Abstract
The QuasiAffine Transformation Evolutionary (QUATRE) algorithm is a swarmbased collaborative optimization algorithm, which has drawn attention from researchers due to its simple structure, easy implementation, and powerful performance. However, it needs to be improved regarding the exploration, especially in the late stage of evolution, and the problem of easy falling into a local optimal solution. This paper proposes an improved algorithm named QuasiAffine Transformation Evolutionary with double excellent guidance (QUATREDEG). The algorithm uses not only the global optimal solution but also the global suboptimal solution to guide the individual evolution. We establish a model to determine the guiding force by the distance between the global optimal position and the suboptimal position and propose a new mutation strategy through the double population structure. The optimization of population structure and the improvement of operation mechanisms bring more exploration for the algorithm. To optimize the algorithm, the experiments on parameter settings were made to determine the size of the subpopulation and to achieve a balance between exploration and development. The performance of QUATREDEG algorithm is evaluated under CEC2013 and CEC2014 test suites. Through comparison and analysis with some ABC variants known for their strong exploration ability and advanced QUATRE variants, the competitiveness of the proposed QUATREDEG algorithm is validated.
1. Introduction
Optimization problems have been a hot research field since optimization solutions are crucial to various engineering practices. Scientific theories have indicated that appropriate optimization methods applied in different fields [1–7] will bring significant benefits. Accordingly, powerful and efficient algorithms have drawn the attention of researchers in different application fields.
Inspired by the laws of the biological community and natural world, various intelligent optimization algorithms have been proposed. At the early stages, evolutionary algorithms were first developed, such as the Genetic Algorithm (GA) [8] and Differential Evolution (DE) algorithm [9]. Genetic Algorithm is a global optimization algorithm developed by imitating the genetic and evolutionary characteristics of organisms. The core idea of the algorithm is “survival of the fittest in natural selection.” The other algorithm, Differential Evolution, is also an efficient evolutionary algorithm. In its process of searching, the better individuals are continuously selected with the cooperation and competition among particles in the population to find the optimal solution. At the later stages, numerous swarm intelligence algorithms inspired by biology emerged. Particle Swarm Optimization (PSO) [10] is an optimization algorithm proposed by observing the living habits of birds. The algorithm imitates the behavior of continuous migration and clustering in the process of searching foods. It has the characteristics of simple structure and easy implementation. The artificial bee colony (ABC) algorithm [11] is a swarm intelligence algorithm firstly proposed by Karabora in 2005, inspired by the bee colony behavior of honey source searching. Compared with the premature convergence of PSO, the ABC algorithm is relatively mature in the application of swarm intelligence because of its strong searching ability. In recent years, with the development of computational intelligence, many novel global optimization algorithms have been proposed, such as Multiverse Optimizer (MVO) [12] algorithm, grasshopper optimization algorithm (GOA) [13], and QuasiAffine Transformation Evolutionary (QUATRE) algorithm [14]. These algorithms also have a simple structure and excellent performance.
It is well known that particle exploitability and exploratory are considered of equivalent importance in swarm intelligence algorithms, and they should be treated equally in the process of updating particle position. QUATRE algorithm is evolved from DE algorithm. It solves the position deviation problem of DE algorithm and has a stronger development ability. But it is slightly insufficient in search performance. In this paper, a new QUATREDEG algorithm is proposed, which divides the population into two subpopulations to increase the diversity of the population and uses the positions of the global optimal individual and the global suboptimal individual to establish the guiding force model, which improves the original mutation strategy and enhances the searching ability of the algorithm. The performance of the proposed QUATREDEG algorithm is verified by the experiments on CEC2013 benchmark set [15] and CEC2014 benchmark set [16]. The specific contributions of this paper are as follows: (1)The twopopulation structure of mutation in two central regions is proposed, which maintains the diversity of particles, so that more benchmark environments can be explored(2)The gravity model based on the relationship between the global optimal solution and the global suboptimal solution is established and applied to the new operation strategy, so as to enhance the information interaction between the excellent individuals(3)In the benchmark set of CEC2013 and CEC2014, some famous ABC variants and advanced QUATRE variants are compared and analyzed with our QUATREDEG algorithm. The results show that the performance of the new QUATREDEG algorithm is competitive
The next few sections are arranged as follows. The second section reviews the development of QUATRE and ABC algorithms. The third section mainly introduces the QUATREDEG algorithm proposed in this paper. In the fourth section, the performance of the QUATREDEG algorithm is evaluated by using CEC2013 and CEC2014 benchmarks, respectively. The final section provides the summary.
