Knowledge transfer (KT) plays an essential role in evolutionary multitask optimization (EMTO), whereas ”what to transfer” and ”how to transfer” are the key focuses that influence the KT performance. For the issue ”what to transfer”, many EMTO algorithms often transfer the top individuals directly during the KT process. However, the transferred individual information is not always effective, especially when the tasks own the low similarity. For the issue ”how to transfer”, many KT methods mainly focus on learning from all dimensions in a single individual, causing the lack of learning diversity. Therefore, in this paper, we propose a novel directional combination learning method for multitask optimization, termed as DCLMTO, tries to address these two issues effectively. Specifically, for the issue ”what to transfer”, a directional learning (DL) strategy is developed to deeply extract the useful directional knowledge, which is behind the individual information and shows the trend of moving toward better position, for KT preparation. For the issue ”how to transfer”, a combination learning (CL) strategy is designed to fully combine the superiorities of different individuals for enriching the learning diversity. Moreover, DCLMTO can be easily extended to many-task optimization problems (MaTOPs) and can directly avoid the difficulty of task selection, showing its convenience and scalability. Experimental results demonstrate that DCLMTO significantly outperforms other state-of-the-art EMTO and EMaTO algorithms on the multitask benchmarks CEC17 and CEC22, and the many-task benchmarks CEC19 and WCCI20. Finally, DCLMTO is applied to a real-world multitask planar kinematic arm control application, further demonstrating its applicability.

Large-scale optimization problem (LSOP) is an essential research topic in the field of evolutionary computation community. Many large-scale optimization algorithms often maintain a large population for diversity enhancement. However, updating such a large population consumes a significant number of fitness evaluations (FEs), which may lead to the insufficient evolution of the population. In light of this, this article proposes a small-scale learning particle swarm optimization (SSLPSO) for solving LSOPs. In the small-scale learning mechanism, only up to two representative individuals are updated in every generation to effectively save FEs and prolong the evolutionary generations, so as to refine the solution accuracy. Specifically, we first design a representative individual selection (RIS) strategy to select the convergence representative individual and the diversity representative individual for updating. Then, we develop a representative individual learning (RIL) strategy, which includes a convergence learning method and a diversity learning method for the convergence representative individual and the diversity representative individual, respectively. Meanwhile, we further propose an adaptive strategy adjustment (ASA) method based on evolutionary state assessment to determine whether the representative individuals should be updated, further achieving the adaptive adjustment of the evolutionary behavior in the population. Experimental results on the commonly used large-scale test suites, IEEE CEC2010 and IEEE CEC2013, show that the performance of SSLPSO is significantly better than, or at least comparable to other state-of-the-art large-scale optimization algorithms, including the winners of large-scale competitions. Finally, the application of SSLPSO to a large-scale constrained water distribution network optimization problem further demonstrates its real-world applicability.

evolutionary multitask optimization (EMTO) aims to optimize multiple tasks simultaneously. In recent years, various EMTO algorithms based on knowledge transfer (KT) have been developed to utilize the information from other tasks and promote the optimization of the current task. However, most of them often use the fixed KT probability (ktp) and a single evolutionary search operator (ESO) during the evolution process, which lacks an adaption mechanism and cannot meet the different searching requirements among multiple tasks. Fuzzy system can effectively express the qualitative knowledge with unclear boundaries, which has good adaptability to nonindependent EMTO. Therefore, this article proposes a fuzzy adaptive multitask optimization (FAMTO), which employs a fuzzy adaptive transfer (FAT) strategy for intertask KT to achieve the adaptive adjustment of the ktp by designing a comprehensive evaluation in KT performance from two aspects, including the survival rate and the quality of transferred offspring. In FAT strategy, the fuzzy logical is employed to handle the interdependent relationships among multiple indicators, further achieving the more robust and adaptive ktp adjustment. In addition, an individual-based random selection (IRS) strategy is developed for each individual to choose the suitable ESO for intratask self-evolution in fuzzy adaptive multitasking optimization (FAMTO). Experimental results show that FAMTO achieves significantly better performance than other state-of-the-art EMTO algorithms on two well-known multitask benchmarks, CEC17 and CEC22. Furthermore, FAMTO is applied to a real-world multitask planar kinematic arm control application, demonstrating its applicability. Finally, the extended experiments on many-task optimization problems (MaTOPs) illustrate the scalability of FAMTO.

