# Dynamical Sphere Regrouping Particle Swarm Optimization: A Proposed Algorithm for Dealing with PSO Premature Convergence in Large-Scale Global Optimization

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## Abstract

**:**

## 1. Introduction

#### Contribution

## 2. Materials and Methods

#### 2.1. Biological Inspiration

#### 2.2. GPSO

#### 2.3. Dealing with Local Optimums in PSO

- Stop the search and accept the result.
- Continue the search while hoping to find a better solution.
- Restart the swarm from new locations and search again.
- Mark the areas in the search space that lead to a local optimum and avoid them.
- Reinvigorate the swarm to maintain diversity.

#### 2.4. Regrouping PSO

#### 2.5. Dynamical Sphere Regrouping PSO (DSRegPSO)

#### 2.5.1. DSRegPSO Inspiration

**Definition**

**1.**

#### 2.5.2. The DSRegPSO Algorithm

## 3. Results and Discussion

- $D$, $n$, $f\left(\overrightarrow{{X}_{i}}\right)$, ${L}_{l}$, and ${L}_{u}$ were specified by the requirements for the optimized functions in each function of CEC’13.
- We assumed that the remaining input parameters are linearly independent. Based on this assumption, we chose the values that resulted in the best cost for each benchmark by varying them heuristically within the ranges specified in Section 3.1.

#### 3.1. Results of the CEC’13 Test

- Fully separable functions:
- ${f}_{1}\u2254$ Elliptic with $\overrightarrow{X}\in \left[-100,100\right]$ and ${f}_{1}\left(\overrightarrow{{X}^{\mathrm{opt}}}\right)=0$.
- ${f}_{2}\u2254$ Rastrigin with $\overrightarrow{X}\in \left[-5,5\right]$ and ${f}_{2}\left(\overrightarrow{{X}^{\mathrm{opt}}}\right)=0$.
- ${f}_{3}\u2254$ Ackley with $\overrightarrow{X}\in \left[-32,32\right]$ and ${f}_{3}\left(\overrightarrow{{X}^{\mathrm{opt}}}\right)=0$.

- Partially Additively Separable Functions:
- Functions with a separable subcomponent:
- ${f}_{4}\u2254$ Elliptic with $\overrightarrow{X}\in \left[-100,100\right]$ and ${f}_{4}\left(\overrightarrow{{X}^{\mathrm{opt}}}\right)=0$.
- ${f}_{5}\u2254$ Rastrigin with $\overrightarrow{X}\in \left[-5,5\right]$ and ${f}_{5}\left(\overrightarrow{{X}^{\mathrm{opt}}}\right)=0$.
- ${f}_{6}\u2254$ Ackley with $\overrightarrow{X}\in \left[-32,32\right]$ and ${f}_{6}\left(\overrightarrow{{X}^{\mathrm{opt}}}\right)=0$.
- ${f}_{7}\u2254$ Schwefels Problem 1.2 with $\overrightarrow{X}\in \left[-100,100\right]$ and ${f}_{7}\left(\overrightarrow{{X}^{\mathrm{opt}}}\right)=0$.

- Functions with no separable subcomponents:
- ${f}_{8}\u2254$ Elliptic with $\overrightarrow{X}\in \left[-100,100\right]$ and ${f}_{8}\left(\overrightarrow{{X}^{\mathrm{opt}}}\right)=0$.
- ${f}_{9}\u2254$ Rastrigin with $\overrightarrow{X}\in \left[-5,5\right]$ and ${f}_{9}\left(\overrightarrow{{X}^{\mathrm{opt}}}\right)=0$.
- ${f}_{10}\u2254$ Ackley with $\overrightarrow{X}\in \left[-32,32\right]$ and ${f}_{10}\left(\overrightarrow{{X}^{\mathrm{opt}}}\right)=0$.
- ${f}_{11}\u2254$ Schwefels Problem 1.2 with $\overrightarrow{X}\in \left[-100,100\right]$ and ${f}_{11}\left(\overrightarrow{{X}^{\mathrm{opt}}}\right)=0$.

- Overlapping Functions:
- ${f}_{12}\u2254$ Rosenbrock’s with $\overrightarrow{X}\in \left[-100,100\right]$ and ${f}_{12}\left(\overrightarrow{{X}^{\mathrm{opt}}}+1\right)=0$.
- ${f}_{13}\u2254$ Schwefels with Conforming Overlapping Subcomponents with $\overrightarrow{X}\in \left[-100,100\right]$ and ${f}_{13}\left(\overrightarrow{{X}^{\mathrm{opt}}}\right)=0$.
- ${f}_{14}\u2254$ Schwefels with Conflicting Overlapping Subcomponents with $\overrightarrow{X}\in \left[-100,100\right]$ and ${f}_{14}\left(\overrightarrow{{X}^{\mathrm{opt}}}\right)=0$.

- Non-separable Functions:
- ${f}_{15}\u2254$ Schwefels Problem 1.2 with $\overrightarrow{X}\in \left[-100,100\right]$ and ${f}_{15}\left(\overrightarrow{{X}^{\mathrm{opt}}}\right)=0$.

- $f{d}_{min}=\left\{1\mathrm{E}-200,1\mathrm{E}-100,1\mathrm{E}-50,1\mathrm{E}-25,1\mathrm{E}-10,1\mathrm{E}-5,1\mathrm{E}-1,1\right\}$.
- $f{d}_{max}=\left\{1\mathrm{E}-200,1\mathrm{E}-100,1\mathrm{E}-50,1\mathrm{E}-25,1\mathrm{E}-10,1\mathrm{E}-5,1\mathrm{E}-1,1\right\}$.
- $\mathsf{\zeta}=\left\{1\mathrm{E}-5,1\mathrm{E}-4,1\mathrm{E}-3,1\mathrm{E}-2,5\mathrm{E}-2,1\mathrm{E}-1,5\mathrm{E}-1\right\}$.
- ${M}_{max}=\left\{0.0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1.0,1.1,1.2,1.3\right\}$.
- ${c}_{2}=\left\{0.0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1.0,1.1,1.2,1.3,1.5,2.0\right\}$.
- $\mathsf{\lambda}=\left\{0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1.0,1.1,1.2,1.3\right\}$.
- ${S}_{max}=\left\{0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1.0\right\}$.
- ${S}_{min}=\left\{0.01,0.02,0.03,0.04,0.05,0.06,0.07,0.08,0.09,0.1\right\}$.
- $n=\left\{1,5,10,20,30,40,50,60,70,80,90,100\right\}$.