2. Related Works
QUATRE is a relatively new global optimization algorithm first proposed in 2016. The term QUATRE is the abbreviation of QuasiAffine Transformation Evolution. The algorithm is based on the DE algorithm and uses particle cooperative evolution in the update process. It replaces the parameter CR in the DE algorithm with a new collaborative search matrix , which solves the position deviation problem in the DE algorithm. It is a powerful and simple optimization algorithm. Its operation mechanism is shown in the following formula: where is an adjustable mutation size factor. In the original QUATRE algorithm, its optimal value is 0.7. is a global optimal matrix, and all row vectors consist of global optimal individuals. and are all randomly generated matrices. They are generated by randomly replacing the row vector of , but the total elements remain unchanged. The is a cooperative search matrix, which is the key to the evolutionary performance of the algorithm. The required matrix can be generated in three steps. Firstly, the lower triangular matrix is initialized, and the elements in the matrix are 0 or 1. Then, all the elements in the row vector of the matrix are randomly displaced. Finally, the matrix can be obtained by randomly displacing the row vector in the matrix. is the binary inverse operation of all elements in , and represents the component multiplication.
The QUATRE algorithm has the characteristics of fewer control parameters, simple to use, and powerful performance, which attracts more and more scholars to study it. Several successful variants have been proposed [17–21], including the competitionbased QUATRE algorithm (CQUATRE) [22] which employs a pairing competition. The winning individuals are directly retained to the next generation, while the losers need to evolve like the original QUATRE algorithm. It effectively improves the performance of the algorithm. Like the DE algorithm, the QUATRE has the disadvantages of premature convergence and easy to fall into a local optimal solution. The diversity of particles can effectively reduce premature convergence, but the convergence of particles is also important, so finding the balance between particle convergence and diversity is the key. A new sortingbased QUATRE algorithm (SQUATRE) [23] was proposed, which not only keeps the convergence of particles but also enhances the diversity of particles. Its main improvement scheme is to divide the population into better and worse groups by sorting strategy and to evolve the individuals in the worse group, which improves the search performance of the algorithm. In addition, the performance of the algorithm is greatly affected by the disturbance strategy and the parameters of the disturbance. For example, in QUATRE, a larger disturbance factor will enhance the algorithm in search, while a smaller one will pay more attention to development. In [24], the BPQUATRE algorithm is proposed and the original population is divided into different subpopulations by a new sorting strategy. Different subpopulations use different disturbance modes, and the linear descent parameter is used to enhance the global optimization ability of the algorithm.
3. Proposed Approach
The original QUATRE algorithm has an inherent shortcoming of paying too much attention to development, which makes it easy to fall into the local optimal solution. The diversity of the population can make up for this shortcoming and enhance the search algorithm. In the iterative process, the global optimal solution of each iteration is widely used in the operation mechanism of a large number of algorithms. It can greatly enhance the convergence of the algorithm, make it easier to find the real global optimal, and also can improve the operation speed of the algorithm. And the excellent individuals in each iteration are also worthy of reasonable utilization, such as the global suboptimal solution. In this study, we propose a new QUATRE algorithm based on double excellent guidance (QUATREDEG) which divides the population into two subgroups, including the optimal guided group and the suboptimal guided group. It makes the possible to use the global suboptimal solution to enhance the performance of the algorithm. The details of the new QUATRE algorithm are as follows.
3.1. Initialization
Generate a new population and obtain the current optimal solution and suboptimal solution, as well as set a maximum number of iterations.
3.2. Double Population Structure
Firstly, all individuals are divided into two fixed subpopulations, including the optimal solution population and suboptimal solution population . The number of individuals of optimal solution population and suboptimal solution population is determined by Eq. (3). In the population of optimal solution, in Eq. (1) still takes the location of the optimal solution as the center for mutation strategy. But in the population of suboptimal solution, is the mutation centered on the position of suboptimal solution . In this way, the individuals in the population will search in two central regions in each iteration, so as to explore more environment regions. where is the total population and is the parameter which are used to determine the individual number of the optimal solution population and of the suboptimal solution population.