Knowledge transfer (KT) has been regarded as an efficient method in evolutionary multitask optimization (EMTO) by utilizing the information of other tasks to promote the optimization of the current task. Most KT methods achieve information communication across index-aligned dimensions. However, the index-aligned dimensions are not always similar or related, which is not always suitable for communication and causes the low efficiency in KT. Moreover, when the KT occurs in the heterogeneous tasks with different dimensions, the task with lower dimensions often pads the extra dimensions to make their dimensions equal. However, the dimension-padding often involves the redundant or useless information, which may mislead the KT process. In this article, a novel multilevel and multisegment learning multitask optimization (MMLMTO) algorithm based on niching technique is proposed to achieve high-quality KT. First, a multilevel learning strategy is proposed to divide the population into three levels according to fitness values for better selecting the individuals for KT. Second, a multisegment learning strategy is proposed to split some top individuals in each level into several segments, and each segment will find its closest segment to form a niche, where the KT is executed. This ensures that KT occurs in the similar or related dimensions and avoids the dimension-padding to eliminate the influence of the redundant information. Experimental results on IEEE CEC2017 and IEEE CEC2022 multitask benchmarks fully demonstrate the effectiveness of MMLMTO, which can significantly outperform other state-of-the-art multitask algorithms. Finally, MMLMTO is applied to a real-world multitask rover navigation application problem to further demonstrate its applicability.

Ant Colony Optimization (ACO) has witnessed great success in tackling the Traveling Salesman Problem (TSP). In ACO, ants involved in the pheromone update play pivotal roles in its optimization effectiveness. Along this road, this paper designs an ant selection mechanism along with a non-linear weight method for ACO to update the pheromone effectively, leading to a novel ACO, called NLW-ACO. Particularly, NLW-ACO leverages the fitness values of ants to assign each ant a selection probability. Then, it adaptively chooses ants for pheromone update. Subsequently, a nonlinear weight is assigned to each selected ant based on its fitness value to update the pheromone matrix. Resultantly, better ants have higher selection probabilities and larger weights to take part in the pheromone update. This leads to that NLW-ACO compromises search convergence and search diversity appropriately to seek for the optimum. Experiments have been carried out on 10 TSP instances of diverse scales. The experimental findings substantiate that NLW-ACO significantly outperforms the 5 typical ACO methods, especially on large-scale TSP problems.

Knowledge transfer (KT) is crucial for optimizing tasks in evolutionary multitask optimization (EMTO). However, most existing KT methods can only achieve superficial KT but lack the ability to deeply mine the similarities or relationships among different tasks. This limitation may result in negative transfer, thereby degrading the KT performance. As the KT efficiency strongly depends on the similarities of tasks, this article proposes a neural network (NN)-based KT (NNKT) method to analyze the similarities of tasks and obtain the transfer models for information prediction between different tasks for high-quality KT. First, NNKT collects and pairs the solutions of multiple tasks and trains the NNs to obtain the transfer models between tasks. Second, the obtained NNs transfer knowledge by predicting new promising solutions. Meanwhile, a simple adaptive strategy is developed to find the suitable population size to satisfy various search requirements during the evolution process. Comparison of the experimental results between the proposed NN-based multitask optimization (NNMTO) algorithm and some state-of-the-art multitask algorithms on the IEEE Congress on Evolutionary Computation (IEEE CEC) 2017 and IEEE CEC2022 benchmarks demonstrate the efficiency and effectiveness of the NNMTO. Moreover, NNKT can be seamlessly applied to other EMTO algorithms to further enhance their performances. Finally, the NNMTO is applied to a real-world multitask rover navigation application problem to further demonstrate its applicability.