- ${c}_{1}=\left\{0.0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1.0,1.1,1.2,1.3,1.5,2.0\right\}$.
- ${c}_{2}=\left\{0.0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1.0,1.1,1.2,1.3,1.5,2.0\right\}$.
- ${w}_{max}=\left\{0.0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1.0,1.1,1.2,1.3\right\}$.
- $\mathsf{\lambda}=\left\{\mathrm{0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1.0,1.1,1.2,1.3}\right\}$.
- $n=\left\{\mathrm{1,5},\mathrm{10,20,30,40,50,60,70,80,90,100}\right\}$.

## 4. Conclusions

#### Future Work

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Regrouping behavior of cost value across iterations with the RegPSO algorithm [30].

**Figure 3.**Cost value for the first 25,000 function evaluations with DSRegPSO and ${f}_{1}$ of CEC’13.

**Figure 4.**Cost value with the x-axis on a logarithmic scale for the DSRegPSO in ${f}_{1}$ of CEC’13.

**Figure 5.**Convergence diagram of Function 1 of CEC’13 for DSRegPSO with 25 runs and 3.00+E06 function evaluations.

**Figure 7.**Convergence diagram of Function 2 of CEC’13 for DSRegPSO with 25 runs and 3.00+E06 function evaluations.

**Figure 9.**Convergence diagram of Function 3 of CEC’13 for DSRegPSO with 25 runs and 3.00+E06 function evaluations.

**Figure 11.**Convergence diagram of Function 4 of CEC’13 for DSRegPSO with 25 runs and 3.00+E06 function evaluations.

**Figure 13.**Convergence diagram of Function 5 of CEC’13 for DSRegPSO with 25 runs and 3.00+E06 function evaluations.

**Figure 15.**Convergence diagram of Function 6 of CEC’13 for DSRegPSO with 25 runs and 3.00+E06 function evaluations.

**Figure 17.**Convergence diagram of Function 7 of CEC’13 for DSRegPSO with 25 runs and 3.00+E06 function evaluations.

**Figure 19.**Convergence diagram of Function 8 of CEC’13 for DSRegPSO with 25 runs and 3.00+E06 function evaluations.

**Figure 21.**Convergence diagram of Function 9 of CEC’13 for DSRegPSO with 25 runs and 3.00+E06 function evaluations.

**Figure 23.**Convergence diagram of Function 10 of CEC’13 for DSRegPSO with 25 runs and 3.00+E06 function evaluations.

**Figure 25.**Convergence diagram of Function 11 of CEC’13 for DSRegPSO with 25 runs and 3.00+E06 function evaluations.

**Figure 27.**Convergence diagram of Function 12 of CEC’13 for DSRegPSO with 25 runs and 3.00+E06 function evaluations.

**Figure 29.**Convergence diagram of Function 13 of CEC´13 for DSRegPSO with 25 runs and 3.00+E06 function evaluations.

**Figure 31.**Convergence diagram of Function 14 of CEC´13 for DSRegPSO with 25 runs and 3.00+"E" 06 function evaluations.

**Figure 33.**Convergence diagram of Function 15 of CEC´13 for DSRegPSO with 25 runs and 3.00+"E" 06 function evaluations.

**Figure 36.**TACO comparison across all algorithms, including DSRegPSO and GPSO in the non-separable CEC’13 function.

$\mathit{f}\left(\overrightarrow{\mathit{X}}\right)$ | $\mathit{f}{\mathit{d}}_{\mathit{m}\mathit{i}\mathit{n}}$ | $\mathit{f}{\mathit{d}}_{\mathit{m}\mathit{a}\mathit{x}}$ | $\mathsf{\zeta}$ | ${\mathit{c}}_{2}$ | ${\mathit{M}}_{\mathit{m}\mathit{a}\mathit{x}}$ | $\mathsf{\lambda}$ | ${\mathit{S}}_{\mathit{m}\mathit{a}\mathit{x}}$ | ${\mathit{S}}_{\mathit{m}\mathit{i}\mathit{n}}$ | $\mathit{n}$ |
---|---|---|---|---|---|---|---|---|---|

${f}_{1}$ | $1.0\mathrm{E}-200$ | $1.0\mathrm{E}-200$ | $5.0\mathrm{E}-02$ | $1.5\mathrm{E}+00$ | $0.0\mathrm{E}+00$ | $7.0\mathrm{E}-01$ | $1.5\mathrm{E}+00$ | $2.0\mathrm{E}-02$ | $5.0\mathrm{E}+01$ |

${f}_{2}$ | $1.0\mathrm{E}-50$ | $1.0\mathrm{E}-01$ | $1.0\mathrm{E}-02$ | $1.5\mathrm{E}+00$ | $1.0\mathrm{E}-01$ | $1.9\mathrm{E}+00$ | $5.0\mathrm{E}-01$ | $5.0\mathrm{E}-02$ | $5.0\mathrm{E}+00$ |

${f}_{3}$ | $1.0\mathrm{E}-100$ | $1.0\mathrm{E}-05$ | $5.0\mathrm{E}-01$ | $1.5\mathrm{E}+00$ | $1.0\mathrm{E}-01$ | $2.0\mathrm{E}-01$ | $1.0\mathrm{E}-01$ | $5.0\mathrm{E}-02$ | $2.0\mathrm{E}+01$ |

