# Niching-Driven Divide-and-Conquer Hill Exploration
## 3 Methodology
Individuals in the population are denoted by solid gray circles. Once the population is stagnant, candidate solutions, which are denoted by pentagrams, are selected from the population. Seed solutions, which are denoted by solid colorful circles, are updated according to the candidate solutions. After that, in any hill region that contains more than one seed solution, such as the blue hill region in our example, random solutions will be sampled, which are denoted by black points, and then grouped. Solutions belonging to the group of each seed solution are denoted by the points of the same color as the seed solution. After clustering, the hill region is divided into multiple hill regions, any of which contains only one group of solutions. To meet the requirements of an ablation study, two modes have been designed for NDDCHE: the omniscient mode and the heuristic mode. The remaining details of each mode of NDDCHE including how to identify candidate solutions from the population, how to check whether two solutions are in the same BoA, and how to divide the hill region to make one hill region only contain one group of solutions, which will be provided in the corresponding subsection. The major difference between the two modes is that, in omniscient mode, the optima and BoAs are given, which make each step of NDDCHE obtain the most ideal result. So after integrating the NDDCHE in omniscient mode with a niching method, the potential improvement of NDDCHE on this niching method can be investigated thoroughly. In heuristic mode, no prior knowledge is given. So, after integrating the NDDCHE in heuristic mode with a niching method, the practical improvement of NDDCHE on this niching method can be demonstrated. ### Omniscient Mode

In omniscient mode, the method for identifying candidate solutions from the population is simple. As shown in Algorithm 4, since the optima and BoAs are given, individuals in each BoA can be identified, then the nearest individuals to the optimum can be found

Figure 2: An example cycle of NDDCHE. Different hill regions are denoted by rectangles of different colors. Individuals in the population are denoted by solid gray circles. Candidate solutions are denoted by pentagrams. Seed solutions are denoted by solid colorful circles. Random solutions are denoted by black points. Solutions belonging to the group of each seed solution are denoted by points of the same color as the seed solution. and added into the set of candidate solutions. To prevent an individual not participating in the evolution from being added to the candidates, the objective distance between the individual and the optimum is calculated. If the distance is greater than \(\epsilon\), which is set to \(10^{-2}\) in this paper, the individual is discarded. ```
Input: population \(\mathbf{P}\), optima \(\mathbf{O}\), BoAs \(\mathbb{B}\) Output: candidate solutions \(\mathbf{C}\)
1 Initialize \(\mathbf{C}\leftarrow\emptyset\);
2 foreach\(\mathbf{o}\in\mathbf{O}\)do
3\(\mathbf{B}\leftarrow\mathbf{B}^{\prime}\in\mathbb{B}\ |\ \mathbf{o}\in\mathbf{B}^{\prime}\);
4 if\(\mathbf{P}\cap\mathbf{B}\
eq\emptyset\)then
5\(\mathbf{x}\leftarrow\underset{\mathbf{x}^{\prime}\in\mathbf{P}}{\arg\min}\|\mathbf{x}-\mathbf{o}\|\);
6 if\(|f(\mathbf{x})-f(\mathbf{o})|\leq\epsilon\)then
7\(\mathbf{C}\leftarrow\mathbf{C}\cup\{\mathbf{x}\}\);
8
9 end if
10
11 end for
```
**Algorithm 4**Identify Candidates (\(\mathbf{P},\mathbf{O},\mathbb{B}\)). The method for checking whether two solutions are in the same BoA in omniscient mode is obvious since the BoAs are given. As for the method for dividing a hill region by the groups of solutions, the process is shown in Algorithm 5. The set of BoAs covered by the hill region to divide \(\mathbf{H}\) is denoted by \(\mathbb{B}^{\prime}\). Each group of individuals corresponds to a new hill region. Then, for each BoA in \(\mathbb{B}^{\prime}\), it is assigned to the new hill region whose corresponding group of individuals occupies the most in this BoA.