Aquila optimizer based on phasor operator and flow direction operator

doi:10.3785/j.issn.1008-973X.2024.02.009

Journal of ZheJiang University (Engineering Science)

2024, Vol. 58

Issue (2): 304-316 DOI: 10.3785/j.issn.1008-973X.2024.02.009

Aquila optimizer based on phasor operator and flow direction operator

Yu ZHOU1(

),Zexuan PEI1,Peichong WANG2,Bo CHEN1

1. College of Electrical Engineering, North China University of Water Resources and Electric Power, Zhengzhou 450045, China
2. College of Information Engineering, Hebei GEO University, Shijiazhuang 050031, China

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Abstract

A multi-strategy improved aquila optimizer (MIAO) was proposed aiming at the disadvantages of low search efficiency and easy to fall into local optimal value in aquila optimization algorithm. Generalized normal distribution optimizer (GNDO) was added to AO, and the result obtained by GNDO was compared with the result of AO in the first stage. Then the best value under the two algorithms was selected. The search space was expanded and the quality of solution was improved. Phasor operators were used to transform the second phase into an adaptive non-parametric optimization in order to improve the high-dimensional optimization ability of the AO. The flow operator was used in the third stage of AO aiming at the problems of reduced population diversity and insufficient local exploitation at late iterations of the AO. Then the information can be transferred between each individual. The utilization rate of population information was improved, and the local exploitation capability of the AO was enhanced. Comparative analysis and optimization results of 16 test functions and Wilcoxon rank sum test showed that MIAO optimization ability and convergence speed were greatly improved. The MIAO algorithm was used to solve reducer design problem in order to verify the practicality and feasibility of MIAO algorithm. Comparative analysis of practical engineering optimization problems shows that MIAO algorithm has certain advantages in processing realistic optimization problems.

Key words： aquila optimizer generalized normal distribution optimization algorithm phasor operator flow direction operator test function Wilcoxon rank sum test

Received: 21 June 2023 Published: 23 January 2024

CLC:

TP 301

Fund: 国家自然科学基金资助项目（U1504622，31671580）；河南省高等学校青年骨干教师培养计划资助项目（2018GGJS079）；河北省高等学校科学技术研究资助项目（ZD2020344）

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Cite this article:

Yu ZHOU,Zexuan PEI,Peichong WANG,Bo CHEN. Aquila optimizer based on phasor operator and flow direction operator. Journal of ZheJiang University (Engineering Science), 2024, 58(2): 304-316.

URL:

https://www.zjujournals.com/eng/10.3785/j.issn.1008-973X.2024.02.009 OR https://www.zjujournals.com/eng/Y2024/V58/I2/304

引入相量算子和流向算子的天鹰优化算法

针对天鹰优化算法搜索效率不足，容易陷入局部最优的缺点，提出多策略改进天鹰优化算法(MIAO). 引入广义正态分布优化算法(GNDO)，将该算法得出的结果与天鹰优化算法第1阶段得出的结果进行比较，筛选出这2种优化算法下的最优值. 该操作扩大了搜索空间，提高了解的质量. 引入相量算子，将第2阶段变为自适应的非参数优化，提高算法的高维优化能力. 针对天鹰优化算法在迭代后期存在种群多样性降低、局部开发能力不足的问题，在天鹰算法的第3阶段引入流向算子，使信息可以在每个个体间相互传递，提高种群信息的利用率，增强天鹰优化算法的开发性能. 通过对16个测试函数寻优对比分析以及Wilcoxon秩和检验可知，MIAO的寻优能力和收敛速度都有较大的提升. 为了验证MIAO算法的实用性和可行性，采用所提算法求解减速器设计问题，通过实际工程优化问题的实验对比分析可知，MIAO算法在处理现实优化问题上具有一定的优越性.

关键词： 天鹰优化算法, 广义正态分布优化算法, 相量算子, 流向算子, 测试函数, Wilcoxon秩和检验

Fig.1 Function behavior of

$ p\left( {\theta _i^t} \right) $

and

$ g\left( {\theta _i^t} \right) $

Fig.2 Changes of

$ W $

Tab.1 Parameter setting of MIAO and comparison algorithm

Tab.2 Test functions

Tab.3 Results of test function (dim=30，

${f_1} - {f_8}$

)

Tab.4 Results of test function (dim=30，

$ {f_9} , {f_{10}},{f_{15}} - {f_{16}} $

)

