Today, the modern world has entered a post peta scale era; the requirements are growing exponentially for computation and data processing, and the need for high-performance computation is increasing day by day; thus, the trend has changed from serial execution to high-performance computation. For achieving high-performance computation, several hurdles need to be tackled. Examples are those problems where the solution is very hard to find, or the solution merely exists or is very hard to achieve, e.g., NP-complete problems. To achieve the solution to those problems, wresearchers have very well-known heuristic techniques which provide the solution to these types of problems, but those solutions are not completely optimized. However, using optimization algorithms, such as the differential evolution (DE) and particle swarm optimization (PSO), optimized solutions to such problems can still be found.

Moreover, wresearchers will observe and discuss its variants, as it is already known that the original DE was first proposed by Storn and Price ^[1] in 1995; this drew the attention of many researchers as it was the simplest algorithm that provided the optimized solutions to many real-world problems. Thus, based on the original DE algorithm, different variants were introduced later. Some of the well-known approaches were the hybridization with other techniques, modification of mutation strategies, adaptation of mutation strategy and parameter settings, and use of neighbor information.

2. Hybridization with Other Techniques

The study [18]^[2] proposed a hybrid algorithm CADE which combines a customized canonical version of CA and DE. The canonical CA uses the ’Accept()’ function which selects the best individual from the population; then, it is updated in the belief space knowledge source by using the ’Update()’ function. The ’Influence()’ function selects the knowledge source that affects the evolution of the next generation of the population. The authors state that in CA, the major source of exploration is topographic knowledge, which is the knowledge about the functional landscape. Moreover, DE can also provide a complementary source of exploration knowledge hence it makes the perfect complement of CA. Both algorithms share the same population space and hence follow high-level teamwork. The study [19]^[3] proposed a mechanism, called ADE-ALC, which is abbreviated to the adaptive DE algorithm with an aging leader and challenges, which is helpful to solve optimization problems. It is introduced in the framework of DE, which helps in maintaining the diversity of the population. Moreover, in the DE algorithm, it is critical to retain the diversity of the evolutionary population in solving multimodal optimization problems. ADE-ALC achieves the optimal solution with fast-converging speed. In the ADE-ADC approach, the key parameters are updated that depend on the given probability distribution that could learn from their successful experience in the next generation. In the end, the effectiveness of the ADE-ALC algorithm is checked by numerical results of twenty-five benchmark test functions, where they found that ADE-ALC shows better or at least competitive optimization performance in terms of statistical performance. The authors proposed a hybrid technique in [20]^[4] to provide a statistically better performance in the optimization problems. The authors used a combination of the DE algorithm and the stochastic fractal search algorithm. As the hybrid approach is used, the combination of both algorithms has the strength of both competent algorithms and produces better results than the single algorithm. Moreover, to test the performance of the hybrid approach, they used the IEEE 30 benchmark suite, IEEE CEC2014. The results show a better performance of the hybrid approach compared to a single algorithm, and results show the statistical superiority of the hybrid approach.

3. Modification of Mutation Strategies

The study [21]^[5] proposed an approach to improve the search efficiency of the DE algorithm. The performance of DE is badly affected by parameter settings and evolutionary operators, e.g., the mutation, crossover, and selection process. To overcome this issue, the authors proposed a new technique, called a combined mutation strategy. A guiding individual-based parameter setting method and a diversity-based selection strategy are used. The proposed algorithm uses the concept of sub-population and divides the population into two subcategories, superior and inferior. Experiments are performed using CEC 2005 and CEC 2014 benchmarks. Moreover, their algorithm is different from greedy selection strategies; hence, they proved their algorithm produced more efficient results than previous proposed techniques. The study [22]^[6] points out that DE uses only the best solution to deal with global optimization problems. Similarly, mutation strategies in the existing literature utilize only one best solution. The authors challenged this concept and introduced the concept of m best candidates. The authors proposed that m best candidates should be selected to obtain the better gain or better achievement. A technique called the collective information-powered DE (CIPDE) algorithm is proposed to obtain the m best candidates and enhance the power of DE. The CEC2013 benchmark functions are used for experiments that prove that the CIPDE technique is much better than existing mutation strategies. The study [23]^[7] proposed a new technique in which they improved the structure of the DE algorithm. The authors argue that the performance of DE is based on control parameters and the mutation strategy; if wresearchers enhance both the selection of proper mutation strategy and control parameter, weresearchers can obtain better results. An automated system is proposed to produce an evolution matrix that later takes the place of the control parameter crossover rate, Cr. Furthermore, parameter F is renewed in the evolution process. The mutation strategy along with the time stamp system is also progressive in this study. The experiment results showed that the proposed technique is very competitive with the existing strategies.

