8 A Review of Major Application Areas of Differential Evolution <s> Test Problem 4. <s> In this paper, we provide a systematic comparison of various evolutionary approaches to multiobjective optimization using six carefully chosen test functions. Each test function involves a particular feature that is known to cause difficulty in the evolutionary optimization process, mainly in converging to the Pareto-optimal front (e.g., multimodality and deception). By investigating these different problem features separately, it is possible to predict the kind of problems to which a certain technique is or is not well suited. However, in contrast to what was suspected beforehand, the experimental results indicate a hierarchy of the algorithms under consideration. Furthermore, the emerging effects are evidence that the suggested test functions provide sufficient complexity to compare multiobjective optimizers. Finally, elitism is shown to be an important factor for improving evolutionary multiobjective search. <s> BIB001 </s> 8 A Review of Major Application Areas of Differential Evolution <s> Test Problem 4. <s> Though it has been claimed that elitism could improve evolutionary multi-objective search significantly, a thorough and extensive evaluation of its effects is still missing. Guidelines on how elitism could successfully be incorporated have not yet been developed. This paper presents a unified model of multi-objective evolutionary algorithms, in which arbitrary variation and selection operators can be combined as building blocks, including archiving and re-insertion strategies. The presented model enables most specific multi-objective (evolutionary) algorithm to be formulated as an instance of it, which will be demonstrated by simple examples. We further show how elitism can be quantified by the model's parameters and how this allows an easy evaluation of the effect of elitism on different algorithms. <s> BIB002
This test problem is defined as: and it has 21 9 local Pareto fronts. All experiments were performed in parallel, using the PVM communication library. For the maintenance of the Pareto optimal set, the archiving technique described in  , which uses an external archive, was employed. The obtained results were compared to that of VEGA algorithm. For this purpose, two established measures, namely the C measure  BIB001 , and the V measure  BIB002 were employed. Metric C(A, B) measures the fraction of members of the Pareto front B that are dominated by members of the Pareto front A, while V(A, B) is the fraction of the volume of the minimal hypercube containing both fronts, that is strictly dominated by members of A but is not dominated by members of B  . Following the analysis presented in BIB001 , a total number of 100 individuals divided in several populations, as well as a maximum of 250 iterations per population per run, were used. We performed 30 experiments for each test problem, using the DE with the mutation operator of Equation (1), because it suits better the migration scheme described in the previous section. This variant is denoted as VEDE1. The results are reported in the boxplots of Figures. 7-10. Each boxplot depicts the obtained values of the corresponding measure, in 30 experiments. The box has lines at the lower quartile, median, and upper quartile values. The lines extending from each end of the box (whiskers) show the extent of the rest of the data. The outliers, i.e. the values that lie beyond the ends of the whiskers, are denoted with crosses. DE is quite sensitive to population size, especially when the number of individuals becomes too small. This was verified in our preliminary experiments with VEDE. Dividing the 100 individuals into more than 5 populations (less than 20 individuals per population) resulted in substantial performance decline. Thus, our experiments were performed using 2 up to 5 populations. Standard values for the μ and ρ parameters, equal to 0.7 and 0.9, respectively, were used. The speedup gained from the parallel implementation using up to 5 nodes is depicted in Figure 6 . As illustrated, there is an almost linearly increasing speedup rate using up to 4 nodes. Beyond 4 nodes, the speedup rate increases marginally. This effect can be attributed to the small number of individuals per population, which falls under 20. Additionally, we have tested the remaining DE mutation operators and in all cases, VEDE outperformed the VEGA with respect to the two metrics, C and V. However, our results support the claim that VEDE, just like DE, is sensitive to population size.