求解高维优化问题的改进鲸鱼优化算法

doi:10.3778/j.issn.1673-9418.2104029

计算机科学与探索 ›› 2022, Vol. 16 ›› Issue (12): 2890-2902.DOI: 10.3778/j.issn.1673-9418.2104029

• 理论与算法 • 上一篇

求解高维优化问题的改进鲸鱼优化算法

王永贵¹, 李鑫¹^,⁺(), 关连正²

1.辽宁工程技术大学软件学院，辽宁葫芦岛 125105
2.北京工业大学信息学部，北京 100124

收稿日期:2021-04-09 修回日期:2021-05-27 出版日期:2022-12-01 发布日期:2021-06-03
通讯作者: +E-mail: lntu_lixin@163.com
作者简介:王永贵（1967—），男，内蒙古宁城人，硕士，教授，CCF会员，主要研究方向为大数据、智能数据处理等。
李鑫（1996—），女，辽宁朝阳人，硕士研究生，主要研究方向为智能数据处理、深度学习等。
关连正（1995—），男，辽宁鞍山人，硕士研究生，主要研究方向为深度学习、智能数据处理等。
基金资助:
国家自然科学基金(61772249)

Improved Whale Optimization Algorithm for Solving High-Dimensional Optimiza-tion Problems

WANG Yonggui¹, LI Xin¹^,⁺(), GUAN Lianzheng²

1. School of Software, Liaoning Technical University, Huludao, Liaoning 125105, China
2. Faculty of Information Technology, Beijing University of Technology, Beijing 100124, China

Received:2021-04-09 Revised:2021-05-27 Online:2022-12-01 Published:2021-06-03
About author:WANG Yonggui, born in 1967, M.S., professor, member of CCF. His research interests include big data, intelligent data processing, etc.
LI Xin, born in 1996, M.S. candidate. Her research interests include intelligent data processing, deep learning, etc.
GUAN Lianzheng, born in 1995, M.S. candidate. His research interests include deep learning, intelligent data processing, etc.
Supported by:
National Natural Science Foundation of China(61772249)

摘要/Abstract

摘要：

针对鲸鱼优化算法在处理高维优化问题时全局勘探能力不足和易陷入局部极值的问题，提出一种改进的鲸鱼优化算法。首先，在搜索空间中采用Fuch混沌映射和优化的对立学习相结合的初始化策略，利用Fuch映射较高的搜索效率产生多样性良好的优质混沌初始种群，然后结合优化的对立学习策略在保证种群多样性的同时产生优良鲸鱼种群，为算法全局搜索奠定基础；其次，在全局勘探阶段对参数A进行调整，帮助鲸鱼种群更有效地进行全局搜索，在平衡全局勘探和局部开发的同时避免早熟收敛；最后，在局部开发阶段引入拉普拉斯算子对最优个体进行动态交叉操作，迭代前期产生距离父代较远的子代提高全局搜索能力摆脱局部极值束缚，迭代后期产生距离父代较近的子代精细搜索范围提高求解精度。选取10个标准测试函数在100维、500维、1 000维下进行仿真实验，结果表明该算法在收敛速度、求解精度和稳定性方面明显优于其他对比算法，能够有效处理高维优化问题。

关键词: 鲸鱼优化算法（WOA）, 拉普拉斯交叉, 混沌对立学习策略

Abstract:

Aiming at the problems of insufficient global exploration ability and easy to fall into local extremes when dealing with high-dimensional optimization problems, an improved whale optimization algorithm is proposed. Firstly, an initialization strategy combining Fuch chaos mapping and optimized oppsition-based learning is used in the search space to generate good quality chaotic initial populations with good diversity by using higher search effi-ciency of Fuch mapping, and then combined with the optimized oppsition-based learning strategy to generate good whale populations while ensuring population diversity, laying foundation for the global search of the algorithm. Secondly, the parameter A is adjusted in the global exploration phase to help the whale populations to perform global search more effectively and avoid premature convergence while balancing global exploration and local exploitation.Finally, the Laplace operator is introduced in the local exploitation stage to perform dynamic crossover operation for optimal individual. Children generation is produced farther away from the parent generation in the early iteration to improve the global search ability to get rid of local extreme value binding, and points are produced closer to the parent generation in the late iteration to refine the search range to improve the solution accuracy. Ten standard test functions are selected for simulation in 100, 500 and 1000 dimensions. The results show that this algorithm is significantly better than other comparative algorithms in terms of convergence speed, solution accuracy and sta-bility, and can effectively deal with high-dimensional optimization problems.