${f}_{4}$ | $1.0\mathrm{E}-50$ | $1.0\mathrm{E}-01$ | $1.0\mathrm{E}-03$ | $1.0\mathrm{E}+00$ | $0.0\mathrm{E}+00$ | $1.0\mathrm{E}+00$ | $5.0\mathrm{E}-01$ | $5.0\mathrm{E}-02$ | $2.0\mathrm{E}+01$ |

${f}_{5}$ | $1.0\mathrm{E}-50$ | $1.0\mathrm{E}+00$ | $5.0\mathrm{E}-02$ | $2.0\mathrm{E}+00$ | $1.3\mathrm{E}+00$ | $1.3\mathrm{E}+00$ | $3.0\mathrm{E}-01$ | $5.0\mathrm{E}-02$ | $4.0\mathrm{E}+01$ |

${f}_{6}$ | $1.0\mathrm{E}-50$ | $1.0\mathrm{E}-25$ | $1.0\mathrm{E}-03$ | $8.0\mathrm{E}-01$ | $7.0\mathrm{E}-01$ | $7.0\mathrm{E}-01$ | $8.0\mathrm{E}-01$ | $8.0\mathrm{E}-02$ | $5.0\mathrm{E}+01$ |

${f}_{7}$ | $1.0\mathrm{E}-50$ | $1.0\mathrm{E}-10$ | $1.0\mathrm{E}-02$ | $1.5\mathrm{E}+00$ | $4.0\mathrm{E}-01$ | $3.0\mathrm{E}-01$ | $9.0\mathrm{E}-01$ | $5.0\mathrm{E}-02$ | $3.0\mathrm{E}+01$ |

${f}_{8}$ | $1.0\mathrm{E}-25$ | $1.0\mathrm{E}-10$ | $5.0\mathrm{E}-02$ | $6.0\mathrm{E}-01$ | $6.0\mathrm{E}-01$ | $4.0\mathrm{E}-01$ | $4.0\mathrm{E}-01$ | $5.0\mathrm{E}-02$ | $5.0\mathrm{E}+01$ |

${f}_{9}$ | $1.0\mathrm{E}-25$ | $1.0\mathrm{E}-25$ | $1.0\mathrm{E}-01$ | $2.0\mathrm{E}+00$ | $1.2\mathrm{E}+00$ | $1.3\mathrm{E}+00$ | $3.0\mathrm{E}-01$ | $5.0\mathrm{E}-02$ | $3.0\mathrm{E}+01$ |

${f}_{10}$ | $1.0\mathrm{E}-25$ | $1.0\mathrm{E}-01$ | $1.0\mathrm{E}-03$ | $8.0\mathrm{E}-01$ | $3.0\mathrm{E}-01$ | $7.0\mathrm{E}-01$ | $5.0\mathrm{E}-01$ | $5.0\mathrm{E}-02$ | $5.0\mathrm{E}+01$ |

${f}_{11}$ | $1.0\mathrm{E}-50$ | $1.0\mathrm{E}+00$ | $1.0\mathrm{E}-02$ | $1.3\mathrm{E}+00$ | $2.0\mathrm{E}-01$ | $5.0\mathrm{E}-01$ | $5.0\mathrm{E}-01$ | $4.0\mathrm{E}-02$ | $3.0\mathrm{E}+01$ |

${f}_{12}$ | $1.0\mathrm{E}-25$ | $1.0\mathrm{E}-01$ | $1.0\mathrm{E}-02$ | $1.0\mathrm{E}-01$ | $3.0\mathrm{E}-01$ | $1.3\mathrm{E}+00$ | $1.0\mathrm{E}-01$ | $1.0\mathrm{E}-01$ | $1.0\mathrm{E}+00$ |

${f}_{13}$ | $1.0\mathrm{E}-50$ | $1.0\mathrm{E}-01$ | $1.0\mathrm{E}-02$ | $1.3\mathrm{E}+00$ | $3.0\mathrm{E}-01$ | $5.0\mathrm{E}-01$ | $5.0\mathrm{E}-01$ | $5.0\mathrm{E}-02$ | $3.0\mathrm{E}+01$ |

${f}_{14}$ | $1.0\mathrm{E}-25$ | $1.0\mathrm{E}+00$ | $5.0\mathrm{E}-01$ | $1.0\mathrm{E}+00$ | $5.0\mathrm{E}-01$ | $4.0\mathrm{E}-01$ | $5.0\mathrm{E}-01$ | $5.0\mathrm{E}-02$ | $4.0\mathrm{E}+01$ |

${f}_{15}$ | $1.0\mathrm{E}-50$ | $1.0\mathrm{E}-25$ | $1.0\mathrm{E}-02$ | $1.3\mathrm{E}+00$ | $4.0\mathrm{E}-01$ | $6.0\mathrm{E}-01$ | $9.0\mathrm{E}-01$ | $5.0\mathrm{E}-02$ | $3.0\mathrm{E}+01$ |

$\mathit{f}\left(\overrightarrow{\mathit{X}}\right)$ | ${\mathit{c}}_{1}$ | ${\mathit{c}}_{2}$ | ${\mathit{w}}_{\mathit{m}\mathit{a}\mathit{x}}$ | $\mathsf{\lambda}$ | $\mathit{n}$ |
---|---|---|---|---|---|

${f}_{1}$ | $2.0\mathrm{E}+00$ | $1.5\mathrm{E}+00$ | $5.0\mathrm{E}-01$ | $1.0\mathrm{E}-01$ | $8.0\mathrm{E}+01$ |

${f}_{2}$ | $2.0\mathrm{E}+00$ | $2.0\mathrm{E}+00$ | $0.0\mathrm{E}+00$ | $1.3\mathrm{E}+00$ | $1.0\mathrm{E}+02$ |

