Dania AlKhulaifi | Maryam AlQahtani | Zenab AlSadeq | Atta-ur Rahman* | Dhiaa Musleh
© 2022 IIETA. This article is published by IIETA and is licensed under the CC BY 4.0 license (http://creativecommons.org/licenses/by/4.0/).
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Differential Evolution (DE) is a widely used global searching algorithm that solves real-world optimization problems. It is categorized as a stochastic parameter optimization method that has a broad spectrum of applications, notably neural networks, logistics, scheduling, and modeling. In practice, different optimization issues need different parameter settings. Due to DE simplicity, ease of implementation, and dependability, many scientists were interested in examining this algorithm. Nonetheless, the quality of DE and its variations are directly influenced by different mutation techniques and control parameter settings. In this paper, an overview and analogy of some algorithms that employ different mutation techniques will be illustrated. Additionally, a novel strategy that uses different mutation methods is proposed and compared with some existing strategies.
differential evolution, self-adaptive differential evolution, mutation strategy, optimization
The existence of Global Optimization Problems (GOP) motivated researchers to develop methods that employ mathematical optimization and probabilistic algorithms. Most mathematical optimization programs feed on gradient functions, as non-differentiable problems require different methods. Furthermore, when handling complex optimization problems, mathematical programming approaches are vulnerable to being locked in a local optimum which keeps possible better solutions undiscovered [1]. Several real-world problems such as nonlinear, multimodal, hybrid, composite, and large-scale problems are complex by nature [2]. Experimental results showed that the meta-heuristic method performs well in tackling such problems, attracting scholars to conduct a further investigation [3]. Most meta-heuristic algorithms are based on natural phenomena. On the contrary, typical mathematical programming methods replicate natural phenomena using established rules and randomization methods to avoid gradient difficulties in optimization. Many studies on metaheuristic approaches have been done over the last two decades [4, 5]. Additionally, multiple metaheuristic algorithms have been developed to address continuous problems, including the Genetic Algorithm [6, 7], Differential Evolution [8-10], and Artificial Bee Colony [11]. Nonetheless, DE has its own flagship and superiority over the peer approaches due to its applicability in both discrete and continuous spaces, less vulnerability to stuck in the local minima and verstaile nature of adaptability in terms of stochastic operator selection and fine tuning in fusion with other secondary algorithms, a case of memetic computing [1]. The use of "trial-and-error" to obtain optimal answers is the main feature of these algorithms. As a result, these algorithms have shown success in addressing global optimization issues [12]. Storn & Price in 1995 [13] established the DE strategy, which is a basic yet effective population-based non-linear search method that works as a global optimizer in a continuous search area. Engineering, communication [8-10], and pattern recognition are just a few of the fields where DE has been applied successfully. There are various advantages of DE over the other evolutionary algorithms of similar nature like GA,
1.1 Classical differential evolution algorithm
DE is a basic nature-inspired algorithm that solves optimization problems. One of its main features is the ease of implementation and study. It operates in four basic steps [25] as depicted in Figure 1.
Figure 1. Working scheme of classical DE
Step 1: Initialization
DE will find an optimal solution in a multidimensional continuous space by randomly initializing the population of the given problem. Those parameter vectors are thought to be like chromosomes that produce the final solution i.e. Scale Factor (F), Crossover Rate (CR), Population size (NP), and a number of iterations (I).
Step 2: Mutation
In this step, a trial vector $v_i^{w+1}$ is produced by the evolution of a target vector. Typically, $v_i^{w+1}$ is found by one of the below equations:
$v_i^{W+1}=X_{i 3}^K+f\left(x_{i 1}^w-x_{i 2}^w\right)$ (1)
$v_i^{W+1}=X_{\text {best }}^K+f\left(x_{i 1}^w-x_{i 2}^w\right)$ (2)
$v_i^{W+1}=X_i^K+F 1\left(x_{\text {best }}^w-x_i^w\right)+F 2\left(x_{i 1}^w-x_{i 2}^w\right)$ (3)
$v_i^{W+1}=X_{\text {best }}^K+F 1\left(x_{i 1}^w-x_{i 2}^w\right)+F 2\left(x_{i 3}^w-x_{i 4}^w\right)$ (4)
$v_i^{W+1}=X_{i 5}^K+F 1\left(x_{i 1}^w-x_{i 2}^w\right)+F 2\left(x_{i 3}^w-x_{i 4}^w\right)$ (5)
where, F, F1 and F2 are scale factors $x_{\text {beas }}^w$ indicates the ultimate vector in iteration (wi(n)) and those represent different integer values that are randomly selected from the set {1,2…, N}.