Tab.5 Fixed dimension test functions

函数	算法	最优值	平均值	标准差	函数	算法	最优值	平均值	标准差
$ {f_1} $	SOA	86.5542	94.1352	20.9621	$ {f_6} $	SOA	0.0341	1.1236	2.6557
	PSO	0.6220	0.6254	0.0239		PSO	157.3364	163.6361	25.3142
	SSA	3.3646×10^?55	3.3345×10^?31	7.223 4×10^?30		SSA	0	0	0
	GNDO	23.1552	33.3120	16.3251		GNDO	362.36	384.21	32.6520
	PPSO	0.7486	0.8998	0.3654		PPSO	258.24	274.65	61.0320
	FDA	89.7243	93.1086	9.2320		FDA	0	0	0
	AO	9.466 2×10^?78	8.253×10^?75	8.223 6×10^?76		AO	0	0	0
	MIAO	3.371 6×10^?158	6.223 7×10^?129	2.678 4×10^?129		MIAO	0	0	0
$ {f_2} $	SOA	0.1378	1.2355	2.6341	$ {f_7} $	SOA	1.223 4×10^?4	4.337 8×10^?3	3.707 1×10^?3
	PSO	0.2201	0.4436	0.6520		PSO	1.8752	2.0314	0.6321
	SSA	7.3240×10^?3	0.0021	2.3114×10^?3		SSA	8.881 8×10^?16	8.881 8×10^?16	0
	GNDO	5.6488	8.0641	5.6630		GNDO	11.7217	13.4094	5.3241
	PPSO	1.7592	2.0900	0.4354		PPSO	2.3323	2.1354	0.3542
	FDA	4.487 9×10^?4	5.832 5×10^?4	2.665 2×10^?4		FDA	4.440 9×10^?15	4.440 9×10^?15	0
	AO	9.782 8×10^?5	3.521 1×10^?4	4.695 7×10^?4		AO	8.881 8×10^?16	8.881 8×10^?16	0
	MIAO	1.704 1×10^?5	4.122 0×10^?4	5.314 2×10^?4		MIAO	8.881 8×10^?16	8.881 8×10^?16	0
$ {f_3} $	SOA	0.0021	0.9328	1.5631	$ {f_8} $	SOA	0.0143	56.4429	83.9514
	PSO	1.913 5×10^?9	3.556 4×10^?7	2.635 7×10^?7		PSO	0.0527	0.0600	1.3255e-3
	SSA	0	0	0		SSA	0	0	0
	GNDO	5.551 1×10^?17	5.551 1×10^?17	2.392 8×10^?19		GNDO	61.1138	74.3512	22.2511
	PPSO	0	0	0		PPSO	0.1039	0.1173	0.0062
	FDA	0	0	0		FDA	0	0	0
	AO	0	0	0		AO	0	0	0
	MIAO	0	0	0		MIAO	0	0	0
$ {f_4} $	SOA	0.0151	4.1531	8.3361	$ {f_9} $	SOA	?1	?0.9889	0.1264
	PSO	66.3548	85.3642	51.3260		PSO	?1	?1	0
	SSA	7.572 5×10^?80	3.214 6×10^?65	4.012 6×10^?65		SSA	?1	?1	0
	GNDO	1141.2	1148.7	43.2651		GNDO	?1	?0.9632	0.0023
	PPSO	180.23	210.24	7.3521		PPSO	?1	?1	0
	FDA	3.624 7×10^?79	5.624 4×10^?75	9.214 4×10^?75		FDA	?1	?1	0
	AO	2.246 3×10^?157	3.514 4×10^?133	6.331 4×10^?133		AO	?1	?1	0
	MIAO	0	0	0		MIAO	?1	?1	0
$ {f_5} $	SOA	2.460 5×10^?6	0.0553	0.3662	$ {f_{10}} $	SOA	?957.9990	?950.6374	10.5231
	PSO	1.887 8×10^?8	1.223 6×10^?7	3.263 1×10^?8		PSO	?66.8437	?66.8437	0
	SSA	5.341 9×10^?113	1.324 9×10^?50	2.312 6×10^?50		SSA	?894.3315	?888.1634	43.6612
	GNDO	1.135 3×10^?7	4.714 2×10^?10	3.256 4×10^?10		GNDO	?959.6406	?959.6406	4.125 0×10^?4
	PPSO	6.154 3×10^?9	6.734 3×10^?7	2.663 4×10^?9		PPSO	?66.0970	?65.4211	1.3325
	FDA	3.257 8×10^?116	2.647 6×10^?113	3.654 2×10^?113		FDA	?959.4607	?959.4605	4.332 1×10^?3
	AO	1.481 3×10^?95	2.794 9×10^?29	3.654 2×10^?29		AO	?959.6407	?-959.6404	3.412 2×10^?3
	MIAO	4.185×10^?178	4.893 1×10^?152	1.233 0×10^?152		MIAO	?959.6407	?959.6406	1.256 1×10^?3

Tab.6 Results of test function (dim=100，

${f_1} - {f_{10}}$

)

Tab.7 Results of test function (dim=100,

${f_{15}} , \;{f_{16}}$

)

Fig.3 MIAO and convergence curves of seven comparison algorithms

Tab.8 Results of Wilcoxon rank sum test

Fig.4 Reducer design structure

Tab.9 Comparison of results of reducer design problem

Fig.5 Reducer convergence curve


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