${f}_{3}$ | $2.0\mathrm{E}+00$ | $1.0\mathrm{E}+00$ | $7.0\mathrm{E}-01$ | $1.0\mathrm{E}-01$ | $3.0\mathrm{E}+01$ |

${f}_{4}$ | $1.5\mathrm{E}+00$ | $1.5\mathrm{E}+00$ | $7.0\mathrm{E}-01$ | $1.0\mathrm{E}-01$ | $8.0\mathrm{E}+01$ |

${f}_{5}$ | $1.2\mathrm{E}+00$ | $2.0\mathrm{E}+00$ | $7.0\mathrm{E}-01$ | $2.0\mathrm{E}-01$ | $7.0\mathrm{E}+01$ |

${f}_{6}$ | $1.5\mathrm{E}+00$ | $8.0\mathrm{E}-01$ | $7.0\mathrm{E}-01$ | $1.0\mathrm{E}-01$ | $7.0\mathrm{E}+01$ |

${f}_{7}$ | $1.5\mathrm{E}+00$ | $1.3\mathrm{E}+00$ | $7.0\mathrm{E}-01$ | $2.0\mathrm{E}-01$ | $5.0\mathrm{E}+01$ |

${f}_{8}$ | $1.3\mathrm{E}+00$ | $2.0\mathrm{E}+00$ | $7.0\mathrm{E}-01$ | $1.0\mathrm{E}-01$ | $5.0\mathrm{E}+01$ |

${f}_{9}$ | $2.0\mathrm{E}+00$ | $1.0\mathrm{E}+00$ | $7.0\mathrm{E}-01$ | $1.0\mathrm{E}-01$ | $9.0\mathrm{E}+01$ |

${f}_{10}$ | $2.0\mathrm{E}+00$ | $1.0\mathrm{E}+00$ | $6.0\mathrm{E}-01$ | $7.0\mathrm{E}-01$ | $5.0\mathrm{E}+01$ |

${f}_{11}$ | $1.5\mathrm{E}+00$ | $1.5\mathrm{E}+00$ | $7.0\mathrm{E}-01$ | $1.0\mathrm{E}-01$ | $8.0\mathrm{E}+01$ |

${f}_{12}$ | $1.0\mathrm{E}-01$ | $2.0\mathrm{E}+00$ | $1.0\mathrm{E}-01$ | $1.0\mathrm{E}-01$ | $1.0\mathrm{E}+02$ |

${f}_{13}$ | $1.5\mathrm{E}+00$ | $1.5\mathrm{E}+00$ | $7.0\mathrm{E}-01$ | $1.0\mathrm{E}-01$ | $5.0\mathrm{E}+01$ |

${f}_{14}$ | $1.5\mathrm{E}+00$ | $1.5\mathrm{E}+00$ | $7.0\mathrm{E}-01$ | $1.0\mathrm{E}-01$ | $9.0\mathrm{E}+01$ |

${f}_{15}$ | $1.5\mathrm{E}+00$ | $1.0\mathrm{E}+00$ | $7.0\mathrm{E}-01$ | $1.0\mathrm{E}-01$ | $8.0\mathrm{E}+01$ |

$\mathit{f}\left(\overrightarrow{\mathit{X}}\right)$ | Average Time per Iteration in Seconds | Mean | SD | Worst | Best | |||||
---|---|---|---|---|---|---|---|---|---|---|

Algorithms | DSRegPSO | GPSO | DSRegPSO | GPSO | DSRegPSO | GPSO | DSRegPSO | GPSO | DSRegPSO | GPSO |

${f}_{1}$ | $3.16\mathrm{E}-05$ | $1.02\mathrm{E}-05$ | $4.07\mathrm{E}-04$ | $1.44\mathrm{E}+10$ | $1.73\mathrm{E}-04$ | $1.55\mathrm{E}+10$ | $1.07\mathrm{E}-03$ | $4.63\mathrm{E}+10$ | $1.90\mathrm{E}-04$ | $2.91\mathrm{E}+09$ |

${f}_{2}$ | $4.44\mathrm{E}-05$ | $1.19\mathrm{E}-05$ | $8.63\mathrm{E}+02$ | $4.36\mathrm{E}+04$ | $1.29\mathrm{E}+02$ | $8.31\mathrm{E}+02$ | $1.16\mathrm{E}+03$ | $4.54\mathrm{E}+04$ | $6.87\mathrm{E}+02$ | $4.19\mathrm{E}+04$ |

${f}_{3}$ | $4.79\mathrm{E}-05$ | $1.33\mathrm{E}-05$ | $2.00\mathrm{E}+01$ | $2.03\mathrm{E}+01$ | $1.10\mathrm{E}-09$ | $2.74\mathrm{E}-02$ | $2.00\mathrm{E}+01$ | $2.03\mathrm{E}+01$ | $2.00\mathrm{E}+01$ | $2.02\mathrm{E}+01$ |

${f}_{4}$ | $4.44\mathrm{E}-05$ | $1.26\mathrm{E}-05$ | $2.15\mathrm{E}+09$ | $8.63\mathrm{E}+10$ | $7.17\mathrm{E}+08$ | $5.17\mathrm{E}+10$ | $3.84\mathrm{E}+09$ | $2.06\mathrm{E}+11$ | $1.24\mathrm{E}+09$ | $2.75\mathrm{E}+10$ |

${f}_{5}$ | $4.95\mathrm{E}-05$ | $1.46\mathrm{E}-05$ | $7.76\mathrm{E}+06$ | $8.38\mathrm{E}+06$ | $2.64\mathrm{E}+06$ | $1.79\mathrm{E}+06$ | $1.45\mathrm{E}+07$ | $1.35\mathrm{E}+07$ | $3.76\mathrm{E}+06$ | $5.66\mathrm{E}+06$ |