Step 3: Crossover
After generating a donor vector, a crossover step takes place. This step applies exponential and binomial crossover mechanisms. The method of crossover is as follows:
$u_{i j}^{w+1}= \begin{cases}v_{i j}^{w+1} \text { if } \operatorname{rand}()<C R \text { or } j=r \\ v_{i j}^w & \text { else }\end{cases}$ (6)
Step 4: Selection
The final step is to decide if the target and trial vectors would pass to the next generation. When the stopping condition is reached as in the maximum number of iterations, the algorithm terminates, or else it will go back and repeat from step 3. The selection method operates as follows:
$\overrightarrow{x_{l, G+1}}=\vec{U}_{i, L}$
$i F f\left(\overrightarrow{U_{l, L}}\right) \leq f\left(\overrightarrow{X_{l, L}}\right)=\vec{X}_{i, L}$
$i F f\left(\overrightarrow{U_{l, L}}\right)>f\left(\overrightarrow{X_{l, L}}\right)$
The section provides a brief but comprehensive overview of plain DE algorithm as well as it most widely used and promising (in terms of applications) variants.
Fan and Yan [26] proposed a DE-based solution dedicated for constrained optimization problems. The method enables each parent to produce several offspring by applying a different mutation operator. The proposed strategy increases the probability of parents producing better solutions. Results showed an acceptable average computational cost and superior performance with some best previously developed solutions.
DE/rand/1/bin is the standard DE algorithm proposed by [13]. After that, Qin et al. [17] presented (SaDE) which uses a novel mutation mode (DE/current-to-pbest/1). Two new parameters have been added. The c parameter determines the rate of adaptation, while the p parameter determines the greediness of the mutation strategy. In SaDE, trial vector generation procedures, as well as their two control factors, have been probabilistically assigned to each target vector in the present population, depending on the probabilities gradually learned from experiments to produce better solutions. Experiments revealed that the SaDE algorithm could evolve appropriate methods and parameter values as evolution progressed. Also, the reduced root-mean-square deviation was attained by this variant. There are two new parameters added. The rate of adaptation is controlled by the c parameter, whereas the greediness of the mutation strategy is determined by the p parameter. Wu et al. [23] proposed CoDE which uses three trial vector generating procedures which are: (i) “rand/1/bin”; (ii) “rand/2/bin”; (iii) “current-to-rand/1” and sets three control parameter which are: (1) F=1.0 and CR=0.1; (2) F=1.0 and CR=0.9; (3) F=0.8 and CR=0.2. CODE creates trial vectors by combining them at random. All the CEC2005 contest test instances have been tested with CoDE. Results yielded competitive results. Ronkkonen et al. [22] proposed SAMDE, which uses an adaptive mutation operator to provide the benefits of the DE/rand/1/bin method and the DE/best/2/bin approach. This method was made to solve a design problem as a constrained minimization problem. As a result, its convergence performance was significantly enhanced. The findings are supported by numerical experiment data. Parameters in EPSDE which is a DE consisting of an ensemble of mutation and crossover techniques, as well as their control factors. Throughout the evolution process, combinations of different mutation and crossover techniques, in addition to a group of values for each control parameter, compete to create offspring in EPSDE. An attempt to tune Scale Factor (F) and Crossover Rate (Cr) depending on the objective function value in the population level without user input. This method yielded success in accuracy, robustness, convergence, and speed compared with other previously developed variants. Moreover, the ensemble of mutation strategies and Parameters in DE [27] were employed to solve numerical optimization problems. In this variation, different distinct mutation strategies coupled with values for each control parameter are adjusted throughout the evolution process. This set competes to produce offspring. The performance of EPSDE was found to be better on a set of bound-constrained problems and compared with conventional DE and different variations of DE. Zou et al. [28] aim at solving unconstrained optimization problems by proposing a modified version of the DE algorithm (MDE). To tune crossover rates and scale factors, their method increases the diversity of possible solutions by the guess distribution and uniform distribution. Also, an external archive with dynamic probabilistic methods was used to ensure quality. Additionally, new solutions are created during the late evolution stage which improves the convergence of the algorithm. After considering other candidate solutions, a middle solution will be generated. It was proven that