# Niching-Driven Divide-and-Conquer Hill Exploration
## 3 Methodology
```
Input: hill \(\mathbf{H}\), solution groups \(\mathbb{G}\), BoAs \(\mathbb{B}\) Output: divided hills \(\mathbb{H}\)
1 Initialize \(\mathbb{H}\leftarrow\emptyset\);
2\(\mathbb{B}^{\prime}\leftarrow\{\mathbf{B}\in\mathbb{B}\ |\ \mathbf{B}\subset\mathbf{H}\}\);
3 foreach\(\mathbf{B}\in\mathbb{B}^{\prime}\)do Calculate \(\underset{\mathbf{G}\in\mathbb{G}}{\arg\max}|\mathbf{B}\cap\mathbf{G}|\) ;
4 foreach\(\mathbf{G}\in\mathbb{G}\)do
5\(\mathbf{H}^{\prime}\leftarrow\emptyset\);
6\(\mathbb{B}^{\prime\prime}\leftarrow\{\mathbf{B}\in\mathbb{B}^{\prime}\ |\ \underset{\mathbf{G}^{\prime}\in\mathbb{G}}{\arg\max}|\mathbf{B}\cap\mathbf{G}^{\prime}|= \mathbf{G}\}\);
7 foreach\(\mathbf{B}\in\mathbb{B}^{\prime\prime}\)do\(\mathbf{H}^{\prime}\leftarrow\mathbf{H}^{\prime}\cup\mathbf{B}\);
8\(\mathbb{H}\leftarrow\mathbb{H}\cup\{\mathbf{H}^{\prime}\}\);
9 end for
```
**Algorithm 5**Divide Hill (\(\mathbf{H},\mathbb{G},\mathbb{B}\)). ### Heuristic Mode

In heuristic mode, the specific method for identifying candidate solutions depends on some details of the niching method integrated with NDDCHE as follows:

* For niching methods with seed solutions or analogous structures, the archive solutions are directly selected as the candidate solutions;
* For niching methods without seed solutions or analogous structures,If a clustering method is adopted in the niching method, the group centers are selected as the candidate solutions;
* Otherwise, a novel method based on image segmentation model is adopted to select the candidate solutions as the process shown in Figure 3 and Algorithm 6. The idea behind the method based on the image segmentation model is that, after calculating the distribution of NBDs of individuals in the population according to Equations (1) and (3), the variation of the distribution exhibits a clear pattern with the evolution. As the example shown in Figure 3, the left upper graph shows the variation of distribution of individuals in the 2D search space with the evolution, and the left lower graph shows the corresponding variation in NBDs. For most individuals, their NBDs decreased rapidly with the growth of iteration, except for the three individuals. Two of them have their NBDs shown in the graph. The remaining individual is the best individual who does not have a nearest-better solution. For these three individuals whose NBDs are outliers, they are also the three centers that the population is converging towards. To automatically identify these outliers, the variation in the distribution of NBDs is transformed into an image \(\mathbf{I}_{\Lambda\times\left|\mathbb{P}\right|}\) according to Equations (4) and (5). \(\Lambda\) denotes the specified resolution of NBD, which is set to 50 in this paper, and \(i_{(\tau,t)}\) denotes the value of pixel in the \(r\)-th row and \(t\)-th column. Input the image through a 3-level U-Net model [26], an output image \(\mathbf{Y}_{\Lambda\times\left|\mathbb{P}\right|}\) highlighting the pixels corresponding to the outliers is generated, where \(y_{(\tau,t)}\) denotes the value of pixel in the \(r\)-th row and \(t\)-th column. According to the image, the candidate solutions in each generation can be obtained, where the threshold value \(\phi\) is set to 0.5. For the ease of use, the candidate solutions in the last generation is selected. To prevent the loss of potential seed solutions during the evolution, candidate solutions from other generations can also be selected.