Key words: whale optimization algorithm (WOA), Laplace crossing, chaotic oppsition-based learning strategy

中图分类号:

TP301.6

王永贵, 李鑫, 关连正. 求解高维优化问题的改进鲸鱼优化算法[J]. 计算机科学与探索, 2022, 16(12): 2890-2902.

WANG Yonggui, LI Xin, GUAN Lianzheng. Improved Whale Optimization Algorithm for Solving High-Dimensional Optimiza-tion Problems[J]. Journal of Frontiers of Computer Science and Technology, 2022, 16(12): 2890-2902.

图/表 19

图1 鲸鱼优化算法流程图

Fig.1 Whale optimization algorithm flow chart

图2 A值随迭代变化图

Fig.2 Variation of A-value with iteration

图3 J值随迭代变化图

Fig.3 Variation of J-value with iteration

图4 拉普拉斯密度函数曲线

Fig.4 Laplace density function curve

图5 DPWOA算法流程图

Fig.5 DPWOA flow chart

表1 测试函数

Table 1 Benchmark functions

函数名称	函数表达式	搜索空间	全局最优值	特性
Griewank	$f 1 = 1 + ∑ i = 1 n x i 2 4 000 - ∏ i = 1 n c o s x i i b 2 - 4 a c$	[-600,600]	0	单峰
Powell Sum	$f 2 = ∑ i = 1 n \| x i \| i + 1$	[-1,1]	0	单峰
Sphere	$f 3 = ∑ i = 1 n x i 2$	[-5.12,5.12]	0	单峰
Schwefel 2.20	$f 4 = ∑ i = 1 n \| x i \|$	[-100,100]	0	单峰
Schwefel 2.22	$f 5 = ∑ i = 1 n \| x i \| + ∏ i = 1 n \| x i \|$	[-100,100]	0	单峰
Ackley	$f 6 = - 20 e x p - 0.2 1 n ∑ i = 1 n x i 2 - e x p 1 n ∑ i = 1 n c o s (2 π x i) + 20 + e x p (1)$	[-32,32]	0	多峰
Periodic	$f 7 = 1 + ∑ i = 1 n s i n 2 x i - 0.1 e x p ∑ i = 1 n x i 2$	[-10,10]	0.9	多峰
Quartic	$f 8 = ∑ i = 1 n i x i 4 + r a n d o m 0,1$	[-1.28,1.28]	0	多峰
Rastrigin	$f 9 = 10 n + ∑ i = 1 n (x i 2 - 10 c o s (2 π x i))$	[-5.12,5.12]	0	多峰
Salomon	$f 10 = 1 - c o s 2 π ∑ i = 1 D x i 2 + 0.1 ∑ i = 1 D x i 2$	[-5,5]	0	多峰

表1 测试函数

Table 1 Benchmark functions

函数名称	函数表达式	搜索空间	全局最优值	特性
Griewank	$f 1 = 1 + ∑ i = 1 n x i 2 4 000 - ∏ i = 1 n c o s x i i b 2 - 4 a c$	[-600,600]	0	单峰
Powell Sum	$f 2 = ∑ i = 1 n \| x i \| i + 1$	[-1,1]	0	单峰
Sphere	$f 3 = ∑ i = 1 n x i 2$	[-5.12,5.12]	0	单峰
Schwefel 2.20	$f 4 = ∑ i = 1 n \| x i \|$	[-100,100]	0	单峰
Schwefel 2.22	$f 5 = ∑ i = 1 n \| x i \| + ∏ i = 1 n \| x i \|$	[-100,100]	0	单峰
Ackley	$f 6 = - 20 e x p - 0.2 1 n ∑ i = 1 n x i 2 - e x p 1 n ∑ i = 1 n c o s (2 π x i) + 20 + e x p (1)$	[-32,32]	0	多峰
Periodic	$f 7 = 1 + ∑ i = 1 n s i n 2 x i - 0.1 e x p ∑ i = 1 n x i 2$	[-10,10]	0.9	多峰
Quartic	$f 8 = ∑ i = 1 n i x i 4 + r a n d o m 0,1$	[-1.28,1.28]	0	多峰
Rastrigin	$f 9 = 10 n + ∑ i = 1 n (x i 2 - 10 c o s (2 π x i))$	[-5.12,5.12]	0	多峰
Salomon	$f 10 = 1 - c o s 2 π ∑ i = 1 D x i 2 + 0.1 ∑ i = 1 D x i 2$	[-5,5]	0	多峰

