A. Early Work

The first CCEA for function optimization, called CCGA, was proposed by Potter and Jong [9], where the algorithm was empirically evaluated on six test functions of up to 30 dimensions. However, no attempt was made in using the cooperative coevolution (CC) framework on higher dimensional problems. More recently, the idea of using CC in optimization has attracted much attention and was incorporated into several algorithms, including evolutionary programming [15], evolution strategies [16], PSO [8], and DE [10], [12], [17].

In their original CCGA, Potter and Jong [9] decomposed a problem into several smaller subcomponents, each evolved by a separate GA subpopulation. As each subpopulation is evolved, the remaining subpopulations are held fixed. The subpopulations are evolved in a round-robin fashion. For a function optimization problem of $n$ variables, Potter and Jong [9] decomposed the problem into $n$ subcomponents, corresponding to $n$ subpopulations (one for each variable). The fitness of a subpopulation member is determined by the $n$-dimensional vector formed by this member and selected members from other subpopulations. In a way, the fitness of a subpopulation member is assessed by how well it “cooperates” with other subpopulations. Two models of cooperation were examined. In the first model CCGA-1, the fitness of a subpopulation member is computed by combining it with the current best members of other subpopulations. It was found that CCGA-1 performed significantly better than a conventional GA on separable problems, but much worse on nonseparable problems. To improve CCGA's performance on nonseparable problems, CCGA-2 was proposed where members were randomly selected from other subpopulations in the fitness evaluation. On a 2-D Rosenbrock function, CCGA-2 was shown to perform better than CCGA-1. In summary, Potter and Jong's original study [9] demonstrated the efficacy of the CC framework applied to function optimization. However, the CCGA framework was tested only on problems of up to 30 dimensions.

Liu et al. [15] applied the CC framework to their fast evolutionary programming (FEP) algorithms. The new algorithm FEP with CC (FEPCC) was able to optimize benchmark functions with 100 to 1000 real-valued variables. However, for one of the nonseparable functions, FEPCC performed poorly and was trapped in a local optimum, confirming the deficiency of handling variable interactions in Potter and Jong's decomposition strategy [9].

Van den Bergh and Engelbrecht [8] first introduced the CC framework to PSO.Two cooperative PSO algorithms, ${\rm CPSO}\hbox{-}{\rm S}_{K}$ and ${\rm CPSO}\hbox{-}{\rm H}_{K}$, were developed. ${\rm CPSO}\hbox{-}{\rm S}_{K}$ adopts the same framework as that of Potter's CCGA, except that it allows a vector to be split into $K$ subcomponents, instead of each subcomponent consisting of a single dimension. ${\rm CPSO}\hbox{-}{\rm H}_{K}$ is a hybrid approach combining both a standard PSO with the ${\rm CPSO}\hbox{-}{\rm S}_{K}$. These two CPSO algorithms were tested on some benchmark problems of up to 30 dimensions (and 190 dimensions in [11]). Some rotated test functions with variable interactions were also used. Their results demonstrated that correlation among variables in such problems reduces the effectiveness of the two CPSO algorithms. However, no new decomposition strategies were proposed for handling high dimensional nonseparable problems. A similar cooperative approach was also adopted in [18] but was implemented for bacterial foraging optimization.

New decomposition strategies were proposed and investigated for DE with CC [10], [12], [17], [19]. A splitting-in-half strategy was proposed by Shi et al. [17], which decomposed the search space into two subcomponents, each evolved by a separate subpopulation. Clearly, this strategy does not scale up very well and loses its effectiveness quickly when the number of dimensions becomes very large. Yang et al. [10], [12] proposed a decomposition strategy based on random grouping of variables, and applied it to a CC DE, on high-dimensional nonseparable problems with up to 1000 real-valued variables. The proposed algorithms, DECC-G [10] and subsequently MLCC [19], outperformed several existing algorithms significantly. This random grouping strategy represents an important step forward in handling nonseparable high-dimensional problems, and will be incorporated into our proposed CCPSO algorithm. We will describe this random grouping technique in detail in Section III.

B. Random Grouping of Variables

In ${\rm CPSO}\hbox{-}{\rm S}_{K}$ [8], the $n$-dimensional search space is decomposed into $K$ subcomponents, each corresponding to a swarm of $s$-dimensions (where $n=K\ast s$). However, the $s$ variables in any given swarm remain in the same swarm over the course of optimization. Since it is not always known in advance how these $K$ subcomponents are related for any given problem, it is likely that such a static grouping method places some interacting variables into different subcomponents. Because CCEAs work better if interacting variables are placed within the same subcomponent, instead of across different subcomponents, this static grouping method is likely to encounter difficulty in dealing with nonseparable problems.

