A hybrid butterfly and Newton-Raphson swarm intelligence algorithm based on opposition-based learning

In response to the issues of local optima entrapment, slow convergence, and low optimization accuracy in Butterfly optimization algorithm (BOA), this paper proposes a hybrid Butterfly and Newton-Raphson swarm intelligence algorithm based on Opposition-based learning (BOANRBO). Firstly, by Opposition-based learning, the initialization strategy of the butterfly algorithm is improved to accelerate convergence. Secondly, adaptive perception modal factors are introduced into the original butterfly algorithm, controlling the adjustment rate through the adjustment factor alpha to enhance the algorithm's global search capability. Then, the exploration probability p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p$$\end{document} is dynamically adjusted based on the algorithm's runtime, increasing or decreasing exploration probability by examining changes in fitness to achieve a balance between exploration and exploitation. Finally, the exploration capability of BOA is enhanced by incorporating the Newton-Raphson-based optimizer (NRBO) to help BOA avoid local optima traps. The optimization performance of BOANRBO is evaluated on 65 standard benchmark functions from CEC-2005, CEC-2017, and CEC-2022, and the obtained optimization results are compared with the performance of 17 other well-known algorithms. Simulation results indicate that in the 12 test functions of CEC-2022, the BOANRBO algorithm achieved 8 optimal results (66.7%). In CEC-2017, out of 30 test functions, it obtained 27 optimal results (90%). In CEC-2005, among 23 test functions, it secured 22 optimal results (95.6%). Additionally, experiments have validated the algorithm's practicality and superior performance in 5 engineering design optimization problems and 2 real-world problems.