# Niching Global Optimisation: Systematic Literature Review
## 2 Niching Paradigms
### Parallel Niching Paradigms
#### 2.2.1 Fitness Sharing  
Fitness sharing is an approach which aims to penalise individuals in a crowd-populated region while rewarding individuals which are scattered in the search space in a bid to encourage search agents to discover new areas [32; 33]. The original fitness function of each individual is thus penalised or modified based upon the density of its location region, forcing the particle to move away from the dense region and thus increasing exploration:  
\[f_{i}^{\prime}=\frac{f_{i}}{m_{i}} \tag{4}\]  
where \(m_{i}\) is the niche decimal count which measures the number of individuals with whom the fitness is shared such that  
\[m_{i}=\sum_{j=i}^{N}sh(d_{ij}) \tag{5}\]  
where \(N\) is the population size and \(d_{ij}\) the distance between individuals \(i\) and \(j\). The sharing function \(sh(. )\) assigns a count \([0,1]\) based on the proximity of two agents:  
\[sh(x)= \begin{cases}1-\left(\frac{x}{\sigma}\right)^{\alpha}\text{, if }x<\sigma\\ 0\text{, otherwise}\end{cases}  \tag{6}\]  
with a typical value of \(\alpha=1\). Thus, the closer an agent is to another, the closer \(sh(. )\) is to one, the larger \(m_{i}\) is, and the greater the penalty \(f_{i}^{\prime}\) for vacating the region from the least dominant to most dominant agent. Algorithm 2 provides a pseudo-code of the fitness-sharing paradigm. \begin{table}
\begin{tabular}{c c c c} \hline
**Authors** & **Year** & **Algorithm** & **Theoretical Ground** & **Type** \\ \hline
[4] & 1993 & SN-GA & Multi-restart genetic algorithms with derating function. & Sequential \\ \hline
[27] & 2001 & SN-PSO & Landscape transformation via stretching to discover larger optima than previous ones. & Sequential \\ \hline
[1] & 2006 & SN-PSO & Radius-free sequential niching using hill concavity repulsion test. & Sequential \\ \hline
[30; 31] & 2008 & SNMA & Sequential niching memetic algorithm combining genetic algorithms and local search for continuous multimodal function optimisation. It uses a Gaussian derating function and clearing to occupy different niches in the function to be optimised. & Sequential \\ \hline \end{tabular}
\end{table} 
Table 1: A collection of sequential niching algorithms. ```
1 Let \(X_{0}=[x^{l},x^{u}]\);
2 Set pop size N, max_iter \(K\), and \(k=0\);
3 Generate the initial population: \(x_{i}\in X_{0}\);
4 Assign neighbourhood radius \(\sigma\);
5 whileallagents not stalled and iter \(<K\)do
6 for\(i=I\):Ndo
7 compute \(f_{i}^{\prime}\);
8 end for
9 update search-direction as per the meta-heuristic;
10 iter++;
11 end for return niche heads;
```  
**Algorithm 2**Typical fitness sharing procedure. Other more modern fitness-based niching methodologies [6; 17] use a fitness-to-distance ratio ranking score which guides search agents in collaborating with their nearest fittest mates. In swarm intelligence, the fitness Euclidean distance estimator ranks neighbourhoods by computing a fitness distance to reward the closest and fittest neighbour around each search agent:  
\[FER_{(j,i)}=\alpha\frac{f(\vec{p}_{j})-f(\vec{p}_{i})}{||\vec{p}_{j}-\vec{p}_{i }||} \tag{7}\]  
where \(\alpha=\frac{||s||}{f(\vec{p}_{j})-f(\vec{p}_{n})}\), \(||s||\), the estimated size of the search space is \(\sqrt{\sum_{i=1}^{D}(x_{i}^{u}-x_{i}^{l})^{2}}\), and \(x^{u}\) and \(x^{l}\) are the bounds of the search space. Using the fitness Euclidean distance score, search particles are directed to head towards the closest particle which is around the most promising basin in the local neighbourhood. Such an adaptation exists for evolutionary algorithms [6]. Table 2 gives an exhaustive overview of the surveyed methods.