This section reviews the original ICA [10]. However, as the original ICA has some design problems, we adopt Atashpaz-Gargari’s Matlab implementation of ICA [28] whenever appropriate. The original ICA is shown in Figure 1. It proceeds in two stages: initialization and evolution.

During the initialization stage (line 1 in Figure 1), a user-specified number of randomly generated solutions {p¹, p², …, p^N} are created, where each solution pⁱ (1≤ i ≤ N ) is a 1 × n array and is called a country, and N denotes the number of countries in the population. Then, the cost of each country pⁱ is calculated as f(pⁱ). A user-specified number of countries with the lowest cost in the population are chosen as imperialists, and the remaining countries as colonies. Let N_imp denote the number of imperialists. Next, N_imp empires are formed by first assigning an imperialist to each empire, and then randomly assigning all colonies to these empires such that the number of colonies of an empire is proportional to the power of its imperialist. That is, (2) (3) where c_i and P_i denote the cost and power of the imperialist of empire i, respectively, and NC_i denotes the number of colonies assigned to empire i, where 1 ≤ i ≤ N_imp. Notably, Equation (2), proposed by Atashpaz-Gargari [28], calculates the power of an imperialist in a way different from that in the original ICA [10] to avoid the problem of assigning zero colony to the weakest empire in the original ICA.

During the evolution stage (lines 2–10 in Figure 1), assimilation within each empire and competition among all empires occur in every generation until the termination condition (e.g., all countries have converged or a user-specified number of generations has been reached) is satisfied. Assimilation within an empire is achieved by moving all colonies of the empire toward their imperialist (line 4 in Figure 1). Given a colony p^c and its imperialist pⁱ, the assimilation operation moves p^c as follows [28]: (4) where β is a parameter with default value 4, δ is a 1 × n array whose elements are random values between 0 and 1, and “.*” denotes element-by-element multiplication between two 1 × n arrays. For optimization problem with bounded variables, Equation (4) could generate values outside the search space. If this happens, the out-of-bound value, e.g., x_i on the i-th dimension, is simply replaced by its nearest boundary (i.e., U_i or L_i).

After moving all colonies to their new positions through the assimilation operation, their costs are recalculated (line 5 in Figure 1) and then compared against the cost of their imperialists (line 6 in Figure 1). If the cost of a colony is smaller than the cost of its imperialist, the colony and the imperialist swap roles to ensure that the imperialist of an empire is always the country with the lowest cost in the empire.

Next, the cost (denoted by Q_i) and the power (denoted by E_i) of each empire i are calculated using the cost of its imperialist (denoted by c_i) and the average cost of the colonies in empire i, as shown in Equations (5) and (6), respectively (line 8 in Figure 1). (5) (6) Notably, ξ is a parameter with suggested value 0.1 in the original ICA, but with default value 0.02 in [28]. This study uses ξ = 0.02. Competition among all empires (line 9 in Figure 1) is achieved by taking the weakest colony away from the weakest empire and giving it to a chosen empire, where the probability of empire i been chosen is calculated as follows.

(7)

When the last colony of an empire is taken away by another empire, the former empire is eliminated, and its imperialist also becomes a colony of the latter empire.

In the literature, many new operations have been proposed to enhance the basic operations in ICA. For example, Atashpaz-Gargari [28] introduced three new features into the original ICA. In each generation, a portion of the colonies are replaced by the same number of randomly generated new candidate solutions, through the revolution operation. When two imperialist are too close to each other, their empires are united as one, through the unite operation. Finally, instead of competition in every generation, the frequency of competition among empires is reduced to avoid converging too quickly.

Instead of using the revolution operation to replace a portion of colonies in every generation, Duan et al. [29] suggested monitoring the minimal cost and average cost of all countries, and if the minimal cost and the difference between the average cost and the minimal cost are below their respective user-specified thresholds, then a random number of countries in the population are replaced by new countries, generated using a chaotic sequence. Jain and Nigam [30] suggested using a genetic algorithm to generate new countries for the revolution operation. Khorani et al. [31] suggested applying genetic algorithm and ICA recursively to benefit from the search power of both algorithms.

