## 1 Introduction  
Optimization problems exist extensively in various domains, including industry [1], agriculture [2], and finance [3]. The methods for solving optimization problems have two main families: traditional mathematical methods and intelligent optimization methods [4]. However, with the advancement of science and technology and the growth of task requirements, optimization problems have gradually began to present complex characteristics such as non-differentiability, multimodality, and non-linearity. These complex characteristics limit the capability of traditional mathematical methods in addressing optimization problems. In contrast, intelligent optimization algorithms are effective in handling optimization problems with such complex characteristics. Therefore, more and more researchers are beginning to pay attention to intelligent optimization algorithms and propose a series of new intelligent optimization algorithms, such as grey wolf optimizer (GWO) [5], harrishawks optimizer (HHO) [6], slime mould algorithm (SMA) [7], particle swarm optimizer (PSO) [8], doctor and patient optimizer (DPO) [9], and so on. Although numerous intelligent optimization algorithms have been proposed and used to solve various optimization problems, no optimization algorithm can solve all kinds of optimization problems according to the \"No Free Lunch Theorem\" (NFL) [10]. And one optimization algorithm may perform better in solving certain optimization problems than others. Therefore, looking for new intelligent optimization algorithms has always been the research focus of relevant researchers in this field. As a novel intelligent optimization approach based on physical law, equilibrium optimizer (EO) [11] has been recently introduced by [PERSON] et al. EO is implemented by iteratively updating the particle concentrations and generating the equilibrium candidates. Due to its advantages of simple structure, easy realization, fast response speed, and strong optimization capability, EO has been widely used in addressing optimization problems from various fields. In [12], EO was employed in handling the optimal power flow problem by [PERSON] et al. In [13], EO was applied in the prediction of traffic transportation by combining with a convolutional neural network. In [14], EO was utilized to search for the optimal path with human-like driving styles in a structured road environment. In [15], EO was employed in improving the fusion parameters of low-frequency parts in multimodal medical image processing by [PERSON] et al. In [16], EO was applied in integrating with the random vector functional link network to predict laser cutting parameters by [PERSON] et al. Furthermore, to improve EO's optimization performance for specific requirements, various EO variants have been developed by the researchers. In [17], a local minima elimination approach and a linear reduction diversity strategy are embedded in EO to enhance the convergence. The exploration ability of EO was enhanced by integrating the escape mechanism and the opposition-based learning strategy in [18]. In [19], [PERSON] et al. integrated a general learning approach with EO to enhance the ability of searching promising regions. [PERSON] et al. [20] presented an adaptive equilibrium optimizer by integrating EO with the adaptive proportional mutation approach, random walk mechanism, and spiral encirclement method, which strengthens the optimization ability in handling the engineering and complex function optimization problems. By effectively using the information of whole population, neighborhood space, and opposite space, an ameliorated equilibrium optimizer was developed in [21] to enhance the optimization performance of addressing the smooth path planning problem. However, in addressing complex optimization problems, more attention should be paid on how to enhance the population diversity.