## 4 Results

Table 2: _Cont._  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Function** & **DPSO** & **PSO** & **GA** & **DE** \\ \hline \(f_{0}\) Norm & 1.000 & **3.179\(\times\)10\({}^{-26}\)** & _3.198\(\times\)10\({}^{-18}\)_ & _0.000_ \\ \hline \(f_{1}\) Norm & **4.177\(\times\)10\({}^{-2}\)** & 1.000 & _9.584\(\times\)10\({}^{-1}\)_ & _0.000_ \\ \hline \(f_{2}\) Norm & **5.512\(\times\)10\({}^{-27}\)** & 1.000 & _1.790\(\times\)10\({}^{-8}\)_ & _0.000_ \\ \hline \(f_{3}\) Norm & _1.038\(\times\)10\({}^{-1}\)_ & **8.430\(\times\)10\({}^{-7}\)** & 1.000 & _0.000_ \\ \hline \(f_{4}\) Norm & 1.000 & _1.695\(\times\)10\({}^{-15}\)_ & _0.000_ & **4.382\(\times\)10\({}^{-16}\)** \\ \hline \(f_{5}\) Norm & **4.974\(\times\)10\({}^{-26}\)** & 1.000 & _0.000_ & _4.589\(\times\)10\({}^{-18}\)_ \\ \hline \(f_{6}\) Norm & **5.333\(\times\)10\({}^{-1}\)** & 1.000 & _0.000_ & _7.938\(\times\)10\({}^{-1}\)_ \\ \hline \(f_{7}\) Norm & 1.000 & _9.388\(\times\)10\({}^{-1}\)_ & **2.211\(\times\)10\({}^{-1}\)** & _0.000_ \\ \hline \(f_{8}\) Norm & 1.000 & _9.366\(\times\)10\({}^{-1}\)_ & **2.205\(\times\)10\({}^{-1}\)** & _0.000_ \\ \hline \(f_{9}\) Norm & **3.572\(\times\)10\({}^{-2}\)** & 1.000 & _4.059\(\times\)10\({}^{-1}\)_ & _0.000_ \\ \hline \(f_{10}\) Norm & **1.183\(\times\)10\({}^{-2}\)** & _6.927\(\times\)10\({}^{-1}\)_ & 1.000 & _0.000_ \\ \hline \(f_{11}\) Norm & _0.000_ & 1.000 & _7.310\(\times\)10\({}^{-3}\)_ & **6.228\(\times\)10\({}^{-5}\)** \\ \hline \(f_{12}\) Norm & _0.000_ & 1.000 & **1.529\(\times\)10\({}^{-1}\)** & _2.407\(\times\)10\({}^{-1}\)_ \\ \hline \(f_{13}\) Norm & _0.000_ & 1.000 & _2.248\(\times\)10\({}^{-1}\)_ & **6.415\(\times\)10\({}^{-2}\)** \\ \hline \(f_{14}\) Norm & **4.245\(\times\)10\({}^{-1}\)** & 1.000 & _7.824\(\times\)10\({}^{-1}\)_ & _0.000_ \\ \hline \(f_{15}\) Norm & _0.000_ & 1.000 & **4.718\(\times\)10\({}^{-1}\)** & _8.685\(\times\)10\({}^{-1}\)_ \\ \hline \(f_{16}\) Norm & _0.000_ & 1.000 & _2.758\(\times\)10\({}^{-1}\)_ & **2.454\(\times\)10\({}^{-1}\)** \\ \hline \(f_{17}\) Norm & _0.000_ & 9.025\(\times\)10\({}^{-1}\)_ & **8.807\(\times\)10\({}^{-1}\)** & 1.000 \\ \hline \(f_{18}\) Norm & _0.000_ & 1.000 & _3.761\(\times\)10\({}^{-1}\)_ & **2.728\(\times\)10\({}^{-1}\)** \\ \hline \(f_{19}\) Norm & **8.134\(\times\)10\({}^{  
The results of this experiment show that DPSO and DE can be optimized through self-tuning given a sufficient number of iterations and that they both demonstrate satisfactory results. The tuned parameters allow these algorithms to remain largely effective when the target problem deviates from the in-sample function, i.e., when they are able to self-tune on an approximation of the out-of-sample problems and can be expected to perform well. Given the fact that the rules were applied equally to all algorithms, the primary cause for improvement in DPSO over PSO is the use of dimension-wise updates. A likely reason why a number of fitness results are relatively large is that ill-conditioning scales up the dimensional range, making it harder to find the global optimum. Regardless, every algorithm was able to demonstrate some improvement in each problem, e.g., ending with a fitness of \(10^{32}\) is a significant improvement when starting with \(10^{44}\). It is interesting that the Ellipse problems had the worst performance results despite the relative simplicity of its surface--likely attributed to the exponential nature of each axis. PSO and DPSO have momentum factors that can cause them to overshoot or orbit the minima, but this