# Differential Evolution with Linear Bias Reduction in Parameter Adaptation
## 3 Results  
The evaluation of the linear bias reduction approach was performed on two sets of benchmark problems, in particular the CEC 2017 [5] and CEC 2020 [42] competitions on bound-constrained single-objective optimization. These two sets of test problems were chosen because they represent two different scenarios: the CEC 2017 tests algorithms on 30 different 10, 30, 50, and 100\(D\) problems with a limited computational resource, and CEC 2020 has only 10 problems in 5, 10, 15, and 20 dimensions, but allows exponential growth of the computational resource available. For CEC 2017 problems, the resource was set to 10,000\(D\) fitness evaluations, and for CEC 2020 it was set to 50,000, \(10^{6}\), \(3\times 10^{6}\), and \(10^{7}\) for 5D, 10D, 15D, and 20\(D\) respectively. All test problems were shifted and rotated to avoid the cheating of the algorithms. The LBR modification was tested with the L-SHADE, jSO, and DISH algorithms, and the following parameters were set for all algorithms: initial population size \(NP_{max}=25\cdot\log(D)\cdot\sqrt{D}\) for CEC 2017, \(NP_{max}=30\cdot D\cdot\sqrt{D}\) for CEC 2020, final population size \(NP_{min}=4\), \(pbest=0.17\), archive size \(NA=NP\), number of memory cells \(H=5\). The initial value of \(p_{max}\) in LBR was set to 40 for \(F\) update and 32 for \(Cr\) update; these values were chosen according to the experimental results reported in [12]. The algorithm was implemented in C++ with GCC and run on a PC with Ubuntu 20.04, an Intel Core i7 8700 k processor, and 32 GB RAM, with the results' post-processing performed using Python 3.6. The comparison of the results was performed for L-SHADE, jSO, and DISH with and without LBR. In the case that LBR was not used, the \(p\)-values in the Lehmer mean were set to two. The Mann-Whitney statistical test with normal approximation, tie-breaking, and the significance level set to 0.01 was applied to identify the difference between algorithm modifications. Table 1 presents the results of comparing the basic L-SHADE with L-SHADE-LBR on all dimensions of the CEC 2017 test problems. The values in the table show the number of wins (+), non-significant differences (=), and losses (\(-\)) summed over all functions. The results presented in Table 1 show that the LBR gives up to 16 improvements for L-SHADE on 100-dimensional functions and 17 improvements for 50-dimensional functions. The jSO and DISH algorithms, which perform better than L-SHADE, and LBR give smaller improvements for these algorithms. Furthermore, the greater the dimension of the test problems, the more improvements are achieved, i.e., for example, for 10-dimensional problems, the LBR does not deliver any improvements, and for L-SHADE and jSO, there are even several performance deteriorations. Most of the improvements are for multimodal, hybrid, and composition functions. Tables 2-5 contain the results for all CEC 2017 functions for all dimensions. The results presented in Tables 2-5 show that the linear bias reduction has the largest influence on 50- and 100-dimensional functions. Best values for each function are highlighted with bold. Table 6 contains the Mann-Whitney tests for the CEC 2020 benchmark problems. The results presented in Table 6 shows that the LBR allows up to two improvements for the L-SHADE algorithm for 10\(D\), 15\(D\), and 20\(D\), and one decreased performance. The improvements were mainly for functions f5 and f6, i.e., hybrid functions. For the jSO algorithm, there were up to three improvements for 15D, and for the DISH algorithm, which also included jSO adaptations, there were up to two improvements, which were observed for f5, f6, and f7.