In pedagogy, teachers usually separate mixed-level students into different levels, treat them differently and teach them in accordance with their cognitive and learning abilities. Inspired from this idea, we consider particles in the swarm as mixed-level students and propose a level-based learning swarm optimizer to settle large scale optimization, which is still considerably challenging in evolutionary computation. At first, a level-based learning strategy is introduced, which separates particles into a number of levels according to their fitness values and treats particles in different levels differently. Then, a new exemplar selection strategy is designed to randomly select two predominant particles from two different higher levels in the current swarm to guide the learning of particles. The cooperation between these two strategies could afford great diversity enhancement for the optimizer. Further, the exploration and exploitation abilities of the optimizer are analyzed both theoretically and empirically in comparison with two popular particle swarm optimizers. Extensive comparisons with several state-of-the-art algorithms on two widely used sets of large scale benchmark functions confirm the competitive performance of the proposed optimizer in both solution quality and computational efficiency. Finally, comparison experiments on problems with dimensionality increasing from 200 to 2000 further substantiate the good scalability of the developed optimizer.

Seeking multiple optima simultaneously, which multimodal optimization aims at, has attracted increasing attention but remains challenging. Taking advantage of ant colony optimization (ACO) algorithms in preserving high diversity, this paper intends to extend ACO algorithms to deal with multimodal optimization. First, combined with current niching methods, an adaptive multimodal continuous ACO algorithm is introduced. In this algorithm, an adaptive parameter adjustment is developed, which takes the difference among niches into consideration. Second, to accelerate convergence, a differential evolution mutation operator is alternatively utilized to build base vectors for ants to construct new solutions. Then, to enhance the exploitation, a local search scheme based on Gaussian distribution is self-adaptively performed around the seeds of niches. Together, the proposed algorithm affords a good balance between exploration and exploitation. Extensive experiments on 20 widely used benchmark multimodal functions are conducted to investigate the influence of each algorithmic component and results are compared with several state-of-the-art multimodal algorithms and winners of competitions on multimodal optimization. These comparisons demonstrate the competitive efficiency and effectiveness of the proposed algorithm, especially in dealing with complex problems with high numbers of local optima.

Taking the advantage of estimation of distribution algorithms (EDAs) in preserving high diversity, this paper proposes a multimodal EDA. Integrated with clustering strategies for crowding and speciation, two versions of this algorithm are developed, which operate at the niche level. Then these two algorithms are equipped with three distinctive techniques: 1) a dynamic cluster sizing strategy; 2) an alternative utilization of Gaussian and Cauchy distributions to generate offspring; and 3) an adaptive local search. The dynamic cluster sizing affords a potential balance between exploration and exploitation and reduces the sensitivity to the cluster size in the niching methods. Taking advantages of Gaussian and Cauchy distributions, we generate the offspring at the niche level through alternatively using these two distributions. Such utilization can also potentially offer a balance between exploration and exploitation. Further, solution accuracy is enhanced through a new local search scheme probabilistically conducted around seeds of niches with probabilities determined self-adaptively according to fitness values of these seeds. Extensive experiments conducted on 20 benchmark multimodal problems confirm that both algorithms can achieve competitive performance compared with several state-of-the-art multimodal algorithms, which is supported by nonparametric tests. Especially, the proposed algorithms are very promising for complex problems with many local optima.

Surrogate-assisted evolutionary algorithms (SAEAs) have become one popular method to solve complex and computationally expensive optimization problems. However, most existing SAEAs suffer from performance degradation with the dimensionality increasing. To solve this issue, this paper proposes a classifier-assisted level-based learning swarm optimizer on the basis of the level-based learning swarm optimizer (LLSO) and the gradient boosting gradient classifier (GBC) to improve the robustness and scalability of SAEAs. Particularly, the level-based learning strategy in LLSO has a tight correspondence with the classification characteristic by setting the number of levels in LLSO to be the same as the number of classes in GBC. Together, the classification results feedback the distribution of promising candidates to accelerate the evolution of the optimizer, while the evolved population helps improve the accuracy of the classifier. To select informative and valuable candidates for real evaluations, we devise a L1-exploitation strategy to extensively exploit promising areas. Then, the candidate selection is conducted between the predicted L1 offspring and the already real-evaluated L1 individuals based on their Euclidean distances. Extensive experiments on commonly-used benchmark functions demonstrate that the proposed optimizer can achieve competitive or better performance with a very small training dataset compared with three state-of-the-art SAEAs.

Large-scale optimization has become a significant yet challenging area in evolutionary computation. To solve this problem, this paper proposes a novel segment-based predominant learning swarm optimizer (SPLSO) swarm optimizer through letting several predominant particles guide the learning of a particle. First, a segment-based learning strategy is proposed to randomly divide the whole dimensions into segments. During update, variables in different segments are evolved by learning from different exemplars while the ones in the same segment are evolved by the same exemplar. Second, to accelerate search speed and enhance search diversity, a predominant learning strategy is also proposed, which lets several predominant particles guide the update of a particle with each predominant particle responsible for one segment of dimensions. By combining these two learning strategies together, SPLSO evolves all dimensions simultaneously and possesses competitive exploration and exploitation abilities. Extensive experiments are conducted on two large-scale benchmark function sets to investigate the influence of each algorithmic component and comparisons with several state-of-the-art meta-heuristic algorithms dealing with large-scale problems demonstrate the competitive efficiency and effectiveness of the proposed optimizer. Further the scalability of the optimizer to solve problems with dimensionality up to 2000 is also verified.

Large-scale optimization with high dimensionality and high computational cost becomes ubiquitous nowadays. To tackle such challenging problems efficiently, devising distributed evolutionary computation algorithms is imperative. To this end, this paper proposes a distributed swarm optimizer based on a special master-slave model. Specifically, in this distributed optimizer, the master is mainly responsible for communication with slaves, while each slave iterates a swarm to traverse the solution space. An asynchronous and adaptive communication strategy based on the request-response mechanism is especially devised to let the slaves communicate with the master efficiently. Particularly, the communication between the master and each slave is adaptively triggered during the iteration. To aid the slaves to search the space efficiently, an elite-guided learning strategy is especially designed via utilizing elite particles in the current swarm and historically best solutions found by different slaves to guide the update of particles. Together, this distributed optimizer asynchronously iterates multiple swarms to collaboratively seek the optimum in parallel. Extensive experiments on a widely used large-scale benchmark set substantiate that the distributed optimizer could: 1) achieve competitive effectiveness in terms of solution quality as compared to the state-of-the-art large-scale methods; 2) accelerate the execution of the algorithm in comparison with the sequential one and obtain almost linear speedup as the number of cores increases; and 3) preserve a good scalability to solve higher dimensional problems.