# Investigation of the iCC Framework Performance for Solving Constrained LSGO Problems +
## 1 Introduction  
The progress of human activity in different areas does not stand still and optimization problems are no an exception as they have become more complex. For today, the general optimization task is a \"black-box\" optimization problem (there is no knowledge in an explicit form about functional dependence of variables) and it is high dimensional (uses more than hundreds/thousands of variables). Such problems are referred to as large-scale global optimization (LSGO) problems [1]. In the scientific publication [2], the current state of LSGO was discussed. In fact, tasks with a large number of objective variables are a challenging task for a wide range of optimization techniques and algorithms. A typical LSGO is single-objective and boxed constrained (variables are constrained only by lower and upper boundaries). In addition, many real-world optimization tasks exist which are constrained by functions (equalities and inequalities) [3; 4]. The constrained optimization problems (COPs) have additionalrequirements, as opposed to unconstrained optimization problems that separate the search space into feasible and infeasible domains. The majority of modern COPs are dealing with BB objective functions and BB constraints. In other words, we can only evaluate objective functions and constraints in candidate solutions. Constrained LSGO (cLSGO) problems have not been well studied yet. The cLSGO problem can be formulated in the following way:  
\[f(\overline{x})\rightarrow\min_{\overline{x}} \tag{1}\]  
\[x_{k}^{L}\leq x_{k}\leq x_{k}^{U},\ k=\overline{1,N} \tag{2}\]  
\[g_{i}(\overline{x})\leq 0,\ i=\overline{1,t} \tag{3}\]  
\[h_{j}(\overline{x})=0,\ j=\overline{1,p} \tag{4}\]  
where \(\overline{x}\) is a vector of objective variables; \(f(\overline{x}):R^{N}\to R\), \(g(\overline{x})\) and \(h(\overline{x})\) are the objective (fitness) function with \(N\) objective variables, inequality constraints, and equality constraints, respectively; \(t\) and \(p\) are the numbers of inequality and equality constraints; and \(x_{k}^{L}\) and \(x_{k}^{U}\) are the lower and upper bounds for the \(k\)-th variable. In concordance with the LSGO survey [2], there are two popular and effective ways of solving LSGO problems. The first group of methods are called non-decomposition methods. The second group of methods are cooperative coevolution (CC) methods. On the one hand, methods without decomposition use EA principles, which are especially developed for optimizing the whole high-dimensional vector [5; 6]. On the other hand, CC methods [7; 8] decompose the optimization vector into several pieces and evaluate them one by one using some Eas. The CC framework, together with EA as an optimizer, demonstrate a better performance in solving LSGO problems than many standard Eas and other optimization techniques. The main advantage of CC is that it is able to decrease the search space for applying an optimization algorithm, and thus it helps in dealing with the \"curse of dimension\" (CoD) [9]. In this study, an original improvement of CC for solving cLSGO tasks has been proposed and investigated. The main idea of the new approach is that it is permanently increasing the number of variables in subcomponents during an EA run. We have titled it \"iCC\". The SHADE (success-history based parameter adaptation for differential evolution) [10] was used as the main meta-heuristic (so-called \"optimization core\") in CC. The SHADE algorithm self-adapts some inner control parameters. These parameters are called \"scale factor\" (\(F\)) and \"crossover rate\" (\(CR\)). In the last years, many extended EA techniques for handling constraints have been proposed, for example [11; 12; 13; 14]. Additional details about constrained-handling approaches can be found in the following survey [15]. In this study, we have chosen the \(\varepsilon\)-DE technique [16], because it is well studied and popular.