Abstract: The purpose of this paper is to analyze the firefly algorithm (FA) systematically. Firstly, two general convergence conditions are obtained by analyzing the convergence process of FA. One is that mathematical expectation of random disturbance term is equal to 0, the other is that maximum attractiveness value β₀ belongs to (0,2), and usually belongs to (0,1], and the more β₀ value, the faster convergence speed. Nextly, according to the criterion of convergence of random algorithm, this paper proves that the FA is not a globally convergent algorithm. Then, this paper theoretically proves that the FA converges to the local optimal solution by using mathematical induction, sandwich theorem and apagoge. Finally, the convergence processes of FA under different conditions are simulated. The experimental results agree well with the theoretical results and prove the correctness of the theory analysis.

Key words: firefly algorithm, convergence analysis, local convergence

摘要： 为了系统地分析萤火虫算法（firefly algorithm, FA），首先对FA算法的收敛过程进行了分析，得出FA算法收敛的两个一般条件：随机扰动项的数学期望等于0；最大吸引度β₀∈(0,2)，通常取β₀∈(0,1]，并且β₀越大，算法收敛速度越快。接着根据随机算法的收敛准则，证明了FA算法不具有全局收敛特性。然后应用数学归纳法，结合夹逼定理及反证法，从理论上证明了FA算法收敛于群体最优解，是一个局部收敛算法。最后对不同条件下的FA算法收敛性进行了仿真，实验结果与理论结果一致，佐证了理论分析的正确性。

关键词: 萤火虫算法, 收敛分析, 局部收敛