their fitness landscapes are shown in Figure 12. In these problems, the distribution of optima is randomly determined, and some random transformations are used to make the fitness landscape in each BoA ill-conditioned and  
Figure 11: Variation of the average error of 100 runs of different niching methods with NDDCHE in different modes for each optimum of the test function with the evolution. (**a–d**) Niching methods with NDDCHE in omniscient mode. (**e–h**) Niching methods with NDDCHE in half-heuristic mode. (**i–l**) Niching methods with NDDCHE in heuristic mode. (**a,e,i**) kNPSO. (**b,f,j**) NSDE. (**c,g,k**) NEA2. (**d,h,l**) HillValIEA. asymmetric. The exponential difference in size of the BoA between optima is kept, which ensures that there is a significant difference in attractiveness between optima. For each of the five selected niching methods, after 100 independent runs on the above eight test problems, the average improvement in error for each optimum made by NDDCHE is shown in Figure 13. The improvement in error is calculated as the logarithmic error of a niching method integrated with NDDCHE minus the logarithmic error of the method without NDDCHE. Each heat map corresponds to a specified niching method. In the heat map, each column refers to the improvement in error on a specified problem, and each row refers to the difference in error to a specified optimum. The population size of LRDE, kNPSO, NSDE, NEA2 is set to 20. The number of maximum evaluations of LRDE, kNPSO, NSDE, NEA2, and HillVallEA is set to \(1\times 10^{5}\), \(2\times 10^{6}\), \(4\times 10^{6}\), \(1\times 10^{5}\), and \(1\times 10^{5}\), respectively. The radius of repulsion of LRDE is set to 0.35. According to the results, as a lower logarithm of error indicates a smaller error, NDDCHE significantly decreases the errors for optima in most cases. In general, the heat maps of the latter four methods exhibit a distinct pattern: errors for optima with smaller indices have lesser decrements. This is because optima with smaller indices possess larger BoAs, making them more attractive and thus easier to locate even without using NDDCHE. Conversely, for optima with larger indices and consequently smaller BoAs, the application of NDDCHE leads to substantial decrements in the average error to these optima, indicating a significant enhancement in the probability of locating them. Additionally, The results also demonstrate that the improvement achieved by NDDCHE depends on the niching method integrated with it. There is a clear similarity between the heat maps for kNPSO and NSDE, which indicate the similarities between their operators  
Figure 12: Contour lines of fitness of the eight test problems with seven optima. The locations of the optima are denoted by white crosses. (**a–h**): F1 to F8. Figure 13: Heat maps of the average improvement in error for each optimum made by NDDCHE on different niching methods during 100 independent runs on different test problems. (**a**) LRDE. (**b**) kNPSO. (**c**) NSDE. (**d**) NEA2. (**e**) HillVallEA. for local search and their niching techniques. The heat maps for NEA2 and HillVallEA also exhibit remarkably similar patterns, which is primarily due to the similarity between their core algorithms.