# Cyber Firefly Algorithm Based on Adaptive Memory Programming for Global Optimization

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Related Work Materials and Methods

#### 2.1. Glowworm Swarm Optimization (GSO)

#### 2.2. Firefly Algorithm (FA)

#### 2.3. Adaptive Memory Programming (AMP)

#### 2.3.1. Scatter Search (SS)

#### 2.3.2. Path Relinking (PR)

## 3. Proposed Methods

#### 3.1. Multiple Guiding Points

#### 3.2. Luciferin-Proportional Movement

#### 3.3. Adaptive Local Search

_{1}fitness function evaluations. To leverage the balance between search efficiency and effectiveness, we adaptively vary the frequency parameter (t

_{1}) for the performed local search. The adaptive local search strategy is based on the analysis of the landscape observed in the objective space as will be noted.

#### 3.4. Multi-Start with PR for Firefly Swarm Rebuilding

_{2}, a signal for detection of a critical event is returned. The parameter t

_{2}is also made adaptive by the landscape analysis approach.

#### 3.5. Responsive Strategies Based on Landscape Analysis

_{1}, d

_{1}), …, (c

_{m}, d

_{m})}, the correlation coefficient is defined as

_{1}), the local objective landscape is considered to be asymptotical single-modal, and we perform two responsive strategies as follows: (1) increasing the distance, on average, of the firefly movement by lb = lb/λ and ub = ub/λ (0 < λ < 1); (2) increasing the elapsed iterations for checking the feasibility of executing the local search and multi-start search by t

_{1}= t

_{1}/λ and t

_{2}= t

_{2}/λ. The implication of the two responsive strategies is that when the local objective landscape is single-modal, the search may be more effective if the firefly movement distance is greater and the frequency for conducting the local search and the multi-start search is lower. On contrary, If the absolute FDC value for the current time period is less than a threshold (|FDC| < h

_{2}), the local objective landscape is considered to be asymptotical multi-modal, and we perform two responsive strategies as follows: (1) decreasing the distance, on average, of the firefly movement by lb = lb × λ and ub = ub × λ; (2) decreasing the elapsed iterations for checking the feasibility of executing the local search and multi-start search by t

_{1}= t

_{1}× λ and t

_{2}= t

_{2}× λ.

#### 3.6. Pseudo-Code of CFA

## 4. Experimental Results and Discussions

#### 4.1. Benchmark Test Functions and Algorithm Parameter Settings

#### 4.2. Analysis of CFA Strategies

#### 4.2.1. Selection Strategy for Multiple Guiding Solutions

#### 4.2.2. Adaptive Strategy for Local Search

#### 4.2.3. Multi-Start Strategy for Swarm Rebuilding

#### 4.2.4. Responsive Strategies Based on Landscape Analysis

#### 4.2.5. Worst-Case Analysis

#### 4.3. Performance Evaluation

_{p}− f* + ε)/(f

_{q}− f* + ε)

^{−7}, f

_{p}and f

_{q}are the mean best function values obtained by the competing algorithms p and q, and f* is the global minimum of the test function. As all the test functions involve minimization, we realize p outperforms q if Merit(p, q) < 1.0, p is inferior to q if Merit(p, q) > 1.0, and p and q perform equally well if Merit(p, q) = 1.0.

#### 4.3.1. Comparison with CFA Counterparts

^{−43}and 9.51 × 10

^{−32}in relation to the GSO and the FA, respectively. We further compare the CFA to the better one of the best function values obtained by the GSO and the FA, giving a merit product of 8.99 × 10

^{−29}as shown in the last column of Table 7. The result suggests that the CFA significantly outperforms its counterparts over the benchmark dataset, and that the CFA also beats a hybrid which takes the better one of the best function values obtained by the two counterparts.

_{a}(X

_{b}) and we want to testify the confidence regarding that X

_{a}is less than X

_{b}. A classic statistical test based on the central limit theorem for comparing two proportions is to approximate the mean X

_{a}− X

_{b}as a normal distribution if the collected number of samples is sufficiently large. Therefore, 30 independent runs of each competing algorithm were conducted. For each run, the online best function value obtained at a particular FE, say e, is a sample for X

_{a}(e), the random variable for the result obtained by method A at FE e. We tally the samples at every instance of FEs during the whole duration of executing the algorithm. By examining the mean curve of X