${f}_{6}$ | $4.72\mathrm{E}-05$ | $1.10\mathrm{E}-05$ | $1.01\mathrm{E}+06$ | $1.03\mathrm{E}+06$ | $1.26\mathrm{E}+04$ | $7.76\mathrm{E}+03$ | $1.04\mathrm{E}+06$ | $1.04\mathrm{E}+06$ | $9.96\mathrm{E}+05$ | $1.01\mathrm{E}+06$ |

${f}_{7}$ | $2.24\mathrm{E}-05$ | $7.74\mathrm{E}-06$ | $5.85\mathrm{E}+04$ | $3.84\mathrm{E}+09$ | $1.26\mathrm{E}+04$ | $3.29\mathrm{E}+09$ | $9.24\mathrm{E}+04$ | $1.79\mathrm{E}+10$ | $3.59\mathrm{E}+04$ | $5.73\mathrm{E}+08$ |

${f}_{8}$ | $9.60\mathrm{E}-05$ | $1.50\mathrm{E}-05$ | $3.34\mathrm{E}+13$ | $4.24\mathrm{E}+14$ | $2.31\mathrm{E}+13$ | $3.30\mathrm{E}+14$ | $1.12\mathrm{E}+14$ | $1.80\mathrm{E}+15$ | $1.06\mathrm{E}+13$ | $1.32\mathrm{E}+14$ |

${f}_{9}$ | $4.56\mathrm{E}-02$ | $1.77\mathrm{E}-05$ | $4.65\mathrm{E}+08$ | $9.12\mathrm{E}+08$ | $1.93\mathrm{E}+08$ | $1.50\mathrm{E}+08$ | $1.28\mathrm{E}+09$ | $1.26\mathrm{E}+09$ | $3.05\mathrm{E}+08$ | $6.12\mathrm{E}+08$ |

${f}_{10}$ | $4.01\mathrm{E}-03$ | $1.52\mathrm{E}-05$ | $9.25\mathrm{E}+07$ | $9.20\mathrm{E}+07$ | $5.49\mathrm{E}+05$ | $6.64\mathrm{E}+05$ | $9.39\mathrm{E}+07$ | $9.31\mathrm{E}+07$ | $9.17\mathrm{E}+07$ | $9.07\mathrm{E}+07$ |

${f}_{11}$ | $4.32\mathrm{E}-05$ | $1.62\mathrm{E}-05$ | $5.42\mathrm{E}+08$ | $1.25\mathrm{E}+11$ | $8.35\mathrm{E}+07$ | $1.06\mathrm{E}+11$ | $7.57\mathrm{E}+08$ | $3.98\mathrm{E}+11$ | $4.32\mathrm{E}+08$ | $3.44\mathrm{E}+09$ |

${f}_{12}$ | $3.36\mathrm{E}-05$ | $1.10\mathrm{E}-06$ | $2.48\mathrm{E}+03$ | $1.60\mathrm{E}+12$ | $1.36\mathrm{E}+03$ | $3.58\mathrm{E}+10$ | $6.69\mathrm{E}+03$ | $1.67\mathrm{E}+12$ | $1.56\mathrm{E}+03$ | $1.53\mathrm{E}+12$ |

${f}_{13}$ | $6.72\mathrm{E}-05$ | $6.72\mathrm{E}-06$ | $1.37\mathrm{E}+07$ | $1.18\mathrm{E}+10$ | $2.58\mathrm{E}+06$ | $5.31\mathrm{E}+09$ | $2.02\mathrm{E}+07$ | $2.52\mathrm{E}+10$ | $1.03\mathrm{E}+07$ | $3.03\mathrm{E}+09$ |

${f}_{14}$ | $4.32\mathrm{E}-05$ | $7.20\mathrm{E}-06$ | $2.47\mathrm{E}+08$ | $1.09\mathrm{E}+11$ | $2.85\mathrm{E}+07$ | $7.19\mathrm{E}+10$ | $3.08\mathrm{E}+08$ | $2.96\mathrm{E}+11$ | $1.92\mathrm{E}+08$ | $7.87\mathrm{E}+09$ |

${f}_{15}$ | $2.88\mathrm{E}-05$ | $3.96\mathrm{E}-06$ | $6.73\mathrm{E}+05$ | $2.31\mathrm{E}+12$ | $6.04\mathrm{E}+04$ | $3.07\mathrm{E}+12$ | $8.18\mathrm{E}+05$ | $1.44\mathrm{E}+13$ | $5.79\mathrm{E}+05$ | $2.80\mathrm{E}+10$ |

Algorithm | $1.20\mathbf{E}+05$ | $6.00\mathbf{E}+05$ | $3.00\mathbf{E}+06$ |
---|---|---|---|

AMO | 0 | 0 | 0 |

APO | 0 | 0 | 0 |

AQO | 0 | 0 | 0 |

BICCA | 0 | 1 | 1 |

CC-CMA-ES | 0 | 1 | 1 |

DECC-G | 6 | 0 | 0 |

DEEPSO | 0 | 0 | 0 |

DMO | 0 | 0 | 0 |

DPO | 0 | 0 | 0 |

DQO | 1 | 0 | 0 |

DSRegPSO | 4 | 1 | 1 |

IHDELS | 1 | 0 | 0 |

MLSHADE-SPA | 0 | 4 | 4 |

MOS | 1 | 0 | 0 |

RO | 0 | 0 | 0 |

SACC | 0 | 1 | 0 |

SHADEILS | 2 | 8 | 8 |

VMODE | 0 | 0 | 0 |

Algorithm | ${\mathit{f}}_{1}$ | ${\mathit{f}}_{2}$ | ${\mathit{f}}_{3}$ | ${\mathit{f}}_{4}$ | ${\mathit{f}}_{5}$ | ${\mathit{f}}_{6}$ | ${\mathit{f}}_{7}$ | ${\mathit{f}}_{8}$ | ${\mathit{f}}_{9}$ | ${\mathit{f}}_{10}$ | ${\mathit{f}}_{11}$ | ${\mathit{f}}_{12}$ | ${\mathit{f}}_{13}$ | ${\mathit{f}}_{14}$ | ${\mathit{f}}_{15}$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