表2 求解100维测试函数的算法性能对比

Table 2 Comparison of algorithms for solving 100D functions

Function	Index	WOA	LWOA	LXWOA	DPWOA
	best	0.00E+00	0.00E+00	0.00E+00	0.00E+00
f₁	worst	5.74E-23	3.27E-36	4.93E-52	0.00E+00
f₁	mean	6.79E-35	3.95E-46	2.87E-59	0.00E+00
	std	1.80E-34	9.53E-47	3.53E-56	0.00E+00
	best	1.18E-144	1.06E-203	1.36E-239	0.00E+00
f₂	worst	4.93E-130	1.92E-153	4.10E-225	0.00E+00
f₂	mean	9.25E-131	2.45E-186	9.45E-226	0.00E+00
	std	1.50E-130	4.47E-172	0.00E+00	0.00E+00
	best	3.76E-91	0.00E+00	6.27E-132	0.00E+00
f₃	worst	1.05E-80	6.38E-85	1.16E-115	0.00E+00
f₃	mean	1.05E-81	3.25E-93	2.45E-116	0.00E+00
	std	3.16E-81	1.38E-90	4.57E-116	0.00E+00
	best	2.24E-50	1.52E-61	2.45E-121	0.00E+00
f₄	worst	1.52E-48	5.88E-57	1.67E-61	5.47E-62
f₄	mean	2.44E-49	7.82E-59	4.09E-62	4.11E-64
	std	4.44E-49	1.47E-58	6.36E-62	4.08E-65
	best	1.19E-51	1.18E-68	5.80E-117	0.00E+00
f₅	worst	4.66E-47	4.36E-52	1.25E-61	4.88E-308
f₅	mean	5.76E-48	2.83E-54	5.42E-62	6.78E-309
	std	1.38E-47	9.88E-56	5.20E-62	0.00E+00
	best	8.88E-16	2.57E-16	8.88E-16	0.00E+00
f₆	worst	6.22E-15	1.12E-15	8.88E-16	0.00E+00
f₆	mean	3.73E-15	7.08E-16	8.88E-16	0.00E+00
	std	2.77E-15	2.24E-16	0.00E+00	0.00E+00
	best	9.00E-01	9.00E-01	9.00E-01	9.00E-01
f₇	worst	3.80E+00	1.88E+00	9.00E-01	9.00E-01
f₇	mean	1.27E+00	1.14E+00	9.00E-01	9.00E-01
	std	8.58E-01	8.35E-01	0.00E+00	0.00E+00
	best	6.22E-05	1.94E-06	8.30E-06	1.26E-12
f₈	worst	4.14E-03	8.26E-04	6.72E-04	6.74E-11
f₈	mean	1.19E-03	5.69E-04	2.05E-04	3.54E-11
	std	1.31E-03	6.33E-04	2.44E-04	8.37E-11
	best	0.00E+00	0.00E+00	0.00E+00	0.00E+00
f₉	worst	0.00E+00	4.41E-268	0.00E+00	0.00E+00
f₉	mean	0.00E+00	3.69E-328	0.00E+00	0.00E+00
	std	0.00E+00	0.00E+00	0.00E+00	0.00E+00
	best	9.28E-02	1.23E-03	8.99E-02	0.00E+00
f₁₀	worst	2.00E-01	5.41E-02	2.00E-01	0.00E+00
f₁₀	mean	1.10E-01	2.72E-03	9.99E-02	0.00E+00
	std	5.38E-02	9.00E-01	6.32E-02	0.00E+00