_{a}(e) and X

_{b}(e) as e increases during the evolution, we can testify if method A well outperforms method B. For clear illustration, the boundary and mean curves over the 30 runs are plotted. Figure 5a shows the online performance analysis with 95% confidence interval for the Rosenbrock (30) function. It is seen that the 95% confidence interval of the best function value obtained by the three competing algorithms (GSO, FA, and CFA) converges with various speeds. Our CFA converges at a much faster speed and reach towards a better function value than its two counterparts. The FA is the second-best performer followed by the GSO. To investigate the detailed performance during the second half duration of the execution, we enlarge the plot for this period as shown in Figure 5b. It can be seen that the CFA significantly outperforms the other two algorithms with 95% confidence level during the second half execution duration. We also found that the GSO performs better than the FA after 80,000 FEs, although the GSO may not beat the FA at the early stage of the execution as previously noted. The online performance comparison with 95% confidence level for the Rastrigin (30) function is shown in Figure 6. We observe that during the whole execution period the CFA significantly outperforms the GSO and the FA. The FA performs better than the GSO when the allowed number of consumed FEs is less than 20,000, but the FA is far surpassed by the GSO if more FEs are allowed. Figure 7 shows the online performance variation with 95% confidence level for the Griewank (30) function. Again, the CFA is the best performer among the three algorithms throughout the whole execution duration. However, we see a phenomenon differing from those for the two previous test functions with the GSO and the FA. The GSO and the FA performs about equally well before consuming 50,000 FEs, although the former is a more stable performer because it has a shorter confidence interval. However, after this critical execution period, the FA becomes very effective both in the convergence speed and the function value. As shown in Figure 7b, the FA significantly surpasses the GSO and reaches a comparative performance with the CFA.

#### 4.3.2. Comparison against Other Metaheuristics

^{−21}and 2.11 × 10

^{−36}, respectively. When we compare the CFA to the CSA, the merit product is 1.94 × 10

^{18}. The result seems to suggest that the CSA performs better on the dataset. However, if we take a closer look, the CSA is very effective in solving small functions with less than ten variables, thus CSA gives significantly greater merits for these functions. For the test functions with ten or more variables, the merit value turns to be in favor to the CFA, disclosing that the CFA is more effective than CSA in tackling larger-sized functions. It is worth noting that both CFA and the CSA take advantage of the features contained in the AMP domain, and the two algorithms extremely outperform the other compared algorithms in our experiments. This phenomenon discloses the potential of future research in the direction of marrying the AMP with other types of metaheuristics.

#### 4.3.3. Comparison on the CEC 2005 Dataset

## 5. Concluding Remarks and Future Research

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

EA | evolutionary algorithm |

GSO | glowworm swarm optimization |

FA | firefly algorithm |

AMP | adaptive memory programming |

CSA | cyber swarm algorithm |

CFA | Cyber Firefly Algorithm |

GRASP | greedy randomized adaptive search procedures |

GA | genetic algorithm |

PSO | particle swarm optimization |

VNS | variable neighborhood search |

SS | scatter search |

SS/PR | path relinking |

FDC | fitness distance correlation |

FE | function evaluations |

CMA-ES | covariance matrix adaptation evolution strategy |

GMM | geometric mean of the merits |

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**Figure 2.**Worst-case analysis with the Rosenbrock (30) function. (

**a**) The worst fitness obtained by the CFA program as the number of repetitive runs increases. (

**b**) The number of program runs with which each performance level is reached.

**Figure 3.**Worst-case analysis with the Rastrigin (30) function. (

**a**) The worst fitness obtained by the CFA program as the number of repetitive runs increases. (

**b**) The number of program runs with which each performance level is reached.

**Figure 4.**Worst-case analysis with the Griewank (30) function. (

**a**) The worst fitness obtained by the CFA program as the number of repetitive runs increases. (

**b**) The number of program runs with which each performance level is reached.

**Figure 5.**Online performance analysis with 95% confidence interval for the Rosenbrock (30) function. (

**a**) The convergence of the best function value during the whole duration of the execution. (

**b**) The convergence of the best function value during the second half duration of the execution.

**Figure 6.**Online performance analysis with 95% confidence interval for the Rastrigin (30) function. (

**a**) The convergence of the best function value during the whole duration of the execution. (

**b**) The convergence of the best function value during the second half duration of the execution.

**Figure 7.**Online performance analysis with 95% confidence interval for the Griewank (30) function. (

**a**) The convergence of the best function value during the whole duration of the execution. (

**b**) The convergence of the best function value during the second half duration of the execution.

No. | Function Name | Number of Variables (r) | Domain | Optimal Value |
---|---|---|---|---|