3.3. Relation between Optimal Solution and Suboptimal Solution
For the global optimization algorithm, the task is to find the global optimal solution. In the process of updating and iteration, excellent individuals can play an important role. It should be concerned that the neighborhood of the current global optimal solution may not be better than the one of the global suboptimal solution, which means that both the global suboptimal solution and the global optimal solution contain extremely valuable information. If the information can be shared between them, it can enhance the performance of the algorithm, since there is a strong information interaction between individuals in the grasshopper optimization algorithm (GOA). Inspired by the GOA, this study proposes a model to determine the guiding force between the global optimal solution and the global suboptimal solution by the distance between them, as shown in Eq. (4). where is the Euclidean distance between the global optimal solution and the global suboptimal solution and represents the dimension. In the function of , the parameter values of and are recommended to be set to 0.5 and 1.5, respectively. The represents the attraction of the global optimal solution to the global suboptimal solution. The and represent the upper and lower bounds of the search space, respectively. In the function, when is equal to 2.079, the function value is 0, and when value increases, the function value will first increase to the maximum value of 0.02 and then gradually decrease to 0. A distancebased attractiveness model is constructed by using the characteristics of . When the distance between the global optimal solution and the global suboptimal solution is close, the attractiveness of the global optimal solution and the global suboptimal solution is positively correlated with the distance . However, when the distance between the global optimal solution and the global suboptimal solution is far, the attraction is negatively correlated with the distance .
3.4. Arithmetic Strategy
The model of establishes the relationship between the global optimal solution and the global suboptimal solution so that the algorithm can effectively search the neighborhood between the global optimal solution and the global suboptimal solution. The characteristics of the function can help us balance the search and development of the algorithm. The new arithmetic strategy is shown in Eq. (5) and Eq. (6). where and are the matrices generated, respectively, by taking the position coordinates of the global optimal solution and the global suboptimal solution as the row vector elements, and the number of columns of the matrix is determined by the number of individuals in the population. The matrix can be obtained by the same principle. As shown in Eq. (6), the matrix and the crossover strategy remain unchanged. The algorithm assigns individuals to two fixed subpopulations to increase the diversity of the population. Each subpopulation has different operation strategies and can share information in the process of searching. When searching with the global optimization as the center, it will be guided by the global suboptimal force; similarly, when searching with the global suboptimal as the center, it will be guided by the global optimal force. This expands the algorithm’s search for excellent regions and avoids falling into the local optimum. The double optimal guidance mechanism not only ensures the convergence of both global optimal position and global suboptimal position, but also brings more exploration to benchmark environments. Accordingly, the proposed algorithm can be an effective approach to enhance the search performance of the algorithm.
3.5. Algorithm Process
The process of QUATREDEG algorithm is given as follows.

4. Experiment and Analysis
In order to verify the performance of the algorithm, we carried out experiments with 58 realparameter single objective optimizations in the CEC2013 and CEC2014 test suites. All benchmark functions are considered as blackbox problems, and the search range is set to [100, 100]. In CEC2013, there are 28 test functions, including unimodal functions (fa1fa5), basic multimodal functions (fa6f20), and composition functions (fa21fa28). In CEC2014, there are 30 test functions including four different categories: unimodal function (fb1fb3), simple multimodal function (fb4fb16), hybrid function 1 (fb17fb22), and composition function (fb23fb30).
We have learned that the QUATRE algorithm focuses on the development, but the search is slightly inadequate. Our QUATREDEG algorithm focuses on improving the search ability of the algorithm to improve the overall performance of the algorithm. In the experiments, to verify the overall performance of the algorithm, it is not only compared with the advanced QUATRE variant but also with the famous variants of the ABC algorithm, which is famous for its strong search ability. The new QUATREDEG algorithm is a further improvement of QUATRE algorithm, and the search strategy of ABC algorithm determines its strong exploration performance. In QUATREDEG, the performance of the algorithm is mainly improved by enhancing the exploration ability. Therefore, we use some famous ABC variants and advanced QUATRE variants to compare with our proposed algorithm to verify that the performance of QUATREDEG is reliable. The comparison includes the original QUATRE [14], CQUATRE [22], IABC [25], MABC [26], and QUATREDEG algorithms. In all QUATREbased algorithms, the population size is set to 100, while the population size of all ABC algorithms is set to 50, and the parameter limit is set to , where is the dimension size and SN is the population size. In addition, in the QUATRE, CQUATRE, and QUATREDEG algorithms, the value of parameter is set to the recommended value of 0.7. In the IABC algorithm, MR is set to 0.3.
The experiments were complete on Windows 10 Professional system with 8 GB RAM and Intel i54210 m processor. The evaluation tool is MATLAB 2019a and the maxFES is set to .