One method to alleviate this problem is to dynamically change the grouping structure [10]. We call this method random grouping, which is the simplest dynamic grouping method and does not assume any prior knowledge of the problem to be optimized. Here, if we randomly decompose the $n$-dimensional object vector into $K$ subcomponents at each iteration, i.e., we construct each of the $K$ subcomponents by randomly selecting $s$-dimensions from the $n$-dimensional object vector, the probability of placing two interacting variables into the same subcomponent becomes higher, over an increasing number of iterations. For example, for a problem of 1000 dimensions, if $K=10$ (hence we know $s=n/K=100$), the probability of placing two variables into the same subcomponent in one iteration is $p={{{10}\over{1}}\over{10}}=0.1$. If we run the algorithm for 50 iterations, 50 executions of random groupings will occur. The probability of optimizing the two variables in the same subcomponent for at least one iteration follows a binomial probability distribution, and can be computed as follows: TeX Source $$\eqalignno{P(x\geq 1)=&\, p(1)+p(2)+\cdots+p(50)\cr=&\, 1-p(0)\cr=&\, 1-\left (\matrix{{50}\cr{0}}\right)(0.1)^{0}(1-0.1)^{50}\cr=&\, 0.9948}$$ where $x$ denotes the number of observed “successes” of placing two variables in the same subcomponent over the 50 trials; $p(1)$ denotes the probability of having one such “success” over 50 iterations, and similarly $p(2)$ being the probability of having two such “successes,” and so on. This suggests the random grouping strategy should help when there are some variable interactions present in a problem. Our recent paper [14] further generalizes the above probability calculation to cases where more than two interacting variables are present.

A. ${\rm CPSO}\hbox{-}{\rm S}_{K}$ and ${\rm CPSO}\hbox{-}{\rm H}_{K}$

Van den Bergh and Engelbrecht [8]developed two cooperative PSO algorithms. In the first CPSO variant, ${\rm CPSO}\hbox{-}{\rm S}_{K}$, they adopted the original decomposition strategy from Potter and Jong [9], but allowing a vector to be split into $K$ subcomponents, each corresponding to a swarm of $s$-dimensions (where $n=K\ast s$). Algorithm 1 illustrates ${\rm CPSO}\hbox{-}{\rm S}_{K}$ [8]. In order to evaluate the fitness of a particle in a swarm, a context vector ${\mathhat{\bf y}}$ is constructed, which is a concatenation of all global best particles from all $K$ swarms (as shown in Fig. 1). The evaluation of the $i$th particle in the $j$th swarm is done by calling the function ${\bf b}(j,P_{j}.{\bf x}_{i})$ which returns an $n$-dimensional vector consisting of ${\mathhat{\bf y}}$ with its $j$th component replaced by $P_{j}\cdot{\bf x}_{i}$. The idea is to evaluate how well $P_{j}\cdot{\bf x}_{i}$ “cooperates” with the best individuals from all other swarms.

A. Related Studies

B. Cauchy and Gaussian-Based PSO

To solve large-scale optimization problems, an optimization algorithm needs to maintain its ability to explore effectively as well as to converge. Drawn from the findings of the aforementioned PSO variants, we propose a PSO model that employs both Cauchy and Gaussian distributions for sampling, as well as an $lbest$ ring topology. The update rule for each particle position is rewritten as follows: TeX Source $$x_{i,d}(t+1)=\cases{y_{i,d}(t)+{\cal C}(1)\vert y_{i,d}(t)-{\mathhat{y}}^{\prime}_{i,d}(t)\vert, &if $rand \leq p$\cr{\mathhat{y}}^{\prime}_{i,d}(t)+{\cal N}(0,1)\vert y_{i,d}(t)-{\mathhat{y}}^{\prime}_{i,d}(t)\vert,&{\rm otherwise}}\eqno{\hbox{(4)}}$$ where ${\cal C}(1)$ denotes a number that is generated following a Cauchy distribution, and in this case we also need to set an “effective standard deviation” [22] for the Cauchy distribution, which is the same standard deviation value we would set for the equivalent Gaussian distribution, $\vert y_{i,d}(t)-{\mathhat{y}}^{\prime}_{i,d}(t)\vert$; $rand$ is a random number generated uniformly from $[0, 1]$; $p$ is a user-specified probability value for Cauchy sampling to occur. Here, ${\mathhat{{\bf y}_{i}}}^{\prime}$ denotes a local neighborhood best for the $i$th particle. Since an $lbest$ ring topology is used for defining local neighborhood, ${\mathhat{{\bf y}_{i}}}^{\prime}$ (i.e., the best-fit particle) is chosen among all three particles including the current $i$th particle and its immediate left and right neighbors (imagine that all particles are stored on a list that is indexed and wrapped-around). Since each particle may have a different ${\mathhat{{\bf y}_{i}}}^{\prime}$, the population is likely to remain diverse for a longer period. The chance of prematurely converging to a single global best ${\mathhat{\bf y}}$, as in a GPSO (3), should be reduced. Note that $p$ can be simply set to 0.5 so that half of the time Cauchy is used to sample around the $i$th particle's personal best ${\bf y}_{i}$ (more exploratory), while for the other half of the time Gaussian is used to sample around its neighborhood best ${\mathhat{{\bf y}_{i}}}^{\prime}$ (less exploratory).