1 | Easom (2) | 2 | [−10, 10] | −1.0 |

2 | Shubert (2) | 2 | [−10, 10] | −186.7309 |

3 | Rosenbrock (2) | 2 | [−30, 30] | 0.0 |

4 | Zakharov (2) | 2 | [−5, 10] | 0.0 |

5 | De Jong (3) | 3 | [−5.12, 5.12] | 0.0 |

6 | Shekel (4, 5) | 4 | [0, 10] | −10.1532 |

7 | Shekel (4, 7) | 4 | [0, 10] | −10.4029 |

8 | Shekel (4, 10) | 4 | [0, 10] | −10.5364 |

9 | Sphere (10) | 10 | [−100, 100] | 0.0 |

10 | Rosenbrock (10) | 10 | [−30, 30] | 0.0 |

11 | Rastrigin (10) | 10 | [−5.12, 5.12] | 0.0 |

12 | Griewank (10) | 10 | [−600, 600] | 0.0 |

13 | Zakharov (10) | 10 | [−5, 10] | 0.0 |

14 | Sphere (20) | 20 | [−100, 100] | 0.0 |

15 | Rosenbrock (20) | 20 | [−30, 30] | 0.0 |

16 | Rastrigin (20) | 20 | [−5.12, 5.12] | 0.0 |

17 | Griewank (20) | 20 | [−600, 600] | 0.0 |

18 | Zakharov (20) | 20 | [−5, 10] | 0.0 |

19 | Sphere (30) | 30 | [−100, 100] | 0.0 |

20 | Rosenbrock (30) | 30 | [−30, 30] | 0.0 |

21 | Rastrigin (30) | 30 | [−5.12, 5.12] | 0.0 |

22 | Griewank (30) | 30 | [−600, 600] | 0.0 |

23 | Zakharov (30) | 30 | [−5, 10] | 0.0 |

Notation | Parameter Description | Set Value |
---|---|---|

Common parameters of CFA, GSO, and FA: | ||

Num_Run | Number of independent runs | 30 |

Max_FE | Maximum number of executed FE for each run | 160,000 |

N_{S} | Number of fireflies | 60 |

N* | Ideal number of local neighbors | 10 |

L | Lower bound of decision variable | See Domain in Table 1 |

U | Upper bound of decision variable | See Domain in Table 1 |

r_{max} | Maximum visibility radius of fireflies | (U − L) × 0.05 |

Additional parameters used by CFA: | ||

lb | Initial minimum step size (to be adaptively tuned) | (U − L) × 10^{−6} |

ub | Initial maximum step size (to be adaptively tuned) | (U − L) × 10^{−2} |

t_{1} | Number of evolution iterations for activating adaptive local search (to be adaptively tuned) | 20 |

t_{2} | Number of non-improving iterations for activating multi-start search (to be adaptively tuned) | 50 |

δ | Ratio of swarm fireflies to be rebuilt | 0.3 |

h_{1} | FDC single-modal threshold | 0.5 |

h_{2} | FDC multi-modal threshold | 0.4 |

λ | Scaling factor for performing responsive strategies | 0.5 |

**Table 3.**The mean best function value and the standard deviation obtained by using various selection strategies.

Functions | Roulette-Wheel Selection | Tournament Selection | Rank Selection | |||
---|---|---|---|---|---|---|

Mean | Std | Mean | Std | Mean | Std | |

Easom (2) | −1.0 | 0.0 | −1.0 | 0.0 | −1.0 | 0.0 |

Shubert (2) | −186.726 | 0.011381 | −186.723 | 0.010061 | −186.723 | 0.016331 |

Rosenbrock (2) | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |

Zakharov (2) | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |

De Jong (3) | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |

Shekel (4, 5) | −8.74128 | 2.436384 | −8.64565 | 2.5462 | −8.09613 | 3.003044 |

Shekel (4, 7) | −9.91291 | 1.345608 | −10.1835 | 1.11003 | −10.2611 | 0.440954 |

Shekel (4, 10) | −10.532 | 0.013263 | −10.511 | 0.130888 | −10.3554 | 0.966638 |

Sphere (10) | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |

Rosenbrock (10) | 0.000052 | 0.000031 | 0.000206 | 0.000982 | 0.002511 | 0.009392 |

Rastrigin (10) | 0.851670 | 0.878212 | 0.847894 | 0.930433 | 0.646338 | 0.800856 |

Griewank (10) | 0.023954 | 0.015559 | 0.018670 | 0.010681 | 0.016450 | 0.009914 |

Zakharov (10) | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |

Sphere (20) | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |

Rosenbrock (20) | 0.800503 | 1.594902 | 0.953237 | 1.679062 | 0.267050 | 0.994099 |

Rastrigin (20) | 3.052760 | 1.833682 | 2.607486 | 1.544556 | 2.466102 | 1.765681 |

Griewank (20) | 0.013792 | 0.020267 | 0.011079 | 0.010911 | 0.007213 | 0.012479 |

Zakharov (20) | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |

Sphere (30) | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |

Rosenbrock (30) | 5.924423 | 3.057484 | 2.962683 | 2.773249 | 0.801094 | 1.598004 |

Rastrigin (30) | 4.204611 | 2.066075 | 4.973816 | 1.826780 | 4.680690 | 2.632701 |

Griewank (30) | 0.009349 | 0.012921 | 0.008202 | 0.012808 | 0.003364 | 0.007581 |

Zakharov (30) | 0.000005 | 0.000005 | 0.0 | 0.0 | 0.0 | 0.0 |

**Table 4.**The mean best function value and the standard deviation obtained by CFA with or without adaptive local search.