Multimodal optimization aims at efficiently finding multiple optimal solutions of a problem. Owing to the population‐based search mechanism, evolutionary algorithms (EAs) are becoming increasingly popular in solving multimodal optimization problems (MOPs). Most existing work focuses on designing and incorporating niching techniques into EAs so that multiple subpopulations can be formed and assigned to locate different optima. To further enhance the exploration and exploitation abilities of existing EAs, this paper developed a multimodal level‐based learning strategy. The basic idea is that individuals should be treated differently according to their positions in the subpopulation. In the evolutionary process, a subpopulation is formed for each candidate solution by grouping its neighboring solutions. Then, individuals in the subpopulation are sorted according to their fitness. Subsequently, the multimodal level‐based learning strategy applies different mutation operators to different individuals according to their rankings. Experiments are conducted on a set of benchmark problems to verify the efficacy of the multimodal level‐based learning strategy. The results show that the proposed learning strategy can significantly enhance the performance of the existing algorithm. In addition, the algorithm integrated with the proposed strategy is applied to the task of finding multiple roots of nonlinear equation systems (NESs). The results indicate that with the support of the proposed learning strategy, the integrated algorithm compares favorably with state‐of‐the‐art root finding algorithms.

Large-scale optimization problems (LSOPs) are challenging because the algorithm is difficult in balancing too many dimensions and in escaping from trapped bottleneck dimensions. To improve solutions, this paper introduces targeted modification to the certain values in the bottleneck dimensions. Analogous to gene targeting (GT) in biotechnology, we experiment on targeting the specific genes in candidate solution to improve its trait in differential evolution (DE). We propose a simple and efficient method, called GT-based DE (GTDE), to solve LSOPs. In the algorithm design, a simple GT-based modification is developed to perform on the best individual, comprising probabilistically targeting the location of bottleneck dimensions, constructing a homologous targeting vector, and inserting the targeting vector into the best individual. In this way, all the bottleneck dimensions of the best individual can be probabilistically targeted and modified to break the bottleneck and to provide global guidance for more optimal evolution. Note that the GT is only performed on the globally best individual and is just carried out as a simple operator that is added to the standard DE. Experimental studies compare the GTDE with some other state-of-the-art large-scale optimization algorithms, including the winners of CEC2010, CEC2012, CEC2013, and CEC2018 competitions on large-scale optimization. The results show that the GTDE is efficient and performs better than or at least comparable to the others in solving LSOPs.

Multimodal optimization problems (MMOPs) require algorithms to locate multiple optima simultaneously. When using evolutionary algorithms (EAs) to deal with MMOPs, an intuitive idea is to divide the population into several small "niches," where different niches focus on locating different optima. These population partition strategies are called "niching" techniques, which have been frequently used for MMOPs. The algorithms for simultaneously locating multiple optima of MMOPs are called multimodal algorithms. However, many multimodal algorithms still face the difficulty of population partition since most of the niching techniques involve the sensitive niching parameters. Considering this issue, in this article, we propose a parameter-free niching method based on adaptive estimation distribution (AED) and develop a distributed differential evolution (DDE) algorithm, which is called AED-DDE, for solving MMOPs. In AED-DDE, each individual finds its own appropriate niche size to form a niche and acts as an independent unit to find a global optimum. Therefore, we can avoid the difficulty of population partition and the sensitivity of niching parameters. Different niches are co-evolved by using the master-slave multiniche distributed model. The multiniche co-evolution mechanism can improve the population diversity for fully exploring the search space and finding more global optima. Moreover, the AED-DDE algorithm is further enhanced by a probabilistic local search (PLS) to refine the solution accuracy. Compared with other multimodal algorithms, even the winner of CEC2015 multimodal competition, the comparison results fully demonstrate the superiority of AED-DDE.

Large-scale optimization has become a significant and challenging research topic in the evolutionary computation (EC) community. Although many improved EC algorithms have been proposed for large-scale optimization, the slow convergence in the huge search space and the trap into local optima among massive suboptima are still the challenges. Targeted to these two issues, this article proposes an adaptive granularity learning distributed particle swarm optimization (AGLDPSO) with the help of machine-learning techniques, including clustering analysis based on locality-sensitive hashing (LSH) and adaptive granularity control based on logistic regression (LR). In AGLDPSO, a master-slave multisubpopulation distributed model is adopted, where the entire population is divided into multiple subpopulations, and these subpopulations are co-evolved. Compared with other large-scale optimization algorithms with single population evolution or centralized mechanism, the multisubpopulation distributed co-evolution mechanism will fully exchange the evolutionary information among different subpopulations to further enhance the population diversity. Furthermore, we propose an adaptive granularity learning strategy (AGLS) based on LSH and LR. The AGLS is helpful to determine an appropriate subpopulation size to control the learning granularity of the distributed subpopulations in different evolutionary states to balance the exploration ability for escaping from massive suboptima and the exploitation ability for converging in the huge search space. The experimental results show that AGLDPSO performs better than or at least comparable with some other state-of-the-art large-scale optimization algorithms, even the winner of the competition on large-scale optimization, on all the 35 benchmark functions from both IEEE Congress on Evolutionary Computation (IEEE CEC2010) and IEEE CEC2013 large-scale optimization test suites.