4. Adaptation of Mutation Strategy and Parameter Settings

The study [24]^[8] proposed a new algorithm that can investigate problem landscape information and the performance histories of operators for dynamically selecting the most suitable DE operator during the evolution process. The need for this mutation strategy is justified by the fact that predominantly existing works use a single mutation strategy. The authors present the concept of using multiple mutation strategies. Multiple mutation strategy-based algorithms are reported to provide far better results than single mutation-based algorithms. In such algorithms, the emphasis is to obtain the better performing evolutionary operator, which will be totally based on performance history for creating new offspring. This procedure is carried out dynamically; it selects the most suitable evolutionary operator. Experimental results using 45 optimization problems show the efficacy of the proposed algorithm. The study [21]^[5] proposed a new and improved version of the DE algorithm. Firstly, the search strategy of the previous DE is improved by using the information of individuals to set the parameter of DE and update the population, and the combined mutation strategy is produced by combining two single mutation strategies. Secondly, the fitness value of the original and guiding individual is used. Finally, a diversity-based selection strategy is developed by applying a greedy selection strategy. The performance is evaluated using CEC 2005 and CEC 2014 benchmarks, and better results are reported. The study [25]^[9] investigates the high-level ensemble in the mutation strategies of DE algorithms. For this purpose, a multi-population-based framework (MFT) was introduced. An ensemble of differential evolution variants (EDEV) based on three high, popular, and efficient DE versions is utilized. JADE-adaptive DE with optional external archive, CoDE DE with composite trial vector generation strategies and control parameters, and EPSDE DE algorithm with an ensemble of parameters and mutation strategies are joined. Furthermore, the whole population of EDEV is divided into four subcategories. In the end, the EDEV-based test is run on the CEC 2005 and CEC 2014, which shows better performance of EDEV.

