# A Modified Cloud Particles Differential Evolution Algorithm for Real-Parameter Optimization
## 4 Experiments and Discussion
### Comparison with Nine State-of-the-Art Intelligent Algorithms on 10 and 30 Dimension  
In this part, MCPDE is compared with PSO, PSOcf (PSO with constriction factor) [58], TLBO, DE, JADE, CoDE, jDE, CMA-ES and CPDE. The appropriate parameters are important for the performance of the intelligent optimization algorithms. Therefore, the setting of parameters of different algorithms is given in the following:  
For MCPDE, The population size \(N\) is set to 13 \(\times\)\(D\). The maximum size of the archive is set to 1.5 \(\times\)\(N\). For DE, the population size \(N\) is set to 100. \(F\) and \(CR\) are set to 0.5 and 0.9, respectively. For PSO, the population size \(N\) is set to 40, the linearly decreasing inertia \(\omega\) from 0.9 to 0.4 is adopted over the course of the search, and the acceleration coefficients \(c_{1}\), \(c_{2}\) are both set to 1.49445. For JADE, the population size \(N\) is set to 100, \(p=0.05\) and \(c=0.1\). The parameters of other algorithms are the same as those used in the corresponding references. The statistical results, in terms of \(F_{man}\), \(SD\), \(Max\) and \(Min\) obtained in 30 independent runs by each algorithm, are reported in Tables 1 and 2. (1) _Unimodal problems \(f_{1}\)-\(f_{5}\)_: From the statistical results of Tables 1 and 2, we can see that MCPDE is better than other compared algorithms on unimodal problems \(f_{1}\)-\(f_{5}\) according to the average rank (Avg-rank) for 10 dimensions and 30 dimensions. Considering \(f_{1}\)-\(f_{5}\) with 10 dimensions, for \(f_{1}\), MCPDE, DE, JADE, jDE, CMA-ES and CPDE work well and obtain better results. For \(f_{2}\), MCPDE performs better than other algorithms except JADE and CMA-ES. Moreover, for \(f_{3}\), MCPDE performs significantly better than the compared algorithms. For \(f_{4}\), MCPDE, CMA-ES and CPDE beat other compared algorithms. For \(f_{5}\), MCPDE, DE, JADE, jDE and CPDE are better than other algorithms. On \(f_{1}\)-\(f_{5}\), MCPDE performs better than PSO, PSOcf, TLBO, DE, JADE, CoDE, jDE, CMA-ES and CPDE on 5, 5, 5, 3, 2, 5, 3, 2, 2 test problems, respectively. The overall ranking sequences for unimodal problems are MCPDE, CMA-ES, DE, CPDE, JADE, jDE, TLBO, CoDE, PSO and PSOcf in descending direction. When the search space dimension \(D\) is set to 30, according to Table 2, MCPDE is much better than PSO, PSOcf, TLBO, DE, JADE, CoDE, jDE, CMA-ES and CPDE on 5, 5, 5, 3, 3, 5, 3, 4 test functions, respectively. MCPDE performs better than all compared algorithms for the unimodal problems except \(f_{2}\) and \(f_{4}\). For \(f_{2}\) and \(f_{4}\), MCPDE ranks secondly. The overall ranking sequences for unimodal problems are MCPDE, jDE, DE, CMA-ES, JADE, CPDE, CoDE, PSO, TLBO and PSOcf in descending direction. The reason that MCPDE has the outstanding performance may be the use of the inertia factors, which are helpful for guiding the search direction. (2) _Multimodal problems \(f_{6}\)-\(f_{20}\)_: Considering the multimodal functions \(f_{6}\)-\(f_{20}\) in Table 3, MCPDE is significantly better than other algorithms on\(f_{6}\),\(f_{7}\),\(f_{9}\) and \(f_{20}\). Considering \(f_{8}\), most of the compared algorithms can achieve the similar results except CMA-ES. MCPDE beats most of the compared algorithms except that JADE and jDE have a similar performance on \(f_{11}\). JADE performs best on \(f_{12}\), \(f_{13}\),\(f_{15}\) and \(f_{18}\)-\(f_{19}\). jDE performs best on \(f_{14}\). CMA-ES performs best on \(f_{10}\) and \(f_{16}\). JADE and jDE perform best on \(f_{17}\). On these 15 multimodal problems, MCPDE performs better than PSO, PSOcf, TLBO, DE, JADE, CoDE, jDE, CMA-ES and CPDE on 14, 12, 14, 14, 4, 14, 9, 13 and 11 problems respectively. The overall ranking sequences on multimodal problems are MCPDE, JADE, jDE, CPDE, PSO, TLBO, CoDE, DE, PSOcf and CMA-ES in descending direction.