# Niching-Driven Divide-and-Conquer Hill Exploration
## 4 Experiments
### Effect of NDDCHE in Omniscient Mode  
To prove the generalization ability of the framework, NDDCHE is integrated with representative niching methods with each category of niching technique. The selected niching methods are as follows. * For the _radial repulsion_ technique, two state-of-the-art niching methods with this technique, which are RS-CMSA [6] and PMODE [7], are selected. * For the _nearest neighbors_ technique, in most state-of-the-art niching methods with this technique, it is combined with other techniques. So \(k\)-nearest PSO (kNPSO) [1], which is a recently proposed PSO that only uses the _nearest neighbors_ technique to achieve a great performance, is selected. * For the _specified number of groups_ technique, NSDE [12] is selected as it is a representative niching method that achieves niching only through the _specified number of groups_ technique. * For the _adaptive number of groups_ technique, NEA2 [15], which is a representative niching method with this technique and still competitive, is selected. * For the _valley detection_ technique, a state-of-the-art niching method with this technique, namely HillVallEA [25], is selected. Firstly, 100 independent runs of RS-CMSA and PMODE on the test function are conducted. The random seeds for the 100 runs are set to 1/101,..., 100/101, respectively. The maximum number of evaluations is set to 50,000. The population size in PMODE is set to 20. The average error for each optimum is shown in Figure 6a. The error to an optimum is defined as the objective distance between the optimum and the best solution found within its BoA. The clear boundaries of the BoAs facilitate an accurate differentiation of the errors to different optima. In this heat map, each column corresponds to a niching method. The color of grid in a specific column denotes the average value of the logarithmic error to the corresponding optimum. According to Equation (7), the smaller the index of an optimum, the smaller the size of its BoA. So the results clearly indicate that, for both RS-CMSA and PMODE, the smaller the size of a BoA, the lower the probability of locating the optimum in it. Figure 5: Contour lines of fitness of an instance problem with seven optima. The locations of the optima are denoted by white crosses. As mentioned in the related works, setting the threshold distance for _radial repulsion_ between a more appropriate range can improve the performance. However, the threshold distance is adaptively adjusted in both RS-CMSA and PMODE, which affords no opportunity for us to attempt. To this end, a recently proposed DE with the _radial repulsion_ technique, namely local repulsion DE (LRDE) [1], is adopted. In LRDE, by setting the radius of repulsion to a value in the range of [0:1], the threshold distance for _radial repulsion_ becomes the product of the radius of repulsion and \(V^{1/D}\), where \(V\) and \(D\) denote the volume and dimension of the search space, respectively. After 100 independent runs of LRDE with the radius of repulsion set to different values, where the maximum number of evaluations is set to 50,000 and the population size is set to 20, the error for each optimum is shown in Figure 6b. Each column shows the result of LRDE with a specific value of the radius of repulsion. The results prove that, by setting the threshold distance for _radial repulsion_ within a more suitable range, which is around \(0.35\cdot V^{1/D}\) in the experiment, a better performance can be achieved. However, although setting the radius of repulsion to 0.35 makes LRDE competitive with the two state-of-the-art niching methods, it is still hard to further improve it by finding a more suitable value of the radius of repulsion, especially in practical cases. At this juncture, it is imperative to revisit the perspectives summarized in Section 2.