# A Stigmergy-Based Differential Evolution
## 4 Experimental Results
### Convergence Speed

Table 6: \(Cont.\)  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{3}{c}{**Final Fitness \(\mu(\sigma)\)**} & \multicolumn{3}{c}{**Final Iterations \(\mu(\sigma)\)**} \\ \hline
**Function** & **SDE** & **DE** & **SDE** & **DE** \\ \hline F1: Sphere & 0.0096 (\(3.5238\times 10^{-4}\)) & 0.0096 (\(3.2083\times 10^{-4}\)) & **857.26 (21.30)** & 3832.63 (106.76) \\ F2: Step & 0.00 (0) & 0.00 (0) & **829.8 (28.91)** & 4612.2 (143.3675) \\ F3: Sum Powers & 0.0074 (0.0022) & **0.0073 (0.0021)** & 53.16 (0.95) & **29.86 (5.96)** \\ F4: Sum Squares & 0.0097 (\(3.0754\times 10^{-4}\)) & **0.0095 (4.5164 \(\times\) 10\({}^{-4}\))** & **1266.10 (18.23)** & 5562.43 (122.29) \\ F5: Dixon and Price & 39.2690 (17.583) & **0.6667 (8.0515 \(\times\) 10\({}^{-7}\))** & 20,000.00 (0) & 20,000.00 (0) \\ F6: Rosenbrock & 36.73601 (130.1247) & **90.104 (0.1371)** & 20,000.00 (0) & 20,000.00 (0) \\ F7: Schwefel 2.22 & 0.0098 (1.2780 \(\times\) 10\({}^{-4}\)) & 0.0098 (2.2375 \(\times\) 10\({}^{-4}\)) & **1227.1 (9.36)** & 6314.47 (136.45) \\ F8: Trid & \(7.889\times 10^{6}\) (9.809 \(\times\) 10\({}^{6}\)) & **4.821 \(\times\) 10\({}^{5}\)(1.97\(\times\) 10\({}^{5}\))** & 20,000.00 (0) & 20,000.00 (0) \\ F9: [PERSON] & **0.2932 (0.13712)** & 262.7573 (19.5592) & 20,000.00 (0) & 20,000.00 (0) \\ F10: [PERSON] & 0.0098 (9.9644 \(\times\) 10\({}^{-5}\)) & **0.0099 (1.7791 \(\times\) 10\({}^{-4}\))** & **1367.93 (24.07)** & 6499.70 (131.53) \\ F11: [PERSON] & **4.3725 (2.6246)** & 592.316 (17.7795) & **1884.16 (467.1)** & 20,000.00 (0) \\ F12: Schwefel & **55.5158 (74.2989)** & 3723.5101 (5356.9) & **10405.80 (9128.0)** & 20,000.00 (0) \\ F13: [PERSON] & 0.0097 (3.3617 \(\times\) 10\({}^{-4}\)) & **0.0095 (3.5729 \(\times\) 10\({}^{-4}\))** & **1307.30 (19.59)** & 5935.20 (107.78) \\ F14: [PERSON] & 0.0097 (2.7136 \(\times\) 10\({}^{-4}\)) & **0.0095 (4.6494 \(\times\) 10\({}^{-4}\))** & **988.06 (18.48)** & 6339.90 (175.04) \\ F15: Treccani & 0.0062 (0.0032) & **0.0048 (0.0032)** & 32.53 (11.66) & **6.53 (5.33)** \\ F16: Beale & 0.0058 (0.0034) & 0.0058 (0.0025) & 92.83 (49.57) & 7.73 (**4.69)** \\ F17: Booth & 0.0071 (0.0032) & **0.0044 (0.0032)** & 89.83 (40.78) & **21.4667 (7.64)** \\ F18: Brent & 0.0076 (0.0021) & **0.0048 (0.0029)** & 69.42 (22.60) & 15.74 (4.66) \\ F19: Cube & 0.0062 (0.0031) & **0.0054 (0.0024)** & 243.23 (179.04) & 37 (**12.16)** \\ F20: Davis & 0.0709 (0.0703) & **0.0064 (0.0018)** & 453.33 (59.00) & 153.03 (**10.76)** \\ F21: Matysa & 0.0065 (0.0032) & **0.0050 (0.0034)** & 47.46 (24.72) & 3.16 (**2.61**) \\ F22: [PERSON] & **0.0046 (0.0031)** & 0.0048 (0.0029) & 193.36 (97.32) & **19.07 (10.65)** \\ F23: [PERSON] & 0.0067 (0.0029) & **0.0049 (0.0025)** & 84.83 (40.78) & 36.23 (**5.48)** \\ F24: Egg Crate & 0.0061 (0.0029) & **0.0039 (0.0029)** & 43.86 (11.56) & 15.8 (**5.67**) \\ F25: [PERSON] & 0.0060 (0.0033) & **0.0058 (0.0026)** & 126.66 (45.06) & 37.1 (**5.87**) \\ F26: [PERSON] & 0.0050 (Table 7 shows the experimental instance when \(D=100\) and \(N_{p}=256\). In this case, both algorithms have a good performance with the Sphere, Step, Sum Powers, Sum Squares, Schwefel 2.22, Ackley, Griewank and Levy functions, as they both achieve a good average fitness; however, except for the Sum Powers function where DE converges in fewer iterations (almost in half of the iterations), SDE is faster than DE in all other cases, with a reduction of more than 84% in the number of iterations for the best result (Levy function), as shown in Figure 3. For the functions Trid, Zakharov, Rastrigin and Schwefel, both algorithms have problems finding the global best, but sometimes SDE obtains better fitness values than DE, with fewer iterations on average. Table 8 shows that the SDE behavior for more individuals (\(N_{p}=512\)) is as reported in previous lines. From Tables 9 and 10 (\(D=500\)), it can be seen that DE struggles with high dimensionality, except for the Sum Powers function.