Cloud workflow scheduling is a significant topic in both commercial and industrial applications. However, the growing scale of workflow has made such a scheduling problem increasingly challenging. Many current algorithms often deal with small- or medium-scale problems (e.g., less than 1000 tasks) and face difficulties in providing satisfactory solutions when dealing with the large-scale problems, due to the curse of dimensionality. To this aim, this article proposes a dynamic group learning distributed particle swarm optimization (DGLDPSO) for large-scale optimization and extends it for the large-scale cloud workflow scheduling. DGLDPSO is efficient for large-scale optimization due to its following two advantages. First, the entire population is divided into many groups, and these groups are coevolved by using the master-slave multigroup distributed model, forming a distributed PSO (DPSO) to enhance the algorithm diversity. Second, a dynamic group learning (DGL) strategy is adopted for DPSO to balance diversity and convergence. When applied DGLDPSO into the large-scale cloud workflow scheduling, an adaptive renumber strategy (ARS) is further developed to make solutions relate to the resource characteristic and to make the searching behavior meaningful rather than aimless. Experiments are conducted on the large-scale benchmark functions set and the large-scale cloud workflow scheduling instances to further investigate the performance of DGLDPSO. The comparison results show that DGLDPSO is better than or at least comparable to other state-of-the-art large-scale optimization algorithms and workflow scheduling algorithms.

Due to the increasing complexity of optimization problems, distributed differential evolution (DDE) has become a promising approach for global optimization. However, similar to the centralized algorithms, DDE also faces the difficulty of strategies' selection and parameters' setting. To deal with such problems effectively, this article proposes an adaptive DDE (ADDE) to relieve the sensitivity of strategies and parameters. In ADDE, three populations called exploration population, exploitation population, and balance population are co-evolved concurrently by using the master-slave multipopulation distributed framework. Different populations will adaptively choose their suitable mutation strategies based on the evolutionary state estimation to make full use of the feedback information from both individuals and the whole corresponding population. Besides, the historical successful experience and best solution improvement are collected and used to adaptively update the individual parameters (amplification factor F and crossover rate CR) and population parameter (population size N), respectively. The performance of ADDE is evaluated on all 30 widely used benchmark functions from the CEC 2014 test suite and all 22 widely used real-world application problems from the CEC 2011 test suite. The experimental results show that ADDE has great superiority compared with the other state-of-the-art DDE and adaptive differential evolution variants.

Niching techniques have been widely incorporated into evolutionary algorithms (EAs) for solving multimodal optimization problems (MMOPs). However, most of the existing niching techniques are either sensitive to the niching parameters or require extra fitness evaluations (FEs) to maintain the niche detection accuracy. In this paper, we propose a new automatic niching technique based on the affinity propagation clustering (APC) and design a novel niching differential evolution (DE) algorithm, termed as automatic niching DE (ANDE), for solving MMOPs. In the proposed ANDE algorithm, APC acts as a parameter-free automatic niching method that does not need to predefine the number of clusters or the cluster size. Also, it can facilitate locating multiple peaks without extra FEs. Furthermore, the ANDE algorithm is enhanced by a contour prediction approach (CPA) and a two-level local search (TLLS) strategy. First, the CPA is a predictive search strategy. It exploits the individual distribution information in each niche to estimate the contour landscape, and then predicts the rough position of the potential peak to help accelerate the convergence speed. Second, the TLLS is a solution refine strategy to further increase the solution accuracy after the CPA roughly predicting the peaks. Compared with the other state-of-the-art DE and non-DE multimodal algorithms, even the winner of competition on multimodal optimization, the experimental results on 20 widely used benchmark functions illustrate the superiority of the proposed ANDE algorithm.