For the rest of the paper, this CGPSO is used as the subcomponent optimizer for each swarm in the context of a CC framework.

C. Dynamically Changing Group Size

When applying random grouping to CCEA, a group size $s$ (i.e., the number of variables in each subcomponent) has to be chosen. Obviously, the choice of value for $s$ will have a significant impact on the performance of CCPSO2. Furthermore, it might be desirable to vary $s$ during a run. For example, it is possible to start with a smaller $s$ value and then gradually increase this value over the course of the algorithm, in order to encourage convergence to a final global solution and to consider more potential variable interactions.

This issue was studied in a recently developed multilevel cooperative coevolution (MLCC) [19], where a scheme was proposed to probabilistically choose a group size from a set of potential group sizes. At the end of each coevolutionary cycle, a performance record list is used to update the probability values so that the group sizes associated with higher performances are rewarded with higher probability values accordingly. As a result, these more “successful” group sizes are more likely to be used again in future cycles.

This paper adopts an even simpler approach. At each iteration, we record the fitness value of the global best ${\mathhat{\bf y}}$ before and after a coevolutionary cycle,² and if there is no improvement of the fitness value, a new $s$ is chosen uniformly at random from a set ${\bf S}$, otherwise, the value of $s$ remains unchanged. Here ${\bf S}$ contains several possible $s$ values ranging from small to large, e.g., ${\bf S}=\{2, 5, 50, 100, 200\}$. The idea is to continue to use the same $s$ value as long as it works well, but if it does not, then a different value for $s$ should be chosen. This simple scheme is demonstrated to work well experimentally (see Section VI-B). Although the set ${\bf S}$ still needs to be supplied, specification for the parameter $s$ is no longer required. $s$ values in ${\bf S}$ is applicable to any function of dimensions up to the largest value in ${\bf S}$.

One important difference between CCPSO2 and MLCC [19] is that the frequency of applying the random grouping method in CCPSO2 is likely to be much greater than that of MLCC. In CCPSO2 only one evolutionary step is executed over each subpopulation in a cycle, whereas in MLCC, it is common that several evolutionary steps are applied to each subpopulation. Given a fixed number of evaluations, random grouping is likely to be applied more frequently in CCPSO2 than MLCC. A higher frequency of applying random grouping should provide more benefit on nonseparable high dimensional problems, since it is more likely that interacting variables be captured in the same subcomponent of a CCEA [14].

D. CCPSO2

1) Basic Algorithm

Algorithm 2 summarizes the proposed CCPSO2 employing random grouping with dynamically changing group size $s$, as well as the Cauchy and Gaussian update rule (as given in (4) for a ring topology-based $lbest$ PSO. Two nested loops are used to iterate through each swarm and each particle in that swarm. In the first nested loop, for the $i$th particle in the $j$th swarm, its personal best $P_{j}\cdot{\bf x}_{i}$ is first checked for update, and similarly for the $j$th swarm best $P_{j}\cdot{\mathhat{\bf y}}$. The function $localBest(\cdot)$ returns $P_{j}\cdot{\mathhat{{\bf y}_{i}}}^{\prime}$, which is the best-fit particle in the local neighborhood of the $i$th particle, of the $j$th swarm. The $j$th swarm best $P_{j}\cdot{\mathhat{\bf y}}$ is used to update the context vector ${\mathhat{\bf y}}$ if it is better. In the second nested loop, each particle's personal best, neighborhood best, and its corresponding swarm best are used to calculate the particle's next position using (4).

2) Updating Personal Bests

In order to update personal bests effectively in a coherent fashion over iterations, a new scheme is proposed here. Two matrices ${\bf X}$ and ${\bf Y}$ are used to store all information of particles' current positions and personal bests in all $K$ swarms. For example, the $i$th row of ${\bf X}$ is a $n$-dimensional vector concatenating all current position vectors ${\bf x}_{i}$ from all $K$ swarms (note that $sw$ denotes $swarmSize$) as follows: TeX Source $${\bf X}=\left [\matrix{x_{1,1}& x_{1,2}&\ldots & x_{1,n}\cr x_{2,1}& x_{2,2}&\ldots& x_{2,n}\cr\vdots &\vdots &\ddots &\vdots\cr x_{{sw},1}& x_{{sw},2}&\ldots& x_{{sw},n}}\right]\eqno{\hbox{(5)}}$$ and TeX Source $$\quad\quad{\bf Y}=\left [\matrix{y_{1,1}& y_{1,2}&\ldots & y_{1,n}\cr y_{2,1}& y_{2,2}&\ldots& y_{2,n}\cr\vdots &\vdots &\ddots &\vdots\cr y_{{sw},1}& y_{{sw},2}&\ldots& y_{{sw},n}}\right].\eqno{\hbox{(6)}}$$