表2 求解100维测试函数的算法性能对比

Table 2 Comparison of algorithms for solving 100D functions

Function	Index	WOA	LWOA	LXWOA	DPWOA
	best	0.00E+00	0.00E+00	0.00E+00	0.00E+00
f₁	worst	5.74E-23	3.27E-36	4.93E-52	0.00E+00
f₁	mean	6.79E-35	3.95E-46	2.87E-59	0.00E+00
	std	1.80E-34	9.53E-47	3.53E-56	0.00E+00
	best	1.18E-144	1.06E-203	1.36E-239	0.00E+00
f₂	worst	4.93E-130	1.92E-153	4.10E-225	0.00E+00
f₂	mean	9.25E-131	2.45E-186	9.45E-226	0.00E+00
	std	1.50E-130	4.47E-172	0.00E+00	0.00E+00
	best	3.76E-91	0.00E+00	6.27E-132	0.00E+00
f₃	worst	1.05E-80	6.38E-85	1.16E-115	0.00E+00
f₃	mean	1.05E-81	3.25E-93	2.45E-116	0.00E+00
	std	3.16E-81	1.38E-90	4.57E-116	0.00E+00
	best	2.24E-50	1.52E-61	2.45E-121	0.00E+00
f₄	worst	1.52E-48	5.88E-57	1.67E-61	5.47E-62
f₄	mean	2.44E-49	7.82E-59	4.09E-62	4.11E-64
	std	4.44E-49	1.47E-58	6.36E-62	4.08E-65
	best	1.19E-51	1.18E-68	5.80E-117	0.00E+00
f₅	worst	4.66E-47	4.36E-52	1.25E-61	4.88E-308
f₅	mean	5.76E-48	2.83E-54	5.42E-62	6.78E-309
	std	1.38E-47	9.88E-56	5.20E-62	0.00E+00
	best	8.88E-16	2.57E-16	8.88E-16	0.00E+00
f₆	worst	6.22E-15	1.12E-15	8.88E-16	0.00E+00
f₆	mean	3.73E-15	7.08E-16	8.88E-16	0.00E+00
	std	2.77E-15	2.24E-16	0.00E+00	0.00E+00
	best	9.00E-01	9.00E-01	9.00E-01	9.00E-01
f₇	worst	3.80E+00	1.88E+00	9.00E-01	9.00E-01
f₇	mean	1.27E+00	1.14E+00	9.00E-01	9.00E-01
	std	8.58E-01	8.35E-01	0.00E+00	0.00E+00
	best	6.22E-05	1.94E-06	8.30E-06	1.26E-12
f₈	worst	4.14E-03	8.26E-04	6.72E-04	6.74E-11
f₈	mean	1.19E-03	5.69E-04	2.05E-04	3.54E-11
	std	1.31E-03	6.33E-04	2.44E-04	8.37E-11
	best	0.00E+00	0.00E+00	0.00E+00	0.00E+00
f₉	worst	0.00E+00	4.41E-268	0.00E+00	0.00E+00
f₉	mean	0.00E+00	3.69E-328	0.00E+00	0.00E+00
	std	0.00E+00	0.00E+00	0.00E+00	0.00E+00
	best	9.28E-02	1.23E-03	8.99E-02	0.00E+00
f₁₀	worst	2.00E-01	5.41E-02	2.00E-01	0.00E+00
f₁₀	mean	1.10E-01	2.72E-03	9.99E-02	0.00E+00
	std	5.38E-02	9.00E-01	6.32E-02	0.00E+00

