In this paper, the generalization of the Particle Swarm Optimization (PSO) algorithm is proposed. The new algorithm involves the adoption of complex-order derivatives (CD). Since the CD produce complex-valued results, conjugate pairs of CD are considered for designing the Complex-Order PSO (CoPSO). First, an extensive sensitivity analysis is carried out for studying the influence of the control parameters on the performance of CoPSO. Then, a set of classical benchmark functions are tested to verify the performance of CoPSO. Both valued- and ranked-based methods are conducted to compare the performance of the algorithm on the whole test suite. The Friedman test is applied to determine the average ranking of the algorithms based on their performances. Additionally, the mean and the standard deviation of the best results are examined in each experiment. The results indicate that the CoPSO has outstanding performance in comparison with previous algorithms, including the standard PSO, the fractional order PSO and the linear and nonlinear decreasing inertia weight PSO. The experimental results indicate the feasibility and efficiency of the CoPSO.

本文提出了粒子群优化算法（PSO）的推广。新算法涉及采用复数阶导数（CD）。由于CD产生复数值结果，因此考虑使用CD共轭对设计复数PSO（CoPSO）。首先，进行了广泛的灵敏度分析，以研究控制参数对CoPSO性能的影响。然后，测试了一组经典基准功能以验证CoPSO的性能。进行了基于值和基于排名的方法来比较算法在整个测试套件上的性能。弗里德曼检验用于根据算法的性能确定算法的平均排名。另外，在每个实验中检查最佳结果的平均值和标准偏差。结果表明，与以前的算法相比，CoPSO具有出色的性能，包括标准PSO，分数阶PSO和线性和非线性递减惯性权重PSO。实验结果表明了CoPSO的可行性和有效性。