4.1. Parameter Setting
There is an important parameter in the QUATREDEG algorithm, which determines the size of the population allocated to the global optimal and global suboptimal subpopulations. The size of the population will affect the performance of the algorithm, especially when the two populations affect each other. Accordingly, it is necessary to determine the size of the two subpopulations first. Ideologically, it usually does not allow the size of the global optimal subpopulation to be lower than that of the global suboptimal. The population should be dominated by the global optimal subpopulation, which conforms to the basic logic of finding the global optimal solution. In our setting, when value is larger, the more individuals are allocated to the global optimal subpopulation and more development will be brought to the global optimal position; on the contrary, when the value is smaller, the less individuals are allocated to the global optimal subpopulation and the less the development will be. Therefore, in this experiment, the parameter is set to 0.1, 0.2, 0.3, 0.4, 0.5, and 0.6, respectively, and the dimension is set to 30D. The algorithms run 10 times for each different value. The results are shown in Table 1. The evaluation compares the average and variance of the running results. It is obvious that when is set to 0.4, the algorithm QUATREDEG is the most competitive. It can also be analyzed that when is larger, the algorithm will pay more attention to exploration, and when is smaller, the algorithm will have a better development. It also shows that when the exploration and development of the algorithm will tend to a balance; it is undoubtedly a good choice of the value.
4.2. Comparison of Optimization Accuracy
For the data reliability, 51 independent experiments were carried out with the comparison of the five algorithms. The mean and variance of the outcome of each algorithm are also compared and analyzed. Firstly, the dimensions were set to 10D, 30D, and 50D for the CEC2013 test suite, and then, 28 benchmark functions were carried out. The comparative experimental results are shown in Tables 2–4 for the dimensions of 10D, 30D, and 50D, respectively. Table 2 shows that the proposed QUATREDEG obtains better performance in 19 functions compared with the CQUATRE algorithm, better performance in 21 functions compared with the QUATRE algorithm, and better performance in 17 functions compared with both of the MABC and IABC algorithms. Table 3 shows the QUATREDEG obtains better performance in 22 functions as well as similar performance in 1 function compared with the MABC, better performance in 21 functions compared with the QUATRE, better performance in 19 functions as well as similar performance in 1 function compared with the IABC algorithms, and better performance in 18 functions as well as similar performance in 1 function compared with the CQUATRE. Table 4 shows the QUATREDEG obtains better performance in 23 functions as well as similar performance in 1 function compared with the MABC, better performance in 21 functions as well as similar performance in 1 function compared with the QUATRE, better performance in 21 functions compared with the CQUATRE, and better performance in 19 functions as well as similar performance in 1 function compared with the IABC. Based on the above results obtained from the CEC2013 test suite, it is obvious the proposed QUATREDEG performs better than the other algorithms.
The CEC2014 test suite also was used in the experiment for the performance evaluation of the proposed QUATREDEG. The results of 30 benchmark functions of CEC2014 are shown in Tables 5–7 for the dimensions of 10D, 30D, and 50D, respectively. Table 5 shows that the proposed QUATREDEG obtains better performance in 21 functions and similar performance in 3 function compared with the QUATRE algorithm, better performance in 20 functions and similar performance in 2 function compared with the CQUATRE algorithm, better performance in 18 functions and similar performance in 1 function compared with the MABC algorithm, and better performance in 15 functions and similar performance in 1 function compared with the IABC algorithm. Table 6 shows the QUATREDEG obtains better performance in 23 functions and similar performance in 4 functions compared with the QUATRE, better performance in 23 functions and similar performance in 3 functions compared with the MABC, better performance in 19 functions and similar performance in 2 functions compared with the IABC, and better performance in 19 functions and similar performance in 3 functions compared with the CQUATRE. Table 7 shows the QUATREDEG obtains better performance in 23 functions and similar performance in 2 functions compared with the QUATRE, better performance in 23 functions and similar performance in 1 function compared with the MABC, better performance in 22 functions and similar performance in 1 function compared with the CQUATRE, and better performance in 21 functions compared with the IABC. Based on the above results obtained from the CEC2014 test suite, it is also obvious the proposed QUATREDEG performs better than the other algorithms.
