# Investigation of Improved Cooperative Coevolution for Large-Scale Global Optimization Problems +
## 1 Introduction  
Evolutionary computation [1; 2; 3] includes a research field, where bio-inspired algorithms are used for solving global optimization problems with different levels of complexity. This family of algorithms demonstrates high performance in solving hard real-world [4; 5; 6; 7] and benchmark [8; 9] problems when the number of variables is not large (less than a hundred). In many modern real-world LSGO problems, the number of continuous variables starts from hundreds or thousands [10]. The performance of classic EAs rapidly decreases when the number of variables increases because of the curse of dimensionality [11]. Many LSGO techniques [12] use the CC approach [13] to overcome the effect of the high-dimensional search space. CC splits a vector of solution variables into the predefined number of subcomponents and optimizes them independently using an EA. CC-based metaheuristics perform well when variables are grouped correctly. The variable grouping is a challenging task for several reasons: (1) metaheuristics are limited in the number of fitness evaluations (FEVs); (2) the landscape of real-world LSGO problems is complex, (3) LSGO problems are presented by a \"black-box\" (BB) model. This paper proposes a modification of the classic CC algorithm, which changes the number of variables in subcomponents dynamically during the optimization process forimproving the performance of the search process. The proposed metaheuristic based on iCC uses the SHADE optimizer and is titled iCC-SHADE. iCC-SHADE performs five optimization stages. At each stage, the algorithm solves an optimization problem using 20% of the total FEVs budget. When a budget of a stage is exhausted, iCC decreases the number of subcomponents and, consequently, increases the number of variables in each subcomponent. The proposed approach efficiently implements the exploration and exploitation strategies. The exploration strategy forces an EA to discover new regions of the search space. The exploitation is performed using small subcomponents. The exploitation strategy provides an improvement of the best-found solutions. The exploration involves subcomponents of a large size, which represent an efficient problem decomposition. In [14], a survey on the balance between these two strategies is proposed. In general, an LSGO problem can be stated as a continuous optimization problem:  
\[f(\overline{x})\rightarrow\min_{\overline{x}} \tag{1}\]  
\[g_{i}(\overline{x})\leq 0,i=1,\ldots,m \tag{2}\]  
\[h_{j}(\overline{x})=0,j=1,\ldots,p \tag{3}\]  
\[a_{k}\leq x_{k}\leq b_{k},k=\overline{1,N}, \tag{4}\]  
where \(f\) is an objective/fitness function to be minimized, \(\overline{x}\) is an \(N\)-dimensional vector of continuous variables, \(f:R^{N}\to R^{1}\), \(g_{i}\) and \(h_{j}\) are inequality and equality constraints, respectively, \(a_{k}\) and \(b_{k}\) are the left and right borders of the \(k\)-th variable, respectively. In our previous studies, we have analyzed the performance of the iCC approach when solving constrained LSGO (cLSGO) problems [15; 16]. We have proposed a new benchmark set with high dimensional cLSGO problems, and have compared the performance \(\varepsilon\)-iCC-SHADE and \(\varepsilon\)-CC-SHADE algorithms. The results have demonstrated that \(\varepsilon\)-iCC-SHADE outperforms on average \(\varepsilon\)-CC-SHADE with a fixed number of subcomponents. Moreover, the results have shown that the performance of the iCC approach depends more on the decomposition of the objective than on a constraint-handling technique. In this study, we have focused on and investigated the performance of the proposed algorithms using unconstrained optimization problems only. Based on our previous study [17], this paper has been extended with a detailed analysis of the performance of iCC-SHADE by four classes of LSGO problems, namely fully-separable, partially additively separable, overlapping, and non-separable problems. The tuned iCC-SHADE has been compared with seven state-of-the-art LSGO algorithms.