# Investigation of the iCC Framework Performance for Solving Constrained LSGO Problems +
## 4 Numerical Experiments and Results  
### Benchmark Set for Constrained Large-Scale Global Optimization Problems  
Today, there is no special cLSGO benchmark set to evaluate optimization algorithms. A new benchmark has been proposed based on scalable problems from the IEEE CEC 2017 Competition on constrained single objective real-parameter optimization (CSORPO) [26]. We selected and modified those problems from the IEEE CEC 2017 benchmark which did not use high-dimensional transformation matrixes (these matrixes have been manually designed for the specific number of variables only). The selected problems were scaled up to one thousand variables and were included in the proposed set of cLSGO problems. Table 2 contains information about the enumeration of functions that are included in the cLSGO benchmark and about types of objective functions. Here, N and S define non-separable and separable problems, respectively. Software Implementation and Setups, Benchmark Set for Constrained Large-Scale Global Optimization Problems  
To evaluate our algorithms, we used the following computational system. The operating system was Ubuntu Linux 18.04 LTS. To decrease computation time, we used two CPUs with 32 threads (Ryzen 7 1700x (8C/16T) and Ryzen 2700 (8C/16T)). The software was developed using the C++ language (g++(gcc) compiler) and Code:Blocks 17.12. To investigate the performance of each algorithm correctly, it is necessary to ensure equal experimental conditions. The dimension for all benchmark problems is \(D\) = 1000. The number of independent runs is 25 per benchmark problem. _MaxFEV_ is equal to 3E+6 for an independent run for each algorithm and for each problem. Investigation of the Performance of iCC Framework versus Classic Cooperative Coevolution with the Fixed Number of Subcomponents  
We use the following notation \"\(\varepsilon\)-CC-SHADE (m)\", where \(m\) is the total fixed number of subcomponents that the algorithm uses in the optimization procedure. We investigate \(\varepsilon\)-CC-SHADE (m) with different numbers of subcomponents (1, 2, 4, 8, and 10) and different numbers of population sizes (25, 50, 75, and 100). In addition, if \(m\) is equal to 1, then the algorithm optimizes a problem without the CC framework, because it uses only one subcomponent, which has a size equal to the optimization \begin{table}
\begin{tabular}{c l} \hline \hline
**Line** & **Pseudocode** \\ \hline
1: & Set _MaxFEV_, _pop\_size_ \\
2: & **while** (termination criterion is not satisfied) **do** \\
3: & Calculate m = Equation (10); \\
4: & **while** (i \(<\) m +1) **do** \\
5: & Divide optimization vector into m subcomponents (0); \\
6: & i = 1; \\
7: & Evaluate the i-th subcomponent with \(\varepsilon\)-SHADE\({}_{i}\) algorithm (0); \\
8: & i = i + 1; \\
9: & end **while** \\
10: & end **while** \\
11: & Return best-found value and finish optimization procedure (0); \\ \hline \hline \end{tabular}
\end{table} 
Table 1: \(\varepsilon\)-iCC-SHADE pseudocode. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c c} \hline \hline
**cLSGO Problem** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** & **10** & **11** & **12** & **13** & **14** & **15** & **16** & **17** & **18** \\ \hline CEC-2017 CSORPO & 1 & 3 & 4 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 \\ \hline Objective Type & N & N & S & S & S & S & S & S & S & N & N & S & S & N & S & S & N \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Scaled optimization problems for the constrained large-scale global optimization (cLSGO) benchmark set. vector. The \(\varepsilon\)-iCC-SHADE was investigated with the different numbers of the population size (25, 50, 75, and 100). The rule of increasing the total number of variables in groups is described in Equation (10). It is worth mentioning that, to evaluate some generation with \(m\) subcomponents and \(pop\_size\) number of individuals, it is required to calculate \(m\_pop\_size\) solutions.