Differential evolution (DE) is a widely recognized method to solve complex optimization problems as shown by many researchers. Yet, non-adaptive versions of DE suffer from insufficient exploration ability and uses no historical information for its performance enhancement. This work proposes Fractional Order Differential Evolution (FODE) to enhance DE performance from two aspects. Firstly, a bi-strategy co-deployment framework is proposed. The population-based and parameter-based strategies are combined to leverage their respective advantages. Secondly, the fractional order calculus is first applied to the differential vector to enhance DE’s exploration ability by using the historical information of populations, and ensures the diversity of population in an evolutionary process. We use the 2017 IEEE Congress on Evolutionary Computation (CEC) test functions, and CEC2011 real-world problems to evaluate FODE’s performance. Its sensitivity to parameter changes is discussed and an ablation study of multi-strategies is systematically performed. Furthermore, the variations of exploration and exploitation in FODE are visualized and analyzed. Experimental results show that FODE is superior to other state-of-the-art DE variants, the winners of CEC competitions, other fractional order calculus-based algorithms, and some powerful variants of classic algorithms.

The coordination of population structure is the foundation for the effective functioning of evolutionary algorithms. An efficient population evolution structure can guide individuals to engage in successful and robust exploitative and exploratory behaviors. However, due to the black-box property of the search process, it is challenging to assess the current state of the population and implement targeted measures. In this paper, we propose a dynamic population structures-based differential evolution algorithm (DPSDE) to uncover the real-time state of population continuous optimization. According to the exploitation and exploration state of population, we introduce four structural modules to address the premature convergence and search stagnation issues of the current population. To effectively utilize these modules, we propose a real-time discernment mechanism to judge the population's current state. Based on the feedback information, suitable structural modules are dynamically invoked, ensuring that the population undergoes continuous and beneficial evolution, ultimately exploring the optimal population structure. The comparative outcomes with numerous cutting-edge algorithms on the IEEE Congress on Evolutionary Computation (CEC) 2017 benchmark functions and 2011 real-world problems verify the superiority of DPSDE. Furthermore, parameters, population state, and ablation study of modules are discussed.

Manta ray foraging optimization (MRFO) tends to get trapped in local optima as it relies on the direction provided by the previous individual and the best individual as guidance to search for the optimal solution. As enriching population diversity can effectively solve this problem, in this paper, we introduce a hierarchical structure and weighted fitness-distance balance selection to improve the population diversity of the algorithm. The hierarchical structure allows individuals in different groups of the population to search for optimal solutions in different places, expanding the diversity of solutions. In MRFO, greedy selection based solely on fitness can lead to local solutions. We innovatively incorporate a distance metric into the selection strategy to increase selection diversity and find better solutions. A hierarchical manta ray foraging optimization with weighted fitness-distance balance selection (HMRFO) is proposed. Experimental results on IEEE Congress on Evolutionary Computation 2017 (CEC2017) functions show the effectiveness of the proposed method compared to seven competitive algorithms, and the proposed method has little effect on the algorithm complexity of MRFO. The application of HMRFO to optimize real-world problems with large dimensions has also obtained good results, and the computational time is very short, making it a powerful alternative for very high-dimensional problems. Finally, the effectiveness of this method is further verified by analyzing the population diversity of HMRFO.