When random grouping is applied, the indices of all columns are randomly permutated. These permutated indices are then used to construct $K$ swarms over $n$ dimensions, from ${\bf X}$ and ${\bf Y}$. This is achieved by simply taking out every $s$ columns (or dimensions) of ${\bf X}$ and ${\bf Y}$ to form a new swarm. In other words, in each swarm a particle's position and personal best vectors are constructed according to the new permutated dimension indices. As a result of random grouping, the personal best of each newly formed particle in a swarm needs to be re-evaluated in order to obtain its correct “coevolving” fitness. This is a key difference from the conventional personal best update. All the experiments in later sections take these evaluations into account.

Similarly, ${\mathhat{\bf y}}$ should also be reconstructed according to the permutated dimension indices. This way, when forming new swarms based on the new permutated indices, the better position values found so far in certain dimensions can be used meaningfully to guide the search in future iterations.

Note that the original dimension indices are recorded so that when evaluating the $i$th particle of the $j$th swarm via function ${\bf b}(j,P_{j}\cdot{\bf x}_{i})$(which returns an $n$-dimensional vector consisting of ${\mathhat{\bf y}}$ with its $j$th component replaced by $P_{j}\cdot{\bf x}_{i}$), the order of the original dimension indices of this $n$-dimensional vector can be always restored before invoking an evaluation function.

After evaluations of all particles in the $j$th swarm, if the swarm best $P_{j}\cdot{\mathhat{\bf y}}$ is found to be better than the context vector ${\mathhat{\bf y}}$, then the corresponding part of ${\mathhat{\bf y}}$ gets replaced by $P_{j}\cdot{\mathhat{\bf y}}$. This will ensure good information is recorded in ${\mathhat{\bf y}}$, and utilized in future iterations.

At the end of an iteration, after applying (4), the new current and personal best vectors are saved back to ${\bf X}$ and ${\bf Y}$. This will ensure that the next iteration can use the updated ${\bf X}$ and ${\bf Y}$ to start the process of applying random grouping again.

A. Experimental Setup

We adopt the seven benchmark test functions proposed for the CEC'08 special session on large-scale global optimization (LSGO) [5]. For each of these functions the global optimum is shifted with a different value in each dimension. While $f_{1}$ (ShiftedSphere), $f_{4}$ (ShiftedRastrigin), and $f_{6}$ (ShiftedAckley) are separable functions, $f_{2}$(SchwefelProblem), $f_{3}$(ShiftedRosenbrock), $f_{5}$ (ShiftedGriewank), and $f_{7}$ (FastFractal) are nonseparable, presenting a greater challenge to any algorithm that is prone to variable interactions. Note that $f_{5}$ (ShiftedGriewank) becomes easier to optimize as the number of dimensions increases, because the product component of $f_{5}$ becomes increasingly insignificant [29], making it more like a separable function (as variable interactions are almost negligible). In addition to the above seven CEC'08 functions, we also include four more functions, $f_{3r}$, $f_{4r}$, $f_{5r}$, and $f_{6r}$, which are rotated versions of $f_{3} f_{4}$, $f_{5}$, and $f_{6}$, respectively. The effect of rotation introduces further variable interactions, making them nonseparable.³ Rotations are performed in the decision space, on each plane using a random uniform rotation matrix [30], [31]. A new random uniform rotation matrix is generated for each individual run for the purpose of an unbiased assessment. By using these benchmark test functions, and the proposed set of evaluation criteria, we were able to compare CCPSO2 with other existing EAs.

Experiments were conducted on the above 11 test functions of 100, 500, and 1000 dimensions. The same performance measurements used for CEC'08 special session on LSGO were adopted [5]. For each test function, the averaged results of 25 independent runs were recorded. For each run, Max_FES (i.e., the maximum number of fitness evaluations) was set to $5000^{\ast}n$ (where $n$ is the number of dimensions). A two-tailed $t$-test was conducted with a null hypothesis stating that there is no difference between two algorithms in comparison. The null hypothesis was rejected if the $p$-value was smaller than the significance level $\alpha=0.05$.

The population size for each swarm that participates in coevolution was set to 30. For random grouping of the variables, we used ${\bf S}=\{2, 5, 10, 50, 100\}$ for 100-D functions, and ${\bf S}=\{2,5, 10, 50, 100, 250\}$ for 500-D and 1000-D functions, where ${\bf S}$ includes a range of possible group size values that can be dynamically chosen. A few further experiments on $f_{1}$, $f_{3}$, and $f_{7}$ of 2000 dimensions were also carried out, using the same setup as mentioned above.