the MDE algorithm gets better objective function values, hence it can be employed to solve unconstrained optimization problems. Wu et al. [29] presented a multi-population-based approach that ensembles multiple strategies, resulting in the Multi-Population Ensemble DE (MPEDE), in which the population is divided into three subpopulations of similar size and one larger subpopulation The three populations are exposed to three different mutation strategies: "current-to-pbest/1," "current-to-rand/1," and "rand/1," with the controlling parameters of each method made adaptively. A self-adaptive DE method with discrete mutation control parameters was proposed by Fan and Yan [26] (DMPSADE). This ensures that each variable has its mutation control parameter, as well as a crossover control parameter and mutation technique for each individual. For handling unconstrained global real parameter optimization in the continuous domain, Li and Yin [30] suggested a modified (DE) with self-adaptive parameter sets. This method employs two mutation criteria based on the population's rand and best individuals. Mutation, crossover, and greedy selection are three types of operators used in this approach to identify solutions. The parameter selection and mutation strategy are significant in resolving an issue. As a result, this algorithm can be superior to others. Mallipeddi et al. [31] proposed (MPMSDE) which instead of using the (MPEDE) grouping method, will create a new grouping approach that uses the ranking of strategies to allocate resources to the most suitable strategy. In addition, to avoid dropping into a local optimum, an information-sharing mechanism will be used in the largest sub-population. Additionally, the "DE/pbad-to-pbest-to-gbest/1" mutation will be used to replace the MPEDE method. To update individuals, the proposed method shows the global best solution which is a strong factor compared to MPEDE. The newly developed technique has proven to balance exploration and exploitation while yielding increased convergence. The results of the experiments reveal that the MPMSDE algorithm's performance is competitive on specific functions. The proposed MPMSDE method has outstanding performance in solving some issues, hence it is important.
Wang et al. [32] proposed a Self-adaptive DE algorithm with enhanced mutation mode (IMMSADE). Each individual has their control settings which are dynamically and rigourously changed/updated based on the population diversity and the individual differences among the produced offsprings. Sun et al. [33] proposed a novel simple variant of the DE Time Varying (TVDE) algorithm. It is used in numerical optimization problems and can be applied for multimodal optimization, and dynamic optimization. In TVDE, Three functions with time-varying features are used to develop a new mutation operator. Also, two parameters are automatically set during the process. For quality assurance, the proposed TVDE was evaluated against seven DE versions. The result demonstrates that among the eight DE algorithms, the TVDE method achieves the best overall performance. Sun et al. [34] proposed a new DE version (CSDE). Two mutation operators with altered features were used to generate the mutant vector. A coordination mechanism based on the past success rate was employed to adjust the two mutation operators. Results show that CSDE was better than seven other DE variants in most cases. Ali et al. [35] proposed a new technique that enables differential evolution algorithms to solve binary-based problems, such as binary knapsack. The method includes a mapping mechanism, representation of possible solutions, and diversity technique. Additionally, the fitness evaluation method was enhanced along with improved means of nominating candidate solutions. The algorithm was tested on 44 binary knapsack instances, and it has shown success in terms of computational time and finding better solutions for bigger knapsack problems. There are plenty of applications of DE in different areas of research [36-46].
The research of multi-objective DE algorithm and hybrid algorithm is a relatively new topic in DE. That is why it is among the most research intensive area to be explored in terms of its application in the multi-objective problems domain of hybrid [47-50] and fused intelligent systems such as proposed by Mahmud et al. [51] respectively. In multi-objective domain, the solution search spaces become huge due to the lenght of objective vector’s dimension [52-60]. For example, in case of binary/discrete problem, its complexity become exponential O(2n) and in case of continuous domain its complexity is even worst that is super-exponential O(nn). This actually makes the room for DE like algorithms to be investigated with all their poweful varaints. Table 1 presents the comparison of different DE based algorithm.