表3 求解500维测试函数的算法性能对比

Table 3 Comparison of algorithms for solving 500D functions

Function	Index	WOA	LWOA	LXWOA	DPWOA
	best	0.00E+00	0.00E+00	0.00E+00	0.00E+00
f₁	worst	4.78E-21	5.28E-35	4.85E-50	0.00E+00
f₁	mean	7.54E-32	8.68E-43	3.68E-56	0.00E+00
	std	3.68E-32	1.36E-45	8.78E-55	0.00E+00
	best	9.52E-144	9.28E-201	7.17E-237	0.00E+00
f₂	worst	6.47E-122	3.68E-152	3.43E-224	0.00E+00
f₂	mean	6.47E-123	4.15E-187	6.95E-225	0.00E+00
	std	1.94E-122	5.26E-168	0.00E+00	0.00E+00
	best	7.49E-88	6.85E-207	1.26E-115	0.00E+00
f₃	worst	2.02E-80	8.21E-82	4.78E-110	0.00E+00
f₃	mean	3.66E-81	5.28E-88	9.57E-111	0.00E+00
	std	7.21E-81	4.75E-87	1.90E-110	0.00E+00
	best	3.94E-50	2.06E-60	6.06E-117	0.00E+00
f₄	worst	2.79E-45	6.13E-56	1.33E-58	7.71E-62
f₄	mean	3.53E-46	6.88E-58	3.24E-59	6.50E-64
	std	8.42E-46	4.29E-57	5.13E-59	5.62E-65
	best	5.75E-51	4.52E-62	9.13E-116	0.00E+00
f₅	worst	1.58E-46	1.38E-51	4.49E-47	6.54E-306
f₅	mean	3.79E-47	4.37E-52	9.24E-48	2.55E-307
	std	5.15E-47	3.51E-54	1.78E-47	0.00E+00
	best	8.88E-16	2.86E-16	8.88E-16	0.00E+00
f₆	worst	6.22E-15	1.98E-15	8.88E-16	0.00E+00
f₆	mean	3.82E-15	7.41E-16	8.88E-16	0.00E+00
	std	2.84E-15	2.62E-16	0.00E+00	0.00E+00
	best	9.00E-01	9.00E-01	9.00E-01	9.00E-01
f₇	worst	5.46E+00	3.58E+00	9.00E-01	9.00E-01
f₇	mean	2.86E+00	2.37E+00	9.00E-01	9.00E-01
	std	9.87E-01	8.95E-01	0.00E+00	0.00E+00
	best	4.07E-04	4.59E-06	5.49E-05	3.58E-12
f₈	worst	3.06E-03	8.39E-03	1.73E-03	9.58E-11
f₈	mean	1.24E-03	9.68E-04	7.34E-04	9.98E-11
	std	9.58E-04	7.85E-04	6.61E-04	8.85E-11
	best	0.00E+00	0.00E+00	0.00E+00	0.00E+00
f₉	worst	3.43E-203	4.06E-252	5.39E-265	0.00E+00
f₉	mean	9.68E-269	7.56E-306	8.12E-278	0.00E+00
	std	4.02E-253	0.00E+00	0.00E+00	0.00E+00
	best	1.56E-01	2.72E-02	8.99E-02	0.00E+00
f₁₀	worst	9.35E-01	8.25E-01	2.00E-01	0.00E+00
f₁₀	mean	5.10E-01	6.57E-02	1.20E-01	0.00E+00
	std	4.98E-01	7.58E-01	7.48E-02	0.00E+00

表3 求解500维测试函数的算法性能对比

Table 3 Comparison of algorithms for solving 500D functions

Function	Index	WOA	LWOA	LXWOA	DPWOA
	best	0.00E+00	0.00E+00	0.00E+00	0.00E+00
f₁	worst	4.78E-21	5.28E-35	4.85E-50	0.00E+00
f₁	mean	7.54E-32	8.68E-43	3.68E-56	0.00E+00
	std	3.68E-32	1.36E-45	8.78E-55	0.00E+00
	best	9.52E-144	9.28E-201	7.17E-237	0.00E+00
f₂	worst	6.47E-122	3.68E-152	3.43E-224	0.00E+00
f₂	mean	6.47E-123	4.15E-187	6.95E-225	0.00E+00
	std	1.94E-122	5.26E-168	0.00E+00	0.00E+00
	best	7.49E-88	6.85E-207	1.26E-115	0.00E+00
f₃	worst	2.02E-80	8.21E-82	4.78E-110	0.00E+00
f₃	mean	3.66E-81	5.28E-88	9.57E-111	0.00E+00
	std	7.21E-81	4.75E-87	1.90E-110	0.00E+00
	best	3.94E-50	2.06E-60	6.06E-117	0.00E+00
f₄	worst	2.79E-45	6.13E-56	1.33E-58	7.71E-62
f₄	mean	3.53E-46	6.88E-58	3.24E-59	6.50E-64
	std	8.42E-46	4.29E-57	5.13E-59	5.62E-65
	best	5.75E-51	4.52E-62	9.13E-116	0.00E+00
f₅	worst	1.58E-46	1.38E-51	4.49E-47	6.54E-306
f₅	mean	3.79E-47	4.37E-52	9.24E-48	2.55E-307
	std	5.15E-47	3.51E-54	1.78E-47	0.00E+00
	best	8.88E-16	2.86E-16	8.88E-16	0.00E+00
f₆	worst	6.22E-15	1.98E-15	8.88E-16	0.00E+00
f₆	mean	3.82E-15	7.41E-16	8.88E-16	0.00E+00
	std	2.84E-15	2.62E-16	0.00E+00	0.00E+00
	best	9.00E-01	9.00E-01	9.00E-01	9.00E-01
f₇	worst	5.46E+00	3.58E+00	9.00E-01	9.00E-01
f₇	mean	2.86E+00	2.37E+00	9.00E-01	9.00E-01
	std	9.87E-01	8.95E-01	0.00E+00	0.00E+00
	best	4.07E-04	4.59E-06	5.49E-05	3.58E-12
f₈	worst	3.06E-03	8.39E-03	1.73E-03	9.58E-11
f₈	mean	1.24E-03	9.68E-04	7.34E-04	9.98E-11
	std	9.58E-04	7.85E-04	6.61E-04	8.85E-11
	best	0.00E+00	0.00E+00	0.00E+00	0.00E+00
f₉	worst	3.43E-203	4.06E-252	5.39E-265	0.00E+00
f₉	mean	9.68E-269	7.56E-306	8.12E-278	0.00E+00
	std	4.02E-253	0.00E+00	0.00E+00	0.00E+00
	best	1.56E-01	2.72E-02	8.99E-02	0.00E+00
f₁₀	worst	9.35E-01	8.25E-01	2.00E-01	0.00E+00
f₁₀	mean	5.10E-01	6.57E-02	1.20E-01	0.00E+00
	std	4.98E-01	7.58E-01	7.48E-02	0.00E+00