AMO | 0.00E+00 | 8.48E+02 | 0.00E+00 | 1.57E+08 | 6.69E+06 | 1.63E+05 | 2.07E+04 | 8.71E+12 | 4.09E+08 | 8.99E+05 | 5.16E+07 | 3.17E+02 | 3.14E+06 | 2.69E+07 | 2.44E+06 |

APO | 0.00E+00 | 8.32E+02 | 0.00E+00 | 1.62E+08 | 7.01E+06 | 1.45E+05 | 3.31E+02 | 1.70E+13 | 3.96E+08 | 7.07E+05 | 2.54E+07 | 1.06E+02 | 7.88E+05 | 9.92E+06 | 2.08E+06 |

AQO | 0.00E+00 | 8.39E+02 | 0.00E+00 | 1.61E+08 | 6.84E+06 | 1.79E+05 | 1.52E+04 | 7.31E+12 | 4.08E+08 | 9.46E+05 | 4.65E+07 | 1.91E+02 | 3.68E+06 | 2.69E+07 | 2.40E+06 |

BICCA | 0.00E+00 | 8.46E−07 | 7.27E−01 | 8.85E+08 | 2.58E+06 | 1.46E+05 | 1.82E+05 | 3.78E+12 | 2.18E+08 | 1.24E+06 | 2.85E+07 | 1.40E+03 | 1.09E+07 | 4.27E+07 | 3.16E+06 |

CC-CMA-ES | 5.80E−09 | 1.33E+03 | 0.00E+00 | 2.19E+09 | 7.28E+14 | 5.87E+05 | 7.44E+06 | 3.88E+14 | 3.71E+08 | 7.55E+05 | 1.58E+08 | 1.27E+03 | 6.69E+08 | 7.10E+07 | 3.03E+07 |

DECC-G | 0.00E+00 | 1.03E+03 | 3.00E−10 | 2.12E+10 | 5.07E+06 | 6.08E+04 | 4.27E+08 | 3.88E+14 | 4.17E+08 | 1.19E+07 | 1.60E+11 | 1.07E+03 | 3.36E+10 | 6.27E+11 | 6.01E+07 |

DEEPSO | 1.44E+08 | 1.49E+04 | 2.04E+01 | 4.77E+09 | 1.45E+07 | 1.02E+06 | 1.54E+07 | 5.42E+12 | 9.17E+08 | 9.07E+07 | 5.60E+08 | 1.54E+10 | 8.75E+08 | 4.33E+08 | 7.04E+06 |

DMO | 0.00E+00 | 8.16E+02 | 0.00E+00 | 2.20E+08 | 7.12E+06 | 1.50E+05 | 5.26E+04 | 1.07E+13 | 5.28E+08 | 5.70E+05 | 1.16E+08 | 2.45E+02 | 6.55E+06 | 4.57E+07 | 3.02E+07 |

DPO | 0.00E+00 | 1.05E+03 | 0.00E+00 | 2.71E+08 | 6.85E+06 | 1.38E+05 | 2.52E+04 | 2.33E+13 | 4.02E+08 | 1.08E+06 | 9.88E+07 | 3.45E+02 | 4.04E+06 | 2.86E+07 | 2.80E+06 |

DQO | 0.00E+00 | 8.41E+02 | 0.00E+00 | 1.56E+08 | 7.06E+06 | 1.52E+05 | 2.06E+04 | 7.52E+12 | 4.10E+08 | 8.02E+05 | 5.43E+07 | 2.07E+02 | 3.21E+06 | 2.43E+07 | 2.38E+06 |

DSRegPSO | 1.90E−04 | 6.87E+02 | 2.00E+01 | 1.24E+09 | 3.76E+06 | 9.96E+05 | 3.59E+04 | 1.06E+13 | 3.05E+08 | 9.17E+07 | 4.32E+08 | 1.56E+03 | 1.03E+07 | 1.92E+08 | 5.79E+05 |

GPSO | 2.91E+09 | 4.19E+04 | 2.02E+01 | 2.75E+10 | 5.66E+06 | 1.01E+06 | 5.73E+08 | 1.32E+14 | 6.12E+08 | 9.07E+07 | 3.44E+09 | 1.53E+12 | 3.03E+09 | 7.87E+09 | 2.80E+10 |

IHDELS | 4.34E−28 | 1.32E+03 | 2.01E+01 | 3.04E+08 | 9.59E+06 | 1.03E+06 | 3.46E+04 | 1.36E+12 | 6.74E+08 | 9.16E+07 | 1.07E+07 | 3.77E+02 | 3.80E+06 | 1.58E+07 | 2.81E+06 |

MLSHADE-SPA | 1.94E−22 | 7.89E+01 | 0.00E+00 | 6.90E+08 | 1.80E+06 | 1.40E+03 | 5.31E+04 | 9.77E+12 | 1.61E+08 | 6.56E+02 | 4.04E+07 | 1.04E+02 | 7.21E+07 | 1.52E+07 | 2.76E+07 |

MOS | 0.00E+00 | 8.32E+02 | 0.00E+00 | 1.74E+08 | 6.94E+06 | 1.48E+05 | 1.62E+04 | 8.00E+12 | 3.83E+08 | 9.02E+05 | 5.22E+07 | 2.47E+02 | 3.40E+06 | 2.56E+07 | 2.35E+06 |

RO | 0.00E+00 | 8.09E+02 | 0.00E+00 | 2.25E+08 | 6.33E+06 | 1.29E+05 | 3.46E+04 | 8.43E+12 | 3.85E+08 | 6.14E+05 | 8.53E+07 | 4.81E+02 | 4.61E+06 | 3.44E+07 | 1.00E+07 |