Table 1. Comparison of various DE algorithms
|
Ref. |
Proposed Algorithm |
Mutation method |
Results |
|
[13] |
DE |
DE/rand/1/bin |
It was assumed to be the finest at the time, but it would have conveyed an excellent role. |
|
[25] |
Modified DE for constrained optimization |
DE/rand/1/bin |
Improved result quality, reduced average computational cost. |
|
[17] |
SADE |
“DE/rand-to-best/1/bin,” “DE/best/1/bin,” and “DE/best/2/ bin” |
It was more effective in obtaining higher success rates and higher quality solutions, with better stability and reduced value of standard deviation. |
|
[23] |
CoDE |
Composite three vector generation schemes and three control parameters |
The analysis indicated that it performed better than the other competitors in terms of total performance. |
|
[22] |
SAMDE |
E/rand/1/bin strategy and the DE/best/2/bin |
The findings of the computational studies revealed that the SAMDE methodology outperforms earlier design methodologies. |
|
[24] |
EPSDE |
(DE/rand/1/bin) |
The effectiveness of the proposed technique was favorable when compared to classifical DE methods |
|
[28] |
MDE |
DE/rand/1/bin DE/best/1/bin |
Convergence rate is high, and exploitation capacity is increased. |
|
[26] |
DMPSADE |
It dynamically modifies mutation technigues and control parameters by competition. |
The statistical findings reveal that DMPSADE's average performance outperforms all other competitors. |
|
[29] |
MPEDE |
It made up of three different mutation strategies, “current-to-pbest/1” and “current-to-rand/1” and “rand/1”. |
It enhanced the adaptation of DE based on multiple population. |
|
[30] |
MDE |
“DE/rand/1/bin”. |
The algorithm performs much better than the approaches in the literature, or is at least similar to it. |
|
[31] |
IMMSADE |
Improving “DE/rand/1” mutation mode of the basic DE. |
In terms of overall performance, IMMSADE exceeds the base DE and other DE algorithms. |
|
[32] |
TVDE |
DE/rand/1, DE/best/1, DE/current-to-best/1, DE/best/2, and DE/rand/2. |
Among the eight DE algorithms, the TVDE algorithm has the best overall performance. |
|
[33] |
CSDE |
DE/current-to-pbest/1. |
In most cases, CSDE outperforms seven state-of-the-art DE versions. |
|
[35] |
NBin-DE |
DE/rand/1 |
Better average profit value, less computational time up to 94.4% improvement. |
|
[12] |
MPMSDE |
Improving of MPEDE, DE/pbad-to-pbest-to-gbest/1 |
The MPMSDE algorithm performs much better than competition in terms of calculation accuracy and convergence rate, according to the results. |
During different phases of evolution, several mutation strategies with several parameter settings might give better results compared with using a single mutation strategy, as in the typical DE. Based on these insights, the proposed technique is made up of a pool of mutation and crossover techniques, as well as a pool of values for each of the control parameters. Every member of the starting population is given a mutation strategy and associated parameter values from one of the pools at random. Members of the population (the Target Vectors) create children (the Trial Vectors) using the mutation technique with parameter values provided to them. If the created trial vector outperforms the target vector, the proposed mutation technique and consequent parameters’ values are carried over to the following generation's parents (Target Vector) and so on. The proposed mutation technique and parameter adjustment that created a better child vector in contrast to the parent vectors is saved. In case the target vector performs better than the trial vector, it is reinitialized at random with a the proposed mutation technique and consequently the related parameters’ values from the corresponding pools or the winning combinations are saved with an equal/flat probability. This technique results in a higher possibility of producing offsprings (children/output vectors) in the future generations due to a greater mixture of various mutation approaches and/or operators and the related control parameters with more diversity and variations.