The arrival of the era of “big data” has witnessed an urgent need of finding a proper algorithm for large scale optimization. Hence, this paper proposes a competitive swarm optimizer with dynamic grouping (CSO-DG) for large scale optimization. In CSO-DG, particles are separated into many groups to search for different regions of the large landscape. Moreover, the size of the group (the number of particles in each group) changes dynamically to control the swarm competitive strength, so as to balance the convergence and diversity. We also investigate the influences of different group sizes on different problems, and then propose a size range for the dynamic change bounds. By using multiple groups, only the worst particle in each group will be updated by learning from the best particle in the same group and the mean position of the whole swarm. This is helpful to save fitness evaluations. This way, the algorithm is able to work well in high-dimensional optimization. The experimental results show that CSO-DG exhibits a competitive performance than other state-of-the-art algorithms on 20 benchmark functions (CEC`2010) for large scale optimization.

Multimodal optimization problem (MMOP), which targets at searching for multiple optimal solutions simultaneously, is one of the most challenging problems for optimization. There are two general goals for solving MMOPs. One is to maintain population diversity so as to locate global optima as many as possible, while the other is to increase the accuracy of the solutions found. To achieve these two goals, a novel dual-strategy differential evolution (DSDE) with affinity propagation clustering (APC) is proposed in this paper. The novelties and advantages of DSDE include the following three aspects. First, a dual-strategy mutation scheme is designed to balance exploration and exploitation in generating offspring. Second, an adaptive selection mechanism based on APC is proposed to choose diverse individuals from different optimal regions for locating as many peaks as possible. Third, an archive technique is applied to detect and protect stagnated and converged individuals. These individuals are stored in the archive to preserve the found promising solutions and are reinitialized for exploring more new areas. The experimental results show that the proposed DSDE algorithm is better than or at least comparable to the state-of-the-art multimodal algorithms when evaluated on the benchmark problems from CEC2013, in terms of locating more global optima, obtaining higher accuracy solution, and converging with faster speed.

Many real-world optimization problems encounter the presence of uncertainties. Dynamic optimization is a class of problems whose fitness functions vary through time. For these problems, evolutionary algorithm is expected to adapt to the changing environments immediately and find the best solution accurately. Besides, most of the environmental changes may not be too drastic in real-world applications, which indicates the evolutionary information in the past may be helpful for finding the optimum solution in the new environment. As a result, effective reuse of historical evolutionary information is necessary, since it leads a faster convergence after a change has occurred. This paper develops an orthogonal learning particle swarm optimization (OLPSO) with the variable relocation strategy (VRS) to solve the dynamic optimization problem (DOP). The proposed OLPSO-VRS algorithm has two advantages as follows. First and foremost, the orthogonal learning strategy guides particles to fly in better directions by constructing a much promising and efficient exemplar, which can achieve a balance between fast convergence and population diversity. Furthermore, the VRS collects historical information in the stable search stage and then reuse such information to guide the particle variable relocation once the search environment has changed. This operation enables the algorithm to quickly shift to the new promising regions in the changing fitness landscape. We evaluated OLPSO-VRS on several dynamic benchmark problems and compared with several state-of-the-art dynamic algorithms. The results show that OLPSO-VRS obtains very competitive results and enjoys a statistically superior performance on most problems.

Multimodal optimization problem always has several peaks that are all optima of the problem. A promising approach to deal with such kind of problem should locate the peaks as many as possible (e.g., all the peaks) and should obtain high accuracy in each peak. The species-based particle swarm optimization (SPSO) divides the population into several subpopulations. Each subpopulation is gathered around a neighborhood best called species seed within the radius r, trying to locate different peaks. It does well in some low-dimensional multimodal optimization problems. However, the parameter r, which is associated with the efficiency and the accuracy of the algorithm, must be specified by the users. This makes SPSO very difficult for users to determine how much the parameter r should be. In this paper, a method of adaptively choosing radius r in SPSO is proposed, termed as adaptive SPSO (ASPSO). The experimental results show that the performance of ASPSO is more effective and accurate than standard SPSO in dealing with low-dimensional multimodal optimization problems.