SACC | 0.00E+00 | 5.71E+02 | 1.21E+00 | 3.66E+10 | 6.95E+06 | 2.07E+05 | 1.58E+07 | 9.86E+14 | 5.77E+08 | 2.11E+07 | 5.30E+08 | 8.74E+02 | 1.51E+09 | 7.34E+09 | 1.88E+06 |

SHADEILS | 2.69E−24 | 1.00E+03 | 2.01E+01 | 1.48E+08 | 1.39E+06 | 1.02E+06 | 7.41E+01 | 3.17E+11 | 1.64E+08 | 9.18E+07 | 5.11E+05 | 6.18E+01 | 1.00E+05 | 5.76E+06 | 6.25E+05 |

VMODE | 8.51E−04 | 5.51E+03 | 3.41E−04 | 8.48E+09 | 7.28E+14 | 1.99E+05 | 3.44E+06 | 3.26E+13 | 7.51E+08 | 9.91E+06 | 1.58E+08 | 2.34E+03 | 2.43E+07 | 9.35E+07 | 1.11E+07 |

Algorithm | AMO | APO | AQO | BICCA | CCCMA-ES | DECC-G | DEEPSO | DMO | DPO | DQO | DSRegPSO | GPSO | IHDELS | MLSHADE-SPA | MOS | RO | SACC | SHADEILS | VMODE | Accum. Error (%) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

AMO | 1.0E+00 | 2.1E−01 | 9.0E−01 | 8.9E−01 | 8.4E−03 | 3.4E−03 | 8.4E−03 | 2.0E−02 | 2.4E−02 | 4.5E−01 | 8.3E−02 | 4.3E−04 | 1.9E−01 | 9.3E−01 | 6.2E−01 | 6.6E−01 | 5.7E−03 | 4.8E−02 | 6.1E−05 | 6.0E+01 |

APO | 2.1E−01 | 1.0E+00 | 1.7E−01 | 3.3E−01 | 2.0E−03 | 3.4E−03 | 8.4E−03 | 7.9E−02 | 8.4E−03 | 9.0E−02 | 3.9E−01 | 4.3E−04 | 5.5E−02 | 9.3E−01 | 2.1E−01 | 6.2E−01 | 1.2E−02 | 6.4E−02 | 6.1E−05 | 6.0E+01 |

AQO | 9.0E−01 | 1.7E−01 | 1.0E+00 | 8.9E−01 | 8.4E−03 | 3.4E−03 | 8.4E−03 | 1.4E−02 | 2.8E−02 | 9.0E−01 | 8.3E−02 | 4.3E−04 | 1.9E−01 | 9.3E−01 | 1.0E+00 | 1.9E−01 | 5.7E−03 | 5.5E−02 | 6.1E−05 | 6.0E+01 |

BICCA | 8.9E−01 | 3.3E−01 | 8.9E−01 | 1.0E+00 | 3.4E−03 | 3.4E−03 | 6.1E−05 | 3.6E−01 | 6.8E−01 | 8.9E−01 | 2.2E−02 | 6.1E−05 | 4.5E−01 | 6.0E−01 | 8.9E−01 | 9.8E−01 | 2.6E−03 | 4.1E−02 | 1.8E−04 | 6.0E+01 |

CCCMA-ES | 8.4E−03 | 2.0E−03 | 8.4E−03 | 3.4E−03 | 1.0E+00 | 1.3E−01 | 1.5E−01 | 4.3E−03 | 8.4E−03 | 8.4E−03 | 3.0E−01 | 8.3E−02 | 7.3E−02 | 6.1E−05 | 6.7E−03 | 2.0E−03 | 1.9E−01 | 1.0E−02 | 8.9E−01 | 6.0E+01 |

DECC-G | 3.4E−03 | 3.4E−03 | 3.4E−03 | 3.4E−03 | 1.3E−01 | 1.0E+00 | 3.3E−01 | 2.2E−02 | 6.7E−03 | 3.4E−03 | 2.2E−02 | 6.4E−01 | 1.4E−01 | 6.1E−05 | 3.4E−03 | 3.4E−03 | 8.0E−01 | 8.4E−03 | 2.3E−01 | 6.0E+01 |

DEEPSO | 8.4E−03 | 8.4E−03 | 8.4E−03 | 6.1E−05 | 1.5E−01 | 3.3E−01 | 1.0E+00 | 2.6E−02 | 8.4E−03 | 8.4E−03 | 1.8E−02 | 1.2E−02 | 8.5E−04 | 2.6E−02 | 8.4E−03 | 1.8E−02 | 8.0E−01 | 8.5E−04 | 4.2E−01 | 6.0E+01 |

DMO | 2.0E−02 | 7.9E−02 | 1.4E−02 | 3.6E−01 | 4.3E−03 | 2.2E−02 | 2.6E−02 | 1.0E+00 | 4.1E−01 | 1.7E−02 | 6.4E−01 | 4.3E−04 | 1.0E+00 | 9.5E−02 | 1.4E−02 | 3.8E−02 | 2.0E−02 | 4.8E−02 | 2.0E−03 | 6.0E+01 |

DPO | 2.4E−02 | 8.4E−03 | 2.8E−02 | 6.8E−01 | 8.4E−03 | 6.7E−03 | 8.4E−03 | 4.1E−01 | 1.0E+00 | 6.0E−02 | 3.9E−01 | 4.3E−04 | 3.9E−01 | 3.3E−01 | 1.0E−02 | 2.6E−01 | 5.7E−03 | 2.2E−02 | 6.1E−05 | 6.0E+01 |

DQO | 4.5E−01 | 9.0E−02 | 9.0E−01 | 8.9E−01 | 8.4E−03 | 3.4E−03 | 8.4E−03 | 1.7E−02 | 6.0E−02 | 1.0E+00 | 8.3E−02 | 4.3E−04 | 1.9E−01 | 9.3E−01 | 8.5E−01 | 1.9E−01 | 1.4E−02 | 5.5E−02 | 6.1E−05 | 6.0E+01 |