5. Use of Neighbor Information

The study [26]^[10] proposed an adaptive social learning (ASL) strategy for the DE algorithm so that neighborhood relationship information of individuals in the current population can be extracted; this is called the social learning of DE (SL-DE). In the classical DE algorithm, parents in mutation are randomly selected from the population. However, in the ASL strategy, the selection of parents is intelligently guided. In ASL, every individual can only interact with their neighbor and parents. To check the efficacy of SL-DE, it is applied to the advanced DE algorithm. Results demonstrate that SL-DE can achieve a better performance than most of the existing variants of DE. The study [27]^[11] proposed the technique in which the authors applied the global numerical optimization and the index-based neighborhood on DE. In this technique, the authors used information and population to enhance the performance of DE. In the existing literature, neighborhood information of the current population has not been systematically exploited in DE design. The authors proposed neighborhood-adaptive DE (NaDE). The NaDE technique is based on the pool of index-based neighborhood topologies. Firstly, several neighborhood interactions for every discrete individual are recorded and later used adaptively for specific function selection. Secondly, the authors introduced a neighborhood-directional mutation operator in NaDE to obtain the new resolution in the designated neighborhood topology. Finally, NaDE is easy to operate and implement and can be matched with earlier DE versions on different kinds of optimization problems. The authors proposed a new approach called enhancing De with a random neighbors-based strategy in [2]^[12]. Traditionally, DE/rand/1 and DE/best/1 mutation strategies are used with DE. In DE/rand/1, the base vector is chosen from the population randomly for better exploration. On the other hand, the DE/best/1 strategy has better exploitation and poor exploration. To overcome this issue, the authors proposed DE/Neighbor/1. In the proposed technique, for each individual population at every generation, the neighbors are chosen from the population in a random manner and the base factor of the DE/Neighbor/1 mutation strategy should be the best one among neighbors. Xiong et al. [28]^[13] introduced a speciation-based DE algorithm in their research work. The presented algorithm utilizes the mechanism of the adaptive neighborhood by considering multimodal benchmark functions. They used the concept of achievement to store inferior individuals in each iteration and remove similar-performing individuals using the mechanism of crowding relief. In their presented approach, the use can fine-tune the parameters adaptively. Liao et al. [29]^[14] considered the system of non-linear equations using the DE algorithm in their research work. They utilized neighborhood-based information to increase the exploitation capability of the DE algorithm. The size of the neighborhood is dynamically selected with the adjustment of parameter adaption in the state of evolution. The search efficiency of the DE algorithm was enhanced by achieving significant results. The research work [30]^[15] presented binary differential evolution based on a self-adaptive neighborhood method for change detection in super-pixels. The change detection process is carried out by using a binary DE mutation strategy to reduce the dimension of super-pixels. Lio et al. [31]^[16] introduced a variable neighborhood-based DE algorithm by utilizing a history archive in their research work. During the evolution process, the neighborhood size is dynamically controlled in their presented approach. The information exchange process is performed between the current population and the population stored in the achieved research. The information exchange is helpful to escape from local optima during the evolutionary process. Liu et al. [32]^[17] considered the economic dispatch problem by incorporating a direction-inducted strategy in neighborhood-based DE algorithm in their research work. They have used a new mutation strategy named a neighborhood-based non-elite direction strategy that enhances the exploitation capability of the presented algorithm. Sheng el al. [33]^[18] introduced the concept of an adaptive neighborhood-based mutation in the DE algorithm. The presented technique is helpful to focus on an intensive search followed by an initial search by the DE algorithm. They also used a Gaussian local search to evolve promising individuals during the search process. Wang et al. [34]^[19] introduced an adaptive memetic-based neighborhood crossover strategy in their research work. They used the concept of a multi-nitching sampling for the evolution of the sub-population to ensure intensive search. They also presented the design of adaptive elimination-based local search in their research work. Their neighborhood crossover strategy focuses on an exploitation capability in the DE algorithm to encourage a good quality solution. Cai et al. [35]^[20] presented a self-organizing DE algorithm in their research work that is helpful in guiding the search process by utilizing neighborhood information. The adaptive adjustment of various individuals in the explored works use a cosine similarity in the self-organizing map. Segredo et al. [36]^[21] proposed a neighborhood based on proximity in the DE algorithm that is helpful to balance between exploration and exploitation during the evolution process. They used Euclidean-based distance to measure the similarity between neighbors of individuals and termed it a similarity-based neighborhood search. Baioletti et al. [37]^[22] presented algebraic differential evolution based on a variable neighborhood concept in their research work. Their presented algorithm utilizes the information of three neighborhoods for shifting and swapping purposes to form permutations. Tian and Gao [38]^[23] introduced the adaptive evolution method by using the neighborhood mechanism in the DE algorithm. They used a selection probability based on the selection of individuals, as well as two mutation operators based on the neighborhood to improve the evolution process. They also used a simple reduction method to adjust the population size to incorporate diversity in the DE algorithm. Tarkhaneh and Moser [39]^[24] performed a cluster analysis by incorporating a neighborhood search and Archimedean spiral in the DE algorithm in their research work. Mantegna Levy’s flight mechanism was used in the Archimedean spiral by generating robust solutions to balance between exploration and exploitation during the searching process.