Functions | No Adaptive Local Search | With Adaptive Local Search | ||
---|---|---|---|---|

Mean | Std | Mean | Std | |

Easom (2) | −0.99997 | 0.000020 | −1.0 | 0.0 |

Shubert (2) | −185.779 | 0.963969 | −186.723 | 0.016331 |

Rosenbrock (2) | 0.0 | 0.0 | 0.0 | 0.0 |

Zakharov (2) | 0.0 | 0.0 | 0.0 | 0.0 |

De Jong (3) | 0.0 | 0.0 | 0.0 | 0.0 |

Shekel (4, 5) | −6.50369 | 2.613879 | −8.09613 | 3.003044 |

Shekel (4, 7) | −7.40191 | 1.802712 | −10.2611 | 0.440954 |

Shekel (4, 10) | −7.80353 | 1.033856 | −10.3554 | 0.966638 |

Sphere (10) | 0.0 | 0.0 | 0.0 | 0.0 |

Rosenbrock (10) | 0.797320 | 1.594630 | 0.002511 | 0.009392 |

Rastrigin (10) | 0.734548 | 0.827677 | 0.646338 | 0.800856 |

Griewank (10) | 0.015448 | 0.011793 | 0.016450 | 0.009914 |

Zakharov (10) | 0.0 | 0.0 | 0.0 | 0.0 |

Sphere (20) | 0.0 | 0.0 | 0.0 | 0.0 |

Rosenbrock (20) | 0.797828 | 1.594400 | 0.267050 | 0.994099 |

Rastrigin (20) | 2.590780 | 1.391706 | 2.466102 | 1.765681 |

Griewank (20) | 0.012054 | 0.013121 | 0.007213 | 0.012479 |

Zakharov (20) | 0.0 | 0.0 | 0.0 | 0.0 |

Sphere (30) | 0.0 | 0.0 | 0.0 | 0.0 |

Rosenbrock (30) | 0.801375 | 1.599876 | 0.801094 | 1.598004 |

Rastrigin (30) | 4.704743 | 1.842029 | 4.680690 | 2.632701 |

Griewank (30) | 0.005663 | 0.007891 | 0.003364 | 0.007581 |

Zakharov (30) | 0.0 | 0.0 | 0.0 | 0.0 |

**Table 5.**The mean best function value and the standard deviation obtained by CFA with or without multi-start strategy.

Functions | No Multi-Start Rebuilding | With Multi-Start Rebuilding | ||
---|---|---|---|---|

Mean | Std | Mean | Std | |

Easom (2) | −1.0 | 0.0 | −1.0 | 0.0 |

Shubert (2) | −186.03 | 0.708244 | −186.723 | 0.016331 |

Rosenbrock (2) | 0.0 | 0.0 | 0.0 | 0.0 |

Zakharov (2) | 0.0 | 0.0 | 0.0 | 0.0 |

De Jong (3) | 0.0 | 0.0 | 0.0 | 0.0 |

Shekel (4, 5) | −5.56783 | 3.517801 | −8.09613 | 3.003044 |

Shekel (4, 7) | −9.12732 | 2.851160 | −10.2611 | 0.440954 |

Shekel (4, 10) | −7.65202 | 3.724529 | −10.3554 | 0.966638 |

Sphere (10) | 0.0 | 0.0 | 0.0 | 0.0 |

Rosenbrock (10) | 0.000006 | 0.000005 | 0.002511 | 0.009392 |

Rastrigin (10) | 1.790926 | 1.294722 | 0.646338 | 0.800856 |

Griewank (10) | 0.018083 | 0.015917 | 0.01645 | 0.009914 |

Zakharov (10) | 0.0 | 0.0 | 0.0 | 0.0 |

Sphere (20) | 0.0 | 0.0 | 0.0 | 0.0 |

Rosenbrock (20) | 0.931281 | 1.685579 | 0.267050 | 0.994099 |

Rastrigin (20) | 3.349695 | 1.807129 | 2.466102 | 1.765681 |

Griewank(20) | 0.008904 | 0.011122 | 0.007213 | 0.012479 |

Zakharov (20) | 0.0 | 0.0 | 0.0 | 0.0 |

Sphere (30) | 0.0 | 0.0 | 0.0 | 0.0 |

Rosenbrock (30) | 1.205757 | 1.838271 | 0.801094 | 1.598004 |

Rastrigin (30) | 4.974795 | 1.973266 | 4.680690 | 2.632701 |

Griewank (30) | 0.005735 | 0.011983 | 0.003364 | 0.007581 |

Zakharov (30) | 0.000001 | 0.000004 | 0.0 | 0.0 |

**Table 6.**The mean best function value and the standard deviation obtained by CFA with or without landscape analysis.

Functions | No Adaptive Landscape Analysis | With Adaptive Landscape Analysis | ||
---|---|---|---|---|