DSRegPSO | 8.3E−02 | 3.9E−01 | 8.3E−02 | 2.2E−02 | 3.0E−01 | 2.2E−02 | 1.8E−02 | 6.4E−01 | 3.9E−01 | 8.3E−02 | 1.0E+00 | 4.3E−04 | 2.8E−01 | 3.0E−02 | 7.3E−02 | 8.3E−02 | 4.1E−02 | 4.8E−02 | 2.1E−01 | 6.0E+01 |

GPSO | 4.3E−04 | 4.3E−04 | 4.3E−04 | 6.1E−05 | 8.3E−02 | 6.4E−01 | 1.2E−02 | 4.3E−04 | 4.3E−04 | 4.3E−04 | 4.3E−04 | 1.0E+00 | 1.2E−02 | 6.1E−05 | 4.3E−04 | 3.1E−04 | 1.1E−01 | 8.5E−04 | 2.2E−02 | 6.0E+01 |

IHDELS | 1.9E−01 | 5.5E−02 | 1.9E−01 | 4.5E−01 | 7.3E−02 | 1.4E−01 | 8.5E−04 | 1.0E+00 | 3.9E−01 | 1.9E−01 | 2.8E−01 | 1.2E−02 | 1.0E+00 | 8.5E−01 | 1.9E−01 | 8.9E−01 | 3.9E−01 | 1.5E−03 | 1.5E−02 | 6.0E+01 |

MLSHADE-SPA | 9.3E−01 | 9.3E−01 | 9.3E−01 | 6.0E−01 | 6.1E−05 | 6.1E−05 | 2.6E−02 | 9.5E−02 | 3.3E−01 | 9.3E−01 | 3.0E−02 | 6.1E−05 | 8.5E−01 | 1.0E+00 | 9.3E−01 | 8.5E−01 | 2.6E−03 | 1.5E−01 | 1.2E−02 | 6.0E+01 |

MOS | 6.2E−01 | 2.1E−01 | 1.0E+00 | 8.9E−01 | 6.7E−03 | 3.4E−03 | 8.4E−03 | 1.4E−02 | 1.0E−02 | 8.5E−01 | 7.3E−02 | 4.3E−04 | 1.9E−01 | 9.3E─01 | 1.0E+00 | 5.2E−02 | 5.7E−03 | 4.8E−02 | 6.1E−05 | 6.0E+01 |

RO | 6.6E−01 | 6.2E−01 | 1.9E−01 | 9.8E−01 | 2.0E−03 | 3.4E−03 | 1.8E−02 | 3.8E−02 | 2.6E−01 | 1.9E−01 | 8.3E−02 | 3.1E−04 | 8.9E−01 | 8.5E−01 | 5.2E−02 | 1.0E+00 | 5.7E−03 | 4.8E−02 | 6.1E−05 | 6.0E+01 |

SACC | 5.7E−03 | 1.2E−02 | 5.7E−03 | 2.6E−03 | 1.9E−01 | 8.0E−01 | 8.0E−01 | 2.0E−02 | 5.7E−03 | 1.4E−02 | 4.1E−02 | 1.1E−01 | 3.9E−01 | 2.6E−03 | 5.7E−03 | 5.7E−03 | 1.0E+00 | 2.2E−02 | 2.1E−01 | 6.0E+01 |

SHADEILS | 4.8E−02 | 6.4E−02 | 5.5E−02 | 4.1E−02 | 1.0E−02 | 8.4E−03 | 8.5E−04 | 4.8E−02 | 2.2E−02 | 5.5E−02 | 4.8E−02 | 8.5E−04 | 1.5E−03 | 1.5E−01 | 4.8E−02 | 4.8E−02 | 2.2E−02 | 1.0E+00 | 1.0E−02 | 6.0E+01 |

VMODE | 6.1E−05 | 6.1E−05 | 6.1E−05 | 1.8E−04 | 8.9E−01 | 2.3E−01 | 4.2E−01 | 2.0E−03 | 6.1E−05 | 6.1E−05 | 2.1E−01 | 2.2E−02 | 1.5E−02 | 1.2E−02 | 6.1E−05 | 6.1E−05 | 2.1E−01 | 1.0E−02 | 1.0E+00 | 6.0E+01 |

Algorithm | DEEPSO | DSRegPSO | GPSO | Accum. Error (%) |
---|---|---|---|---|

DEEPSO | 1.0E+00 | 1.8E−02 | 1.2E−02 | 9.8E+00 |

DSRegPSO | 1.8E−02 | 1.0E+00 | 4.3E−04 | 9.8E+00 |

GPSO | 1.2E−02 | 4.3E−04 | 1.0E+00 | 9.8E+00 |

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## Share and Cite

**MDPI and ACS Style**

Rivera, M.M.; Guerrero-Mendez, C.; Lopez-Betancur, D.; Saucedo-Anaya, T.
Dynamical Sphere Regrouping Particle Swarm Optimization: A Proposed Algorithm for Dealing with PSO Premature Convergence in Large-Scale Global Optimization. *Mathematics* **2023**, *11*, 4339.
https://doi.org/10.3390/math11204339

**AMA Style**

Rivera MM, Guerrero-Mendez C, Lopez-Betancur D, Saucedo-Anaya T.
Dynamical Sphere Regrouping Particle Swarm Optimization: A Proposed Algorithm for Dealing with PSO Premature Convergence in Large-Scale Global Optimization. *Mathematics*. 2023; 11(20):4339.
https://doi.org/10.3390/math11204339

**Chicago/Turabian Style**

Rivera, Martín Montes, Carlos Guerrero-Mendez, Daniela Lopez-Betancur, and Tonatiuh Saucedo-Anaya.
2023. "Dynamical Sphere Regrouping Particle Swarm Optimization: A Proposed Algorithm for Dealing with PSO Premature Convergence in Large-Scale Global Optimization" *Mathematics* 11, no. 20: 4339.
https://doi.org/10.3390/math11204339