# Niching-Driven Divide-and-Conquer Hill Exploration
## 5 Conclusions  
This paper reviews and analyzes representative niching techniques used in existing methods. Although both of them can enhance their abilities in terms of locating less-attractive optima by tuning their key parameters, it is hard for them to balance search resources over optima with significant differences in attractiveness, which limits their potentials as the difference further increases.  
To address the above issue, this paper proposed a hierarchical divide-and-conquer framework for niching methods, which is named NDDCHE. From the search result of a niching method, the framework identifies seed solutions, any two of which do not share the  
Figure 14: (**a**) Heat map of the average improvement in error for each optimum made by NDDCHE on LRDE with different radii of repulsion during 100 independent runs on the 10-D test problem. (**b–e**) Variation of the average improvement in error for each optimum made by NDDCHE on different niching methods with the evolution during 100 independent runs on the 10-D test problem. (**b**) kNPSO. (**c**) NSDE. (**d**) NEA2. (**e**) HillVallEA.  
same BoA, and gradually divide the search space into smaller hill regions, each of which corresponds to the estimation of the BoA of an optimum that has already been detected. After that, the niching method is reinitialized in one of the hill regions, which contributes to detecting more seed solutions.  
The framework is integrated with five representative niching methods with different categories of niching techniques. The experiments on test problems with exponential differences in the size of BoA between the optima proved that NDDCHE enables each enriching method to achieve a much more balanced distribution of search resources among the optima and thereby improves the ability to find optima with less attractiveness without tuning any parameters.  
Moreover, although NDDCHE was compared with parameter tuning in some experiments, it does not conflict with the latter. Specifically, if desired solutions cannot be found by a niching method, NDDCHE can be initially employed to fully exploit the potential of the niching method under the current parameter settings. When no additional seed solutions can be identified within any hill region, the parameter should then be tuned. If a new seed solution is found after tuning the parameter, NDDCHE can then continue to fulfill its role.  
It is worth noting that, in the current framework, no priority is assigned to different hill regions, as the population is re-initialized in a randomly selected hill region. Clearly, the number of BoAs requiring further subdivision varies across different hill regions, and the fitness values of different local optima also differ. Therefore, if hill regions containing more BoAs or better local optima could be prioritized in the search, the search efficiency could be further improved. Predicting the differences in search potential among various hill regions will be the focus of our next steps.  
Conceptualization, [PERSON] and C.L.; methodology, [PERSON]; software, J.W., Y.D. and C.L.; validation, [PERSON]; formal analysis, [PERSON]; investigation, J.W.; resources, J.W.; data curation, J.W.; writing--original draft preparation, [PERSON]; writing--review and editing, J.W. and C.L.; visualization, [PERSON]; supervision, C.L.; project administration, [PERSON]; funding acquisition, C.L. All authors have read and agreed to the published version of the manuscript.  
This work was supported in part by the National Natural Science Foundation of China under Grant 62476006, in part by the Hubei Provincial Natural Science Foundation of China under Grant 2023 AFA049, and in part by the Fundamental Research Funds of the AUST under Grant 2024 JBZD0007.  
Not applicable.  
The raw data supporting the conclusions of this article will be made available by the authors on request.  
The authors declare no conflicts of interest.