4.3. Analysis of Convergence Rate
The QUATREDEG algorithm was also evaluated from the perspective of convergence speed. Figures 1 and 2 show the convergence curves obtained by the representative benchmark functions in the CEC2013 and CEC2014 test suites, respectively. Figure 1 shows the proposed QUATREDEG performs much better than the other algorithms in functions fa12, fa15, fa16, and fa18 and performs similar to the other algorithms in functions fa2, fa3, fa4, and fa7. Figure 2 shows the convergence speed of the proposed QUATREDEG is significantly better than the other algorithms in functions fb1, fb5, fb12, and fb17 and is slightly better than the other algorithms in fb15, fb18, fb19, and fb27 functions. The ABC variant is easy to converge prematurely and fall into local optimal solution, such as the performance of fa3, fa4, fa7, fa12, fb17, and fb27 benchmark functions. However, the performance of QUATRE variants in the functions of fa2, fa3, fa4, fa7, fb18, and fb19 is only slightly insufficient. In summary, the proposed QUATREDEG can perform with a very competitive convergence speed compared with the QUATRE variants and ABC variants.
4.4. The Wilcoxon Signed Rank Test
Wilcoxon signed rank test is needed when two results present two different populations [27]. Here, we also give the Wilcoxon signed rank test results. Tables 8 and 9 give the Wilcoxon signed rank test results of CEC2013 test suites and CEC2014 test suites, respectively. The results are calculated by using Origin 2019b statistical analysis packages, and our proposed QUATREDEG algorithm is verified to be well performed.
The statistical analysis results of applying Wilcoxon’s test between QUATREDEG and other comparison algorithms using CEC2013 for all dimensions are summarized in Table 8. As the table states, QUATREDEG shows a significant improvement over all comparison algorithms, with a level of significance in 30 and 50 dimensions, over QUATRE and CQUATRE, with a level of significance in 10 dimensions. On the other hand, in all comparisons, QUATREDEG gets higher values than in all dimensions. Therefore, the performance of QUATREDEG algorithm is proved to be better than the other algorithms in this benchmark.
The statistical analysis results of applying Wilcoxon’s test between QUATREDEG and other comparison algorithms using CEC2014 for all dimensions are summarized in Table 9. As the table states, QUATREDEG shows a significant improvement over all comparison algorithms, with a level of significance in 30 and 50 dimensions, over QUATRE and CQUATRE, with a level of significance in 10 dimensions. On the other hand, in all comparisons, QUATREDEG gets higher values than in all dimensions. Therefore, the performance of QUATREDEG algorithm is proved to be better than the other algorithms in this benchmark.
4.5. Comparison of Complexity
The evaluation method of algorithm complexity adopted by the CEC2013 competition was used to evaluate the proposed QUATREDEG algorithm. Let be the standard computation time for the reference benchmark given by CEC2013 and be the standard computation time of the specified function fa14 for 200000 evaluations with a dimension of 30D. Let be the computation time of function fa14 using a specified algorithm for 200000 evaluations with the same dimension of 30D, and is the average of values acquired by 51 runs. Finally, the algorithm complexity is determined by . Table 10 shows the result of comparing the proposed QUATREDEG with the algorithms. It indicates that the QUATREDEG can obtain lower complexity compared with either the MABC or IABC. However, the QUATREDEG takes more computation time compared with the QUATRE and CQUATRE. It is reasonable due to the additional process in the proposed algorithm.
5. Conclusion
QUATRE is a relatively new algorithm proposed in recent years. Although it is powerful, it still can be improved in some concerns. For example, in the later stage of evolution, it is easier to fall into local optimal solutions and could not explore more. This study proposes a new QUATREDEG algorithm to solve this problem. In the proposed algorithm, excellent individuals in the current iteration process are fully utilized for the optimization. Not only the global optimal solution but also the global suboptimal solution is used to guide the evolution of individuals. According to the distance between the global optimal individuals and the global suboptimal individuals, a new mutation strategy is proposed to expand the current search environment. In addition, the population is divided into two subpopulations, including the subpopulation with the global optimal location as the search center and the subpopulation with the global suboptimal location as the search center. The expansion of search coverage and the increase of population diversity provide better conditions for more exploration, so as to avoid falling into the local optimal solution. Although QUATREDEG has stronger search performance and can explore more benchmark environments, it also reduces its search speed and sometimes makes it not easy to completely converge to the global optimal solution, which indicates the possible limitation of the approach. In the experiments, we use CEC2013 and CEC2014 to verify the performance of the proposed QUATREDEG algorithm. The results show that the optimization accuracy and convergence speed are excellent, and the time complexity is tolerable. In future works, the proposed algorithm will be applied to several practical engineering problems such as UAV path planning [28, 29]. Moreover, CEC2017 benchmark suite will be used to evaluate the further works.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was partially supported by the Fujian University of Technology under Grant GYZ20016 and GYZ18183, National Natural Science Foundation of China under Grant 61773415, and Fujian Provincial Natural Science Foundation under Grant 2017J01730.