The CGPSO described in Section IV-B was used in all experiments of CCPSO2, with $p$ in (4) simply set to 0.5.

B. sep-CMA-ES

The CMA-ES algorithm, which makes use of adaptive mutation parameters through computing a covariance matrix and hence correlated step sizes in all dimensions, has been shown to be an efficient optimization method [32], [33]. However, most of the published results of CMA-ES were on functions of up to 100 dimensions [33], [34]. One major drawback of CMA-ES is its cost in calculating the covariance matrix, which has a complexity of $O(n^{2})$. As dimensionality increases, this cost rapidly rises. Furthermore, sampling using a multivariate normal distribution and factorization of the covariance matrix also become increasingly expensive. In a recent effort to alleviate this problem [35], a simple modification to CMA-ES was introduced where only the calculation of the diagonal elements of the covariance matrix ${\bf C}$ is required. Thus, the complexity of updating ${\bf C}$ becomes linear with $n$. Since only ${\bf C}$ diagonal is utilized, the sampling becomes independent in each dimension, making this CMA-ES variant (so called sep-CMA-ES) no longer rotationally invariant. The tradeoff here is the much reduced cost as opposed to the deficiency in handling nonseparable problems. It was shown in [35] that sep-CMA-ES performs not only faster, but also surprisingly well on several separable and nonseparable functions of up to 1000 dimensions, outperforming the original CMA-ES.

In this paper, sep-CMA-ES [35], [36] was chosen to compare against the proposed CCPSO2. In our experiments, for sep-CMA-ES, each individual in the initial population is generated uniformly within the variable bounds $[A,B]^{n}$. The initial step-size $\sigma$ is set to $(B-A)/2$. The population size was determined by $n$ such that $\lambda=4+\lfloor 3 {\rm ln}(n)\rfloor$, $\mu=\lfloor{{\lambda}/{2}}\rfloor$, where $n$ is the dimensions of the problem, $\lambda$ is the number of offspring, and $\mu$ is the number of parents, as suggested in [35]. A number of stopping criteria were used. For example, sep-CMA-ES stops if the range of the best objective function values of the last $10+\lceil 30n/\lambda\rceil$ iterations is zero or below a small value specified, or if the standard deviation of the normal distribution is smaller than a specified threshold value in all coordinates, or if the maximal allowed number of evaluations is reached.

A. Why Cauchy and Gaussian PSO?

In Section IV-B, we provided the rationale of choosing the CGPSO as the subcomponent optimizer for optimizing each swarm under the framework of CC. In this section, we present the empirical evidence to support this choice. For a function of 1000 dimensions, $s$ may range from several dimensions up to several hundreds. In this paper, we used four PSO variants using the ring topology, including constricted PSO (CPSO) [i.e., using (1) and (2)], Bare-bones PSO (3), GPSO [25], and CGPSO (4). We chose $f_{1}$ to $f_{7}$ of 2, 5, 10, 20, 50, and 100 dimensions. Using a small population of 30, each algorithm was run 50 times (with each run allowing 300000 FES), and the mean and standard error of the best fitness were recorded. As shown in Figs. 2 and 3, for $f_{1}$ to $f_{7}$ up to 50 dimensions, CGPSO is the overall best performer. However, when the dimension was increased to 100, CGPSO's lead is no longer obvious. CGPSO outperformed CPSO on $f_{1}$, $f_{4}$, and $f_{5}$, but lost to CPSO on $f_{2}$, $f_{3}$, and $f_{6}$. Both CGPSO and CPSO performed similarly on $f_{7}$. Note that the global optimum of $f_{7}$ is usually unknown (unlike other functions) and may change depending on the dimensionality. Results of the other two variants, Bare-bones PSO and GPSO were not competitive. In particular, GPSO tended to prematurely converge very quickly on several functions. Comparing GPSO and CGPSO, it is noticeable that using a combination of Cauchy and Gaussian distributions is far more effective than using only a Gaussian distribution for sampling. Overall, CGPSO appears to be the most consistent and better performer over other variants. Typically for a function of 500 dimensions, a subcomponent size (i.e., group size $s$) ranging from 2 to 100 dimensions is most likely to be adaptively chosen (see Section VII) during the CC process. Hence, we selected CGPSO as the subcomponent optimizer for each swarm of CCPSO2.

Fig. 2. Averaged best fitness values of four PSO variants (using a ring topology) on $f_{1}$ of 2, 5, 10, 20, 50, and 100 dimensions.

