# Hybrid Multi-Strategy Improved Butterfly Optimization Algorithm
## 1 Introduction  
In recent years, with the development of natural computing and intelligent algorithms, metaheuristic algorithms have emerged. From the perspective of their formation principles, metaheuristic algorithms can be divided into three categories: those based on evolutionary mechanisms, such as Imperialist Competitive Algorithm [1], Geometrical Optimization Algorithm, etc. [2]; those based on physical principles, such as Simulated Annealing Algorithm, Surface Space Optimization, etc.; and those based on swarm intelligence, such as Particle Swarm Optimization, Grey Wolf Optimization Algorithm, etc. Among them, metaheuristic algorithms developed based on swarm intelligence have attracted much attention.  
[PERSON], by observing the foraging and migration behaviors of butterflies in nature, proposed the Butterfly Optimization Algorithm (BOA) [3]. It has been widely used in path planning [4; 5], fault detection [6], image processing [7], and other fields [8; 9]. Although BOA has shown superior performance in various application scenarios, there is no absolute thing in the world, and naturally, there is no algorithm that can be applied to all optimization problems. BOA also has some drawbacks, such as the lack of a disturbance mechanisms within the population, susceptibility to local optimal solutions, and the zero-sum game between new individuals and the optimal individuals, which loses the opportunity to explore other areas. Therefore, since the algorithm was proposed, scholars have proposed different improvement strategies to address the shortcomings of the butterfly algorithm. At present, the common improvement directions of the algorithm can be roughly divided into six categories: heterogeneous integration, adaptive parameter, noise interference, information exchange, guide selection method, and update mechanism [10]. In addition,there are improved methods that refer to the concept of feedback control circuits in circuit analysis, so that butterflies can be affected by positive or negative feedback from other butterflies [11].  
In this paper, an improved butterfly optimization algorithm (IBOA) is proposed, which integrates several strategies. The first step is to use SPM chaotic mapping and reverse learning to initialize the population in the experimental sample, and then select the most suitable individuals according to the fitness to form a new population. The second step is to introduce the Levy flight strategy in the global search stage. The Levy flight strategy is used to make individual butterflies jump long distances in the global search stage. In the third stage of local search, the perturbation characteristic of sine and cosine algorithm is used to expand the search range. Finally, the simulated annealing mechanism is introduced into BOA, and the quality of the global solution is improved by using the characteristic that the simulated annealing algorithm has a certain probability to accept the poor solution. The improved BOA overcomes the limitations of traditional BOA by enhancing its search ability and formulating more efficient search strategies, thereby achieving a faster convergence speed and a higher optimization accuracy.  
The structure of this article is as follows. Section 2 provides a brief introduction to the standard BOA. Section 3 is a detailed description of the IBOA. Section 4 lists five comparison algorithms, test functions, experimental results, and statistical analysis.