The assimilation operation of ICA often converges to a local optimum prematurely [29,30,32], and thus many attempts have been proposed to replace it with more powerful operations. For examples, Bahrami et al. [33] used a chaotic map to determine the moving direction for the assimilation operation. Zhang et al. [34] defined the assimilation operation as a two-step process. First, each colony is moved precisely toward their imperialists. Then, some randomly chosen colonies are moved further in all dimensions to improve diversity. Lin et al. [35] replaced Equation (4) with Equation (8) below to allow some colonies moving away from their imperialists: (8)

They also suggested bouncing the out-of-bound values (e.g., x_i on the i-th dimension) into the feasible space using Equation (9), instead of sticking them to the nearest boundary.

(9)

Lin et al. [26] indicated that the assimilation operation is a primitive form of Particle Swarm Optimization (PSO), and thereby suggested replacing the assimilation operation with PSO. Similarly, Karimi et al. [13] replaced the assimilation operation with an Electromagnetism-like Algorithm. Nozarian and Jahan [36] applied ICA as the local search mechanism in a memetic algorithm. Bahrami et al. [37] proposed an assimilation operation adaptive to the distribution of colonies to balance between the exploration and exploitation abilities of ICA.

This section proposes the idea of artificial imperialist, and describes how it is used in Interaction Enhanced ICA. Let p¹, p², …, p^m denote the imperialists in current population, listed in ascending order of their costs. An artificial imperialist p^a is constructed as the weighted sum of all imperialists, as shown below.

(10)

(11)

The weights of these imperialists form a geometric sequence with ratio 0.9. A better imperialist has a larger weight. Intuitively, the artificial imperialist combines the niche of all imperialists, and allows exploration of global optimum efficiently. To employ artificial imperialists in Interaction Enhanced ICA, the steps in Figure 3 are used for the interaction step in Figure 2. If the artificial imperialist p^a is better than the weakest imperialist p^m, then p^a becomes the new imperialist of empire m, and p^m is simply discarded. We denote the Interaction Enhanced ICA using Artificial Imperialist as ICAAI.

ICAAI only incurs one more calculation of the objective function per generation than the original ICA does. However, the time complexity remains the same for both methods.

With ICA, the imperialist of an empire cannot move unless it is no longer the best country in the empire. This situation is similar to Electromagnetic-like algorithm [4] where all particles except the best particle are allowed to move. However, with ICA, there exist several empires, and consequently, several imperialists. Thus, this motivates the idea of exchanging information among the imperialists to improve the convergence speed of ICA. Based on this motivation, this section proposes crossover imperialists (see Figure 4) for the interaction step in Interaction Enhanced ICA.

Notably, ν in step 1 of Figure 4 is a parameter between 0 and 1 to control the crossover ratio, and m is the number of imperialists in the current population. Given two countries pⁱ and p^j, the steps to generate two new countries q¹ and q² using uniform crossover are shown in Figure 5.

Denote Interaction Enhanced ICA using Crossover Imperialists as ICACI. ICACI incurs ⌊νm⌋ more calculations of the objective function per generation than ICA does. Notably, m equals N_imp initially and is non-monotonic decreasing afterwards. The time complexity of ICACI remains the same as that of ICA.

A set of 13 well-known benchmark functions, listed in Table 1, was used in this experiment [33,38]. The number of dimensions (i.e., n in Table 1) of the search space is set to 30. Functions f₁–f₅ are unimodal, and functions f₆–f₁₃ are multimodal. Notably, the function u in functions f₁₂ and f₁₃ is defined as follows.

(12)

This performance study contains two experiments. In experiment 1, PSO and three ICA variants (Perturbed ICA [35], ICAAI and ICACI) were tested. For the three ICA methods, the number of countries, the number of imperialists, β in Equations (4) and (8), and ξ in Equation (5) are set to 88, 8, 4 and 0.02, respectively. For ICACI, the crossover ratio ν is set to 0.8. For both ICAAI and ICACI, ρ in line 12 of Figure 2 is set to 1. For PSO, the number of particles, inertia weight w, and two acceleration coefficients c₁ and c₂ are set to 88, 0.7, 2 and 2, respectively.

Experiment 2 studied the effect of competition frequency on both ICAAI and ICACI. All experimental settings are the same as those in experiment 1 except that the value of ρ varied from 0 to 1 in step of 0.1. Notably, ρ = 0 means that no competition occurs among empires.

For each setting, 30 runs were conducted, and their average performance was obtained. Each run stops until the maximum number of generations (1000 or 3000 in experiment 1, and 1000 in experiment 2) is reached. For the same run, the same set of initial countries (or particles in PSO) was generated for all four methods.