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Fig. 3. Averaged best fitness values (with one standard error) of four PSO variants using a ring topology on $f_{2}$ to $f_{7}$ of 2, 5, 10, 20, 50, and 100 dimensions. (a) $f_{2}$ SchwefelProblem. (b) $f_{3}$ ShiftedRosenbrock. (c) $f_{4}$ ShiftedRastrigin. (d) $f_{5}$ ShiftedGriewank. (e) $f_{6}$ ShiftedAckley. (f) $f_{7}$ FastFractal.

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B. Effects of Dynamically Changing Group Size

Table I compares the results of CCPSO2 using dynamically changing group size $s$ and those using a prespecified fixed group size. When different fixed group sizes are used, it is noticeable that performance fluctuates a lot, depending heavily on the given $s$ value. On the contrary, CCPSO2, using a dynamically changing $s$ value, consistently gave a better performance than the other variants. The closest rival to CCPSO2 is the $s=50$ variant, which only outperformed CCPSO2 on $f_{2}$. However, in most real-world problems, we do not have any prior knowledge about the optimal $s$ value.

TABLE I RESULTS ON 500-D FUNCTIONS OF CCPSO2 USING RANDOM GROUPING WITH A CHANGING GROUP SIZE AND THOSE CCPSO2 VARIANTS USING A PRESPECIFIED FIXED GROUP SIZE

Among the CCPSO2 variants using a fixed $s$ value, it is interesting to note that it is not necessarily always better to use a small fixed $s$ even for separable functions such as $f_{1}$. The $s=100$ variant in fact is much better than the $s=5$ variant. The better performance of CCPSO2 using a dynamically changing $s$ may be attributed to the diverse range of $s$ values that CCPSO2 can utilize during a run. CCPSO2 can use smaller $s$ values to evolve small subcomponent-based intermediate solutions first and then uses larger $s$ values to evolve the combined intermediate solutions to achieve even fitter overall solutions.

Fig. 4 shows typical CCPSO2 runs with a dynamically changing group size on the CEC'08 test functions. It is noticeable that for separable functions $f_{1}$, $f_{4}$, $f_{6}$, and $f_{5}$ (which is considered as a more separable function for 500-D), CCPSO2 tended to choose a mixture of small and large group sizes at different stages of a run. For nonseparable functions such as $f_{2}$, $f_{3}$, and $f_{7}$, CCPSO2 tended to favor a small group size throughout the run. Since the group size remains unchanged when there is further performance improvement, this suggests CCPSO2 was still able to improve the performance though very slowly.

Fig. 4. Changing group size during a CCPSO2 run on $f_{1}$ to $f_{7}$ of 500 dimensions. (The result of $f_{3}$ ShiftedRosenbrock is not shown as it has a unchanged group size of 5 over the run.) (a) $f_{1}$ ShiftedSphere. (b) $f_{2}$ SchwefelProblem. (c) $f_{4}$ ShiftedRastrigin. (d) $f_{5}$ ShiftedGriewank. (e) $f_{6}$ ShiftedAckley. (f) $f_{7}$ FastFractal.

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C. Comparing CCPSO2 With sep-CMA-ES

Table II shows the results of the 11 test functions of 100-D, 500-D, and 1000-D. The result of sep-CMA-ES scaled very well from 100-D to 1000-D on $f_{1}$, $f_{3}$, and $f_{4r}$, outperforming CCPSO2 on all these cases. On $f_{5}$ and $f_{5r}$, the performance of sep-CMA-ES is better on 100-D, but not statistically different on 500-D and 1000-D. Figs. 5–8 show the convergence plots where it can be noted the sep-CMA-ES achieved very fast convergence on $f_{1}$, $f_{3}$, $f_{5}$, and $f_{5r}$.

TABLE II RESULTS OF CCPSO2 AND SEP-CMA-ES ON TEST FUNCTIONS OF 100-D, 500-D, AND 1000-D

In comparison, CCPSO2 clearly outperformed sep-CMA-ES on $f_{2}$, $f_{4}$, $f_{6}$, and $f_{7}$. On $f_{6r}$, CCPSO2 performed better on 100-D and 500-D, but its performance is not statistically different from sep-CMA-ES on 1000-D. There is also no statistical difference on $f_{3r}$. It can be also noted that rotation of $f_{4}$ into $f_{4r}$ degraded the performance of CCPSO2, but had little effect on sep-CMA-ES.

On $f_{2}$, $f_{4}$, $f_{6}$, and $f_{6r}$, sep-CMA-ES prematurely converged as soon as a run started.⁴ sep-CMA-ES also had a poor performance compared with CCPSO2 on $f_{7}$FastFractal function, which has a more complex and irregular fitness landscape, and highly multimodal. The standard CMA-ES (which uses the full covariance matrix) was also tested on $f_{2}$, $f_{4}$, and $f_{6}$, but CMA-ES suffered from the same problem of premature convergence.

