Contents I. Introduction
Brain storm optimization (BSO) is a population-based swarm intelligence algorithm, which was inspired by one of human being collective problem solving skills, i.e., the brainstorming process [1], [2]. Three kinds of persons are usually involved in a brainstorm process, that is, a facilitator, problem owners, and a group of brainstorming persons [3]. The facilitator is mainly for guiding the brainstorming process to be run smoothly without being biased, which means that a facilitator who has facilitation experience but has no knowledge about the problem to be solved will be preferred. The group of brainstorming persons is the group of persons who actually generate ideas through the brainstorming process facilitated by the facilitator. The problem owners are responsible for picking up ideas which they believe are better ones among what have been generated during each round of idea generations. Generally, each brainstorming process consists of three rounds of idea generation process because human being will be exhausted and will not be efficient anymore after three rounds of idea generation process. The BSO algorithm was designed by abstracting the model that mimics the brainstorming process [2]. Since its introduction in 2011, BSO has found successful applications in solving realworld applications. It has been applied to solve electric power dispatch problems [4]–[8], DC brushless motor efficiency problems [9], optimal satellite formation reconfiguration [10], receding horizon control for multiple UAV formation flight [11], and stock index forecasting [12]. BSO has also been utilized to solve multi-modal optimization problems [13], [14]. A BSO algorithm consists of two major components, which are convergent operation and divergent operation, with other components being similar to other swarm intelligence algorithms. The convergent operation is designed to mimic the better idea picking up operation by problem owners during which better ideas are picked up so that new ideas could be generated by focusing on these picked up better ideas in the next round of idea generation process, therefore, the convergent operation is an operation that maps a population of individuals to a small set of individuals. In the original BSO [1], [2], the k-means clustering algorithm was utilized to implement the convergent operation in which the population of individuals are clustered into several clusters with the best idea in each cluster being used to represent one better idea picked up by one problem owner. The divergent operation is designed to mimic generating more ideas by focusing on better ideas picked up in the previous round of idea generation process. In the original BSO, adding Gaussian random values to a selected idea is utilized to implement the divergent operation in which new ideas are generated by adding Gaussian random values to selected existing ideas. In order to improve the performance of the BSO, modifications have been made to the original BSO to have better performed BSOs in one way or another. Most modifications have been focused on modifications to the divergent operation and convergent operation. For the divergent operation, Ramanand et al. combined BSO with a Teaching-Learning-Based Algorithm and applied the hybrid algorithm to solve electric power dispatch problems [4]. Duan et al. introduced the predator-prey concept into the BSO to better utilize the global information and to improve the population diversity during the evolution process in which the cluster centers act as predators and other ideas act as preys [9]. Similar to differential evolution, in [10], [15], three ideas instead of only one or two ideas in the original BSO are used to generate new ideas (individuals). In [16], chaotic operation is introduced into the divergent operation to improve BSO's population diversity. In [17], [18], Yang et al. considered idea generation based on one idea as intracluster discussion, and idea generation based on two ideas as inter-cluster discussion, and designed discussion mechanism based BSOs. In [19], Zhou et al. adapted the step-size of the divergent operation according to the dynamic range of individuals on each dimension. For the convergent operation, the k-means clustering algorithm utilized in the original BSO is simple to implement but itself is implemented recursively until it converges or is terminated according to some predefined criteria, which is therefore time consuming. In order to reduce its computation time, the number of iterations for clustering algorithm needs to be reduced. Another disadvantages of the k-means clustering algorithm is its sensitive to outliers. In [20], the k-medians clustering algorithm is utilized to replace the k-means clustering algorithm to reduce the impacts of outliers, in addition, the cluster centroids calculated are replaced by individuals closest to them which consequently causes the k- medians clustering algorithm converging within few iterations partially due to small population size, say 50 or 100, so that its computation time is reduced. In [15], a simple grouping method (SGM) is used as the clustering algorithm, in which cluster centroids are randomly selected and all individuals are clustered to these randomly selected cluster centroids so that during the convergent operation only one iteration needs to be executed which brings a big computation time saving. Even with only one iteration of clustering during the convergent operation in [15], distances from each individual to all randomly selected cluster centroids are required to be calculated in order to assign each individual to a cluster, which is still time consuming, especially when dimension of the problem to be solved is large, say 1000. In order to make BSO to be more attractive, a BSO algorithm needs to be simple to implement and to be fast to run. Because population of individuals in any population-based optimization algorithm needs to update individuals iteration over iteration while individual updating in a BSO is realized in the divergent operation in which each individual is generated by adding random values to a selected individual, therefore, further time saving is difficult to achieve in the divergent operation in a BSO. In this paper, we will achieve further computation time saving in the convergent operation by mimicking picking up better idea operation in the objective space instead in the solution space in the original BSO to make the BSO even simpler to implement and faster to run.