Mean | Std | Mean | Std | |

Easom (2) | −0.99997 | 0.000020 | −1.0 | 0.0 |

Shubert (2) | −185.779 | 0.963969 | −186.723 | 0.016331 |

Rosenbrock (2) | 0.0 | 0.0 | 0.0 | 0.0 |

Zakharov (2) | 0.0 | 0.0 | 0.0 | 0.0 |

De Jong (3) | 0.0 | 0.0 | 0.0 | 0.0 |

Shekel (4, 5) | −6.50369 | 2.613879 | −8.09613 | 3.003044 |

Shekel (4, 7) | −7.40191 | 1.802712 | −10.2611 | 0.440954 |

Shekel (4, 10) | −7.80353 | 1.033856 | −10.3554 | 0.966638 |

Sphere (10) | 0.0 | 0.0 | 0.0 | 0.0 |

Rosenbrock (10) | 0.797320 | 1.594630 | 0.002511 | 0.009392 |

Rastrigin (10) | 0.734548 | 0.827677 | 0.646338 | 0.800856 |

Griewank (10) | 0.015448 | 0.011793 | 0.016450 | 0.009914 |

Zakharov (10) | 0.0 | 0.0 | 0.0 | 0.0 |

Sphere (20) | 0.0 | 0.0 | 0.0 | 0.0 |

Rosenbrock (20) | 0.797828 | 1.594400 | 0.267050 | 0.994099 |

Rastrigin (20) | 2.590780 | 1.391706 | 2.466102 | 1.765681 |

Griewank (20) | 0.012054 | 0.013121 | 0.007213 | 0.012479 |

Zakharov (20) | 0.0 | 0.0 | 0.0 | 0.0 |

Sphere (30) | 0.0 | 0.0 | 0.0 | 0.0 |

Rosenbrock (30) | 0.801375 | 1.599876 | 0.801094 | 1.598004 |

Rastrigin (30) | 4.704743 | 1.842029 | 4.680690 | 2.632701 |

Griewank (30) | 0.005663 | 0.007891 | 0.003364 | 0.007581 |

Zakharov (30) | 0.0 | 0.0 | 0.0 | 0.0 |

Functions | (a) GSO | (b) FA | (c) Min (a, b) | (d) CFA | Merit (d, a) | Merit (d, b) | Merit (d, c) |
---|---|---|---|---|---|---|---|

Easom (2) | −0.999999 | −0.999999 | −0.999999 | −1.0 | 1.0 | 1.0 | 1.0 |

Shubert (2) | −184.059055 | −91.464488 | −184.059055 | −186.723 | 9.86 × 10^{−1} | 4.90 × 10^{−1} | 9.86 × 10^{−1} |

Rosenbrock (2) | 0.000007 | 0.0 | 0.0 | 0.0 | 6.50 × 10^{−2} | 1.0 | 1.0 |

Zakharov (2) | 0.0 | 0.0 | 0.0 | 0.0 | 1.0 | 1.0 | 1.0 |

De Jong (3) | 0.0 | 0.0 | 0.0 | 0.0 | 1.0 | 1.0 | 1.0 |

Shekel (4, 5) | −6.883340 | −3.339854 | −6.883340 | −8.09613 | 8.50 × 10^{−1} | 4.13 × 10^{−1} | 8.50 × 10^{−1} |

Shekel (4, 7) | −6.624701 | −2.593236 | −6.624701 | −10.2611 | 6.46 × 10^{−1} | 2.53 × 10^{−1} | 6.46 × 10^{−1} |

Shekel (4, 10) | −7.770920 | −2.211454 | −7.770920 | −10.3554 | 7.50 × 10^{−1} | 2.14 × 10^{−1} | 7.50 × 10^{−1} |

Sphere (10) | 0.021677 | 0.000045 | 0.000045 | 0.0 | 2.31 × 10^{−5} | 1.11 × 10^{−2} | 1.11 × 10^{−2} |

Rosenbrock (10) | 1.658934 | 1.070858 | 1.070858 | 0.002511 | 1.51 × 10^{−3} | 2.35 × 10^{−3} | 2.35 × 10^{−3} |

Rastrigin (10) | 2.499270 | 11.099298 | 2.499270 | 0.646338 | 2.59 × 10^{−1} | 5.82 × 10^{−2} | 2.59 × 10^{−1} |

Griewank (10) | 0.154192 | 17.666979 | 0.154192 | 0.01645 | 1.07 × 10^{−1} | 9.31 × 10^{−4} | 1.07 × 10^{−1} |