The offspring population size $\lambda$ of a sep-CMA-ES run was determined by $\lambda=4+\lfloor 3 {\rm ln}(n)\rfloor$. For $n$ equals to 100, 500, and 1000, $\lambda$ is 17, 22, and 24, respectively. Further tests revealed that these population sizes were too small for sep-CMA-ES (or CMA-ES) to perform well on $f_{2}$, $f_{4}$, $f_{6}$, and $f_{7}$, though smaller population sizes did not pose any trouble on $f_{1}$, $f_{3}$, and $f_{5}$. For sep-CMA-ES to perform well, it requires a much larger population size in order to sample effectively around the mean of the selected fit individuals. This suggests that the formula proposed for determining the population size $\lambda=4+\lfloor 3 {\rm ln}(n)\rfloor$ is not suitable for multimodal functions of higher dimensions $(n>100)$. This observation is also supported in [38], where a study was carried out specifically on CMA-ES over multimodal functions. It shows that on the Rastrigin function of only 80 dimensions and Schwefel of 20 dimensions, CMA-ES requires a $\lambda$ greater than 1000 in order to maintain a success rate of 95% (Figs. 2 and 3 of [38]). In [35], a different formula for determining population size was also studied: $\lambda=2n$. Nevertheless this means that the population will be unrealistically large for functions of 1000 dimensions for instance. Fig. 9(a)shows the results of sep-CMA-ES on $f_{4}$(ShiftedRastrigin) of 500 dimensions, over population sizes ranging from 100 to 1500. It can be seen that as the population size increases, the performance also improves, however, even with a population size of 1500, the best fitness achieved was 14.725, which is still worse than CCPSO2. Fig. 9(b) shows that on the more complex $f_{7}$ (FastFractal), the best result achieved was $-{\rm 7057.21}$ (when a population size of 400 was used), which is still worse than the best result obtained by CCPSO2 (i.e., ${-}{7.23}{\rm E}+03$ as shown in Table II). Clearly, sep-CMA-ES's performance is sensitive to the population size parameter. And even if the optimal population size is chosen, CCPSO2 still outperforms sep-CMA-ES on these multimodal functions. More importantly, the performance of CCPSO2 does not depend on a large population size. For all experiments, a small swarm of 30 particles for a coevolving subcomponent was shown to be sufficient in producing reasonable performances.

Fig. 5. Averaged best fitness values for sep-CMA-ES and CCPSO2 on functions of 500 dimensions. (a) $f_{1}$ ShiftedSphere. (b) $f_{2}$ SchwefelProblem. (c) $f_{3}$ ShiftedRosenbrock. (d) $f_{4}$ ShiftedRastrigin. (e) $f_{5}$ ShiftedGriewank. (f) $f_{6}$ ShiftedAckley. (g) $f_{7}$ ShiftedRastrigin. (h) $f_{3r}$ ShiftedRotatedRosenbrock. (i) $f_{4r}$ ShiftedRotatedRastrigin.

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Fig. 6. Averaged best fitness values for sep-CMA-ES and CCPSO2 on functions of 1000 dimensions. (a) $f_{1}$ ShiftedSphere. (b)$f_{2}$ SchwefelProblem. (c) $f_{3}$ ShiftedRosenbrock. (d) $f_{4}$ ShiftedRastrigin. (e) $f_{5}$ ShiftedGriewank. (f) $f_{4}$ ShiftedRastrigin. (g) $f_{7}$ ShiftedRastrigin. (h) $f_{3r}$ ShiftedRotatedRosenbrock. (i) $f_{4r}$ ShiftedRotatedRastrigin.

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Fig. 7. Averaged best fitness values for sep-CMA-ES and CCPSO2 on $f_{5r}$ and $f_{6r}$ of 500 dimensions. (a) $f_{5r}$. (b) $f_{6r}$.

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Fig. 8. Averaged best fitness values for sep-CMA-ES and CCPSO2 on $f_{5r}$ and $f_{6r}$ of 1000 dimensions. (a) $f_{5r}$. (b) $f_{6r}$.

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Fig. 9. Averaged best fitness values of sep-CMA-ES on $f_{4}$ and $f_{7}$ of 500 dimensions, over a range of population sizes. (a) $f_{4}$ ShiftedRastrigin. (b) $f_{7}$ FastFractal.

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Table II shows that overall, for the 11 test functions, CCPSO2 performed significantly better than sep-CMA-ES on five functions, while losing to sep-CMA-ES on three functions. For the remaining three test functions, there is no statistical difference between the two. Most importantly, CCPSO2 performed better than sep-CMA-ES on high-dimensional multimodal functions with a complex fitness landscape such as $f_{7}$, which arguably more closely resemble real-world problems. The CC framework allowing decomposing a high dimensional problem into small subcomponents is a key contributing factor that CCPSO2 scales well to very high-dimensional problems.