3.1 Optimization strategy
In this paper, the proposed method uses JADE and DE/pbad-to-pbest-to-gbest/1 mutation strategy to grant improved convergence and make use of globally optimal solutions. Binomial and exponential crossover algorithms with 100 iterations are utilized as crossover strategies, (-5.0,5.0) lower and upper bounds, a constant population size of (50) while mutation factor was ∈{0.5} and a crossover rate of ∈{0.7}. The main purpose of this study is to investigate the said combination of the mutation operators and crossover techniques in the traditional DE algorithm to study its convergance rate and the strength to achieve the global optimum. Consequently, the said combination can be used for the problems with multi-objective cost functions optimization with more effectiveness compared to other variants and counterparts.
3.2 Implementation
This method used an objective function which is obj(x). All steps were combined in a differential evolutionary process. The population size, the boundaries of each input variable, the total number of iterations, the mutation scale factor, and the crossover rate are all input variables to this function. The function will return the most suitable option discovered and its evaluation. The coding implementation of the proposed variant of the differential evolution algorithm was conducted in Python language.
After multiple trials, the results were drawn in a graph depicted in Figure 2. It reveals the objective function evaluation after each improvement. That is the algorithm exhibits with large changes initially (transient mode) and very small changes towards the end of the search (steady state) as the algorithm converged to the optima. The two dimensional plot shows the objective function evaluation for each improvement, as the algorithm converged on the optima. There were big changes at first and then very modest changes towards the end of the search. The trend is natural and realistic in terms of its convergence rate and no anomaly can be observed. In the early iterations the error is more nonetheless as the number of iterations increase from 5, the error reduces to zero for the standard objective function obj(x) where x is the trial vector being generated by the proposed methodology.
Figure 2. Performance convergence analysis
As shown in Figure 3, the objective function values are returned at every iteration which can be depicted by modifying the main differential evolution function to keep track of the values of the objective function or the cost function and return them in the list, namely, objective iterative. Finally the solution is found and the error goes to zero ultimately that shows no more need to evolute the algorithm. In Table 2, the proposed approach has been compared with two standard variants namely EPSDE and MPMSDE, in terms of standard deviation and mean. That are the two figure-of-merits to justify the performance of such type of stochastic algorithms in the literature. Consequently, it is evident that the proposed algorithm outperforms both EPSDE and MPMSDE in terms of both metrics. The comparison is fair in a way that same objective function has been used under same simulation environment and conditions.
Figure 3. Convergence and final outcome
Table 2. Comparison of means and standard deviations
|
|
Proposed Algorithm |
EPSDE |
MPMSDE |
|||
|
Mean |
Std |
Mean |
Std |
Mean |
Std |
|
|
F1 |
0 |
0 |
0.2278 |
-0.06017 |
0.01954 |
0.04664 |
|
F2 |
-0.18418 |
-0.0018 |
-0.18418 |
0.03751 |
0.00143 |
0.0008 |
|
F3 |
-0 |
0 |
-0 |
0 |
-4.e-05 |
-5.e-05 |
|
F4 |
0.e+01 |
0.e+01 |
-0.e+00 |
1.e-05 |
0.e+00 |
1.e-05 |
Differential Evolution is one of the famous nature-inspired algorithms dedicated to solving optimization problems especially the problems of multiobjective nature. It has a variety of applications, and it has shown success in solving multidimensional continuous-spaced optimization problems. Much literature was done earlier to enhance this algorithm either by applying different mutation strategies or combining the advantages of best-performing algorithms. In this paper, many DE improvements were reviewed and compared in terms of their used mutation method and overall results to guide future research to enhance such algorithms. Also, a novel variant of DE was developed based on the best features of EPSDE and MPMDE which are some of the latest DE improvements. As a future direction of this work, taking advantage of further best-performing DE variations to create a modified version of the algorithm would be of great interest. Also, incorporating more means of comparison such as disadvantages could enhance this work. In future, the proposed algorithm may be investigated for the real life problems involving multiobjective nature. Moreover, the proposed algorithm being global optimizer may be hybridized with some local optimizer for sake of fine tuning the optimization results in the umbrella of memetic computing. Similarly, other hybrid intelligent techniques can also be investigated.
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