Zakharov (10) | 0.000330 | 0.000115 | 0.000115 | 0.0 | 1.51 × 10^{−3} | 4.31 × 10^{−3} | 4.31 × 10^{−3} |

Sphere (20) | 0.085764 | 0.000264 | 0.000264 | 0.0 | 5.83 × 10^{−6} | 1.89 × 10^{−3} | 1.89 × 10^{−3} |

Rosenbrock (20) | 8.473378 | 21.006502 | 8.473378 | 0.26705 | 3.15 × 10^{−2} | 1.27 × 10^{−2} | 3.15 × 10^{−2} |

Rastrigin (20) | 5.550878 | 32.150683 | 5.550878 | 2.466102 | 4.44 × 10^{−1} | 7.67 × 10^{−2} | 4.44 × 10^{−1} |

Griewank (20) | 0.553463 | 0.600160 | 0.553463 | 0.007213 | 1.30 × 10^{−2} | 1.20 × 10^{−2} | 1.30 × 10^{−2} |

Zakharov (20) | 0.002574 | 0.001387 | 0.001387 | 0.0 | 1.94 × 10^{−4} | 3.60 × 10^{−4} | 3.60 × 10^{−4} |

Sphere (30) | 0.151057 | 0.000761 | 0.000761 | 0.0 | 3.31 × 10^{−6} | 6.57 × 10^{−4} | 6.57 × 10^{−4} |

Rosenbrock (30) | 23.373028 | 31.808803 | 23.373028 | 0.801094 | 3.43 × 10^{−2} | 2.52 × 10^{−2} | 3.43 × 10^{−2} |

Rastrigin (30) | 9.135285 | 59.807848 | 9.135285 | 4.68069 | 5.12 × 10^{−1} | 7.83 × 10^{−2} | 5.12 × 10^{−1} |

Griewank (30) | 1.053057 | 0.010732 | 0.010732 | 0.003364 | 3.19 × 10^{−3} | 3.13 × 10^{−1} | 3.13 × 10^{−1} |

Zakharov (30) | 2.774662 | 0.008677 | 0.008677 | 0.000001 | 4.98 × 10^{−7} | 1.59 × 10^{−4} | 1.59 × 10^{−4} |

Merit Product | 7.32 × 10^{−43} | 9.51 × 10^{−32} | 8.99 × 10^{−29} |

Functions | (a) PSO | (b) GA | (c) CSA | (d) CFA | Merit (d, a) | Merit (d, b) | Merit (d, c) |
---|---|---|---|---|---|---|---|

Easom (2) | −1.0 | −1.0 | −1.0 | −1.0 | 1.0 | 1.0 | 1.0 |

Shubert (2) | −186.7309 | −186.7309 | −186.7309 | −186.7232 | 1.54 × 10^{4} | 1.54 × 10^{4} | 1.54 × 10^{4} |

Rosenbrock (2) | 0.0 | 0.1015 | 0.0 | 0.0 | 1.0 | 4.93 × 10^{−6} | 1.0 |

Zakharov (2) | 0.0 | 0.0 | 0.0 | 0.0 | 1.0 | 1.0 | 1.0 |

De Jong (3) | 0.0 | 0.0 | 0.0 | 0.0 | 1.0 | 1.0 | 1.0 |

Shekel (4, 5) | −6.6329 | −10.0535 | −10.1532 | −8.0961 | 5.84 × 10^{−1} | 2.06 × 10^{1} | 4.11 × 10^{6} |

Shekel (4, 7) | −8.0176 | −10.0637 | −10.4029 | −10.2611 | 5.94 × 10^{−2} | 4.18 × 10^{−1} | 2.84 × 10^{5} |

Shekel (4, 10) | −7.4195 | −10.0750 | −10.5364 | −10.3554 | 5.81 × 10^{−2} | 3.92 × 10^{−1} | 3.62 × 10^{5} |

Sphere (10) | 0.0 | 0.0009 | 0.0 | 0.0 | 1.0 | 5.55 × 10^{−4} | 1.0 |

Rosenbrock (10) | 0.4727 | 8.1009 | 0.1595 | 0.0025 | 5.29 × 10^{−3} | 3.09 × 10^{−4} | 1.57 × 10^{−2} |

Rastrigin (10) | 6.4672 | 0.0004 | 0.7464 | 0.6463 | 9.99 × 10^{−2} | 1.61 × 10^{3} | 8.66 × 10^{−1} |

Griewank (10) | 0.0644 | 0.0493 | 0.0474 | 0.0164 | 2.55 × 10^{−1} | 3.33 × 10^{−1} | 3.46 × 10^{−1} |