CCPSO2 tended to converge slower than sep-CMA-ES, but had more potential to further improve its performance in the later stage of a run. This may be attributed to the use of the Cauchy distribution in CCPSO2 to encourage more exploration. On the contrary, sep-CMA-ES converged extremely fast, however, it either converged very well or tended to become stagnant very quickly, especially on complex high dimensional multimodal functions.

D. Comparing CCPSO2 With Other Algorithms

Table III shows the results of CCPSO2 on the seven CEC'08 benchmark test functions of 1000 dimensions, in comparison with two recently proposed PSO variants, EPUS-PSO [39], DMS-PSO [40], and a CC DE algorithm MLCC [19]. The results of these algorithms were obtained under the same criteria, as set out by the CEC'08 special session on LSGO [5]. A two-tailed $t$-test was only conducted between CCPSO2 and MLCC, since CCPSO2 clearly outperformed EPUS-PSO and DMS-PSO.

TABLE III COMPARISON BETWEEN CCPSO2, EPUS-PSO, DMS-PSO, AND MLCC AND ON 1000-D FUNCTIONS

EPUS-PSO used an adaptive population sizing strategy to adjust the population size according to the search results [39]. DMS-PSO adopted a random regrouping strategy to introduce a dynamically changing neighborhood structure to each particle. Every now and then the population is regrouped into multiple small subpopulations according to the new neighborhood structures [40]. MLCC also adopts a CC approach (like CCPSO2) applying both random grouping and adaptive weighting [19]. Given a fixed number of evaluations, the applications of adaptive weighting is at the expense of random grouping, which proves to be less effective [14]. In addition, MLCC employs a self-adaptive mechanism to favor certain group size over others when invoking the random grouping scheme. All these three algorithms participated in the CEC 2008 competition on LSGO [37].

CCPSO2 outperformed EPUS-PSO on six out of the seven CEC'08 test functions, except $f_{2}$. CCPSO2 also outperformed DMS-PSO on five functions, except $f_{1}$ and $f_{5}$. DMS-PSO found the global optimum for $f_{1}$ and $f_{5}$, but its performance is exceptionally poor in comparison with CCPSO2 on the other five functions. CCPSO2 adopts a CC framework to decompose a high-dimensional problem, whereas DMS-PSO relies on the use of a large number of small sub-swarms (each has only three particles) to maintain its population diversity. DMS-PSO does not employ any dimensionality decomposition strategy. As a result, DMS-PSO uses much larger population sizes to handle high-dimensional problems. For example, for 100, 500, and 1000-D functions, the population sizes were 90, 450, and 900, respectively. In contrast, CCPSO2 uses just a small population size of 30 for all dimensions. This suggests that DMS-PSO may only work well in some unimodal fitness landscapes, but has a very poor ability in handling complex multimodal fitness landscapes.

Comparing with MLCC, CCPSO2 is better on $f_{1}$, $f_{2}$, and $f_{3}$, but statistically not different on $f_{5}$ and $f_{6}$. MLCC is better on $f_{4}$ and $f_{7}$. MLCC uses a self-adaptive neighborhood search DE (SaNSDE) [12] as its core algorithm to coevolve its CC subcomponents. SaNSDE is able to maintain a better diversity of search step sizes and the population, thereby contributing to its better performance on the multimodal problems.

E. Comparisons on Functions of 2000 Dimensions

From the previous Section VI-C (Table II, Figs. 5 and 6), we noticed that on $f_{2}$, $f_{4}$, and $f_{6}$ of 100, 500, and 1000 dimensions, sep-CMA-ES always prematurely converged not long after the run started. sep-CMA-ES was actually outperformed substantially by CCPSO2 on these functions. It is reasonable to expect that sep-CMA-ES be outperformed by CCPSO2 on these functions of even higher dimensions such as 2000-D.

Table IV shows the results of CCPSO2 and sep-CMA-ES on the three other CEC'08 test functions: $f_{1}$, $f_{3}$, and $f_{7}$ of 2000 dimensions. These results were averaged over 25 runs, with each run consuming $1.0{\rm E}+07$ FES. To our knowledge, this paper is the first reporting results on these functions of 2000 dimensions. Overall, we can see that both CCPSO2 and sep-CMA-ES continued to scale well on $f_{1}$ and $f_{3}$ of 2000 dimensions. sep-CMA-ES performed better than CCPSO2 on $f_{1}$ (which is a separable function) and $f_{3}$, but was outperformed by CCPSO2 on the more complex multimodal $f_{7}$. Considering the tendency of sep-CMA-ES converging prematurely on multimodal functions $f_{2}$, $f_{4}$, and $f_{6}$, once again CCPSO2 shows its better search capability in handling complex high-dimensional multimodal functions.

TABLE IV RESULTS OF CCPSO2 AND SEP-CMA-ES ON $f_{1}$, $f_{3}$, AND $f_{7}$ of 2000 DIMENSIONS.