## 2 ARBF-MLPA Method
### Performance Test of Optimization Algorithm
#### 2.5.3 Test of Mathematical Functions of Optimization Algorithm  
This section tests the convergence performance of ARBF-MLAP algorithm using the test functions and performance evaluation indicators described above. In the meantime, in order to verify the effectiveness and fast convergence of the ARBF-MLAP method, the RBF-UDPA method is selected for comparison. The traditional RBF point selection does not have the advantages of ARBF. For the sake of test fairness, 5 sample points are uniformly selected on the optimal front in finding the solution of the multi-objective function established by RBF, in order to be consistent with the point selection method of ARBF. So, this uniform point selection method is called the radial basis function uniform point-adding (RBF-UDPA) method. The selection of MMD and MMD-NB points uses the same method as that of RBF and ARBF.  
For details of the iteration of different evaluation indicators of RBF-UDPA and ARBF-MLPA, function and the comparison between test front and real front edge are shown under each test in Figures 4-7.  
As shown in Figures 4 and 5, under the two test functions (ZDT1 ZDT2), ARBF-MLPA converges at generation 8, while RBF-UDPA does not converge until generation 10. In addition, based on IGD, Spacing and HV evaluation criteria, ARBF-MLPA is superior to RBF-UDPF in all performance indicators.  
As shown in Figure 6, under the test function ZDT3, ARBF-MLPA converges at generation 9, while RBF-UDPA does not converge until generation 13. In addition, based on IGD, Spacing and HV evaluation criteria, ARBF-MLPA is superior to RBF-UDPF in all performance indicators.  
As shown in Figure 7, ARBF-MLPA is closer to the real front edge of three test functions.  
To sum up, ARBF-MLPA is superior to RBF-UDPA in terms of total convergence speed and the final optimal front distribution. Accordingly, ARBF-MLPA can save computation time and obtain more accurate solution in solving multi-objective problems and in its application to practical engineering problems.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Function** & **Range of Variable** & **Function Expression** & **Features of Front** \\ \hline ZDT1 & [0, 1] & \(\left\{\begin{matrix}\min~{}\mathrm{f}_{1}(\mathrm{x}_{1})=\mathrm{x}_{1}\\ \min~{}\mathrm{f}_{2}(\mathrm{x}_{2})=\mathrm{g}(1-\sqrt{\mathrm{f}_{1}/ \mathrm{g}})\\ \mathrm{g}(\mathrm{x})=1+9\sum\limits_{i=2}^{\mathrm{m}}\mathrm{x}_{i}/( \mathrm{m}-1)\end{matrix}\right.\) & Continuous convex \\ ZFDT2 & [0, 1] & \(\left\{\begin{matrix}\min~{}\mathrm{f}_{1}(\mathrm{x}_{1})=\mathrm{x}_{1}\\ \min~{}\mathrm{f}_{2}(\mathrm{x})=\mathrm{g}\Big{(}1-\left(\mathrm{f}_{1}/ \mathrm{g}\right)^{2}\Big{)}\\ \mathrm{g}(\mathrm{x})=1+9\sum\limits_{i=2}^{\mathrm{m}}\mathrm{x}_{i}/( \mathrm{m}-1)\end{matrix}\right.\) & Continuous concave \\ ZDT3 & [0, 1] & \(\left\{\begin{matrix}\min~{}\mathrm{f}_{1}(\mathrm{x}_{1})=\mathrm{x}_{1}\\ \min~{}\mathrm{f}_{2}(\mathrm{x})=\mathrm{g}\big{(}1-\sqrt{\mathrm{f}_{1}/ \mathrm{g}}-\left(\mathrm{f}_{1}/\mathrm{g}\right)\sin(10\pi\mathrm{f}_{1}) \big{)}\\ \mathrm{g}(\mathrm{x})=1+9\sum\limits_{i=2}^{\mathrm{m}}\mathrm{x}_{i}/( \mathrm{m}-Figure 4: (a) IGD under function ZDT1. (b) Spacing under function ZDT1. (c) HV under function ZDT1.  
Figure 5: (**a**) ICD under function ZDT2. (**b**) Spacing under function ZDT2. (**c**) HV under function ZDT2.  
Figure 6: (a) IGD under function ZDT3. (b) Spacing under function ZDT3. (c) HV under function ZDT3.  
Figure 7: (**a**) Front comparison under function ZDT1. (**b**) Front comparison under function ZDT2. (**c**) Front comparison under function ZDT3.