Zakharov (10) | 0.0 | 0.1809 | 0.0 | 0.0 | 1.0 | 2.76 × 10^{−6} | 1.0 |

Sphere (20) | 0.0 | 0.0069 | 0.0 | 0.0 | 1.0 | 7.25 × 10^{−5} | 1.0 |

Rosenbrock (20) | 0.3992 | 9.0326 | 0.4788 | 0.2671 | 6.69 × 10^{−1} | 2.96 × 10^{−2} | 5.58 × 10^{−1} |

Rastrigin (20) | 18.7052 | 0.0036 | 6.8868 | 2.4661 | 1.32 × 10^{−1} | 6.85 × 10^{2} | 3.58 × 10^{−1} |

Griewank (20) | 0.0227 | 0.0533 | 0.0128 | 0.0072 | 3.17 × 10^{−1} | 1.35 × 10^{−1} | 5.63 × 10^{−1} |

Zakharov (20) | 2.683 | 37.1846 | 0.0 | 0.0 | 1.86 × 10^{−7} | 1.34 × 10^{−8} | 1.0 |

Sphere (30) | 0.0 | 0.0226 | 0.0 | 0.0 | 1.0 | 2.21 × 10^{−5} | 1.0 |

Rosenbrock (30) | 9.2089 | 83.0118 | 0.3627 | 0.8011 | 8.70 × 10^{−2} | 9.65 × 10^{−3} | 2.21 |

Rastrigin (30) | 33.9281 | 0.0115 | 11.9425 | 4.6807 | 1.38 × 10^{−1} | 4.07 × 10^{2} | 3.92 × 10^{−1} |

Griewank (30) | 0.0093 | 0.0893 | 0.0052 | 0.0034 | 3.66 × 10^{−1} | 3.81 × 10^{−2} | 6.54 × 10^{−1} |

Zakharov (30) | 5.4347 | 136.7129 | 0.0 | 0.0 | 9.20 × 10^{−8} | 3.66 × 10^{−9} | 1.0 |

Merit Product | 8.80 × 10^{−21} | 2.11 × 10^{−36} | 1.94 × 10^{18} |

Functions | LFA | FAtidal | VSSFA | GDAFA | WFA | CLFA | CFA |
---|---|---|---|---|---|---|---|

Sphere (2) | 0.043 | 0.0 | − | − | − | − | 0.0 |

Rosenbrock (2) | 1.34 | 0.0076 | − | − | − | − | 0.0 |

Zakharov (2) | 0.4950 | 0.0 | − | − | − | − | 0.0 |

Sphere (10) | 17.5 | 0.0 | 0.0 | 0.0 | 0.0690 | 0.0 | 0.0 |

Rosenbrock (10) | 3.15 × 10^{4} | 9.93 | − | − | − | − | 0.0025 |

Rastrigin (10) | 85.5 | 7.20 | 8.6769 | 0.0 | 10.6039 | 11.4284 | 0.6463 |

Griewank (10) | 0.0069 | − | 0.0 | 0.0 | 0.0075 | 0.0 | 0.0164 |

Zakharov (10) | 158.0 | 0.0 | − | − | − | − | 0.0 |

Sphere (30) | 14.0818 | − | 17.0501 | 1.61 × 10^{−5} | 0.2373 | 0.1942 | 0.0 |

Rastrigin (30) | 38.3584 | − | 133.9519 | 0.0619 | 42.7987 | 91.6579 | 4.6807 |

Griewank (30) | 0.4365 | − | 0.5380 | 3.65 × 10^{−6} | 0.0123 | 0.0099 | 0.0034 |

FEs | GAP | Merit | ||||
---|---|---|---|---|---|---|

1000 | 10,000 | 100,000 | 1000 | 10,000 | 100,000 | |

Min./Avg. | Min./Avg. | Min./Avg. | Min./Avg. | Min./Avg. | Min./Avg. | |

CFA | 572.3/751.2 | 317.6/554.4 | 212.4/441.1 | 1.0/1.0 | 1.0/1.0 | 1.0/1.0 |

CSA | 382.0/665.9 | 291.3/506.6 | 234.5/432.6 | 1.498/1.128 | 1.090/1.094 | 0.906/1.020 |

STS | 616.1/759.4 | 348.9/576.6 | 198.3/413.4 | 0.929/0.989 | 0.910/0.961 | 1.071/1.067 |

G-CMA-ES | 269.7/542.0 | 260.0/419.4 | 256.0/265.3 | 2.122/1.386 | 1.222/1.322 | 0.830/1.663 |

EDA | 669.9/1059.1 | 287.1/335.1 | 269.4/300.6 | 0.854/0.709 | 1.106/1.654 | 0.788/1.467 |

BLX-MA | 456.7/711.1 | 315.5/445.1 | 306.2/430.1 | 1.253/1.056 | 1.007/1.246 | 0.694/1.026 |

SPC-PNX | 621.7/750.3 | 279.6/391.0 | 206.0/309.9 | 0.921/1.001 | 1.136/1.418 | 1.031/1.423 |

BLX-GL50 | 676.0/716.3 | 272.8/341.0 | 257.2/319.0 | 0.847/1.049 | 1.164/1.626 | 0.826/1.383 |

L-CMA-ES | 289.0/825.7 | 225.9/655.8 | 202.7/411.1 | 1.980/0.910 | 1.406/0.845 | 1.048/1.073 |

DE | 715.4/914.1 | 396.7/492.4 | 228.8/272.0 | 0.800/0.822 | 0.801/1.126 | 0.928/1.622 |

K-PCX | 671.0/968.5 | 488.0/564.4 | 257.4/475.6 | 0.853/0.776 | 0.651/0.982 | 0.825/0.927 |

Co-EVO | 672.6/799.0 | 437.5/623.5 | 268.3/465.4 | 0.851/0.940 | 0.726/0.889 | 0.792/0.948 |

Merit Product | 2.833/0.663 | 0.949/5.493 | 0.238/8.195 | |||

Geometric Mean | 1.099/0.963 | 0.995/1.167 | 0.878/1.211 |

FEs | GAP | Merit | ||||
---|---|---|---|---|---|---|

1000 | 10,000 | 100,000 | 1000 | 10,000 | 100,000 | |

Min./Avg. | Min./Avg. | Min./Avg. | Min./Avg. | Min./Avg. | Min./Avg. | |

CFA | 792.5/1033.1 | 450.8/778.6 | 418.7/602.5 | 1.0/1.0 | 1.0/1.0 | 1.0/1.0 |

CSA | 629.2/785.0 | 454.4/647.8 | 420.6/578.2 | 1.260/1.316 | 0.992/1.202 | 0.995/1.042 |

STS | 829.3/957.0 | 614.9/747.3 | 431.3/540.3 | 0.956/1.080 | 0.733/1.042 | 0.971/1.115 |

G-CMA-ES | 570.3/658.4 | 414.4/526.8 | 405.7/493.0 | 1.390/1.569 | 1.088/1.478 | 1.032/1.222 |

EDA | 39,742/63,491 | 11,951/26,418 | 653.6/934.7 | 0.020/0.016 | 0.038/0.029 | 0.641/0.645 |

BLX-MA | 792.9/1198.7 | 443.9/502.4 | 410.7/457.2 | 0.999/0.862 | 1.016/1.550 | 1.019/1.318 |

SPC-PNX | 29,793/74,050 | 637.6/850.1 | 414.8/430.0 | 0.027/0.014 | 0.707/0.916 | 1.009/1.401 |

BLX-GL50 | 8545.4/20,008 | 474.8/545.9 | 433.0/507.5 | 0.093/0.052 | 0.949/1.426 | 0.967/1.187 |

L-CMA-ES | 790.8/1009.8 | 447.6/722.6 | 404.6/617.0 | 1.002/1.023 | 1.007/1.077 | 1.035/0.976 |

DE | 3473.3/14,461 | 726.0/781.8 | 558.7/592.0 | 0.228/0.071 | 0.621/0.996 | 0.749/1.018 |

K-PCX | 27,749/108,623 | 27,719/108,602 | 866.1/2257.2 | 0.029/0.010 | 0.016/0.007 | 0.483/0.267 |

Co-EVO | 908.5/1025.8 | 7496/822.0 | 625.3/734.5 | 0.872/1.007 | 0.060/0.947 | 0.670/0.820 |

Merit Product | 4.7 × 10^{−7}/2.0 × 10^{−8} | 1.2 × 10^{−5}/8.0 × 10^{−4} | 1.5 × 10^{−1}/4.3 × 10^{−1} | |||

Geometric Mean | 0.266/0.195 | 0.358/0.523 | 0.846/0.927 |

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## Share and Cite

**MDPI and ACS Style**

Yin, P.-Y.; Chen, P.-Y.; Wei, Y.-C.; Day, R.-F.
Cyber Firefly Algorithm Based on Adaptive Memory Programming for Global Optimization. *Appl. Sci.* **2020**, *10*, 8961.
https://doi.org/10.3390/app10248961

**AMA Style**

Yin P-Y, Chen P-Y, Wei Y-C, Day R-F.
Cyber Firefly Algorithm Based on Adaptive Memory Programming for Global Optimization. *Applied Sciences*. 2020; 10(24):8961.
https://doi.org/10.3390/app10248961

**Chicago/Turabian Style**

Yin, Peng-Yeng, Po-Yen Chen, Ying-Chieh Wei, and Rong-Fuh Day.
2020. "Cyber Firefly Algorithm Based on Adaptive Memory Programming for Global Optimization" *Applied Sciences* 10, no. 24: 8961.
https://doi.org/10.3390/app10248961