表4 求解1 000维测试函数的算法性能对比

Table 4 Comparison of algorithms for solving 1000D functions

Function	Index	WOA	LWOA	LXWOA	DPWOA
	best	3.06E-28	0.00E+00	0.00E+00	0.00E+00
f₁	worst	6.58E+00	8.12E-35	6.51E-49	0.00E+00
f₁	mean	9.31E-19	2.39E-44	4.39E-57	0.00E+00
	std	1.06E-18	6.29E-44	1.01E-56	0.00E+00
	best	3.73E-148	8.27E-201	4.13E-238	0.00E+00
f₂	worst	8.03E-70	1.63E-110	1.11E-115	4.91E-245
f₂	mean	1.61E-131	9.12E-163	2.22E-179	0.00E+00
	std	3.21E-131	2.68E-165	8.14E-171	0.00E+00
	best	1.64E-88	2.09E-205	6.11E-127	0.00E+00
f₃	worst	3.26E-45	5.49E-81	4.25E-107	0.00E+00
f₃	mean	1.31E-80	3.37E-86	8.50E-108	0.00E+00
	std	1.11E-80	3.14E-85	1.70E-107	0.00E+00
	best	1.12E-48	7.28E-58	9.55E-85	0.00E+00
f₄	worst	2.14E-45	6.88E-48	5.70E-37	8.83E-62
f₄	mean	4.55E-46	5.15E-52	1.20E-56	6.52E-64
	std	8.41E-46	5.21E-53	2.25E-58	7.46E-65
	best	2.01E-47	8.52E-62	3.25E-115	0.00E+00
f₅	worst	1.52E-45	3.26E-45	1.99E-37	8.29E-306
f₅	mean	3.60E-46	4.87E-46	5.35E-38	1.34E-307
	std	5.81E-46	3.51E-51	7.55E-38	0.00E+00
	best	8.88E-16	2.86E-16	8.88E-16	0.00E+00
f₆	worst	1.33E-14	2.86E-15	2.86E-15	8.88E-16
f₆	mean	5.86E-15	8.36E-16	8.88E-16	1.23E-16
	std	4.26E-15	6.38E-16	0.00E+00	3.55E-17
	best	9.00E-01	9.00E-01	9.00E-01	9.00E-01
f₇	worst	9.19E+00	3.96E+00	9.00E-01	9.00E-01
f₇	mean	5.02E+00	3.76E+00	9.00E-01	9.00E-01
	std	9.97E-01	9.03E-01	0.00E+00	0.00E+00
	best	9.19E-04	1.37E-05	7.62E-05	4.21E-12
f₈	worst	7.60E-01	9.81E-02	1.65E-02	9.88E-04
f₈	mean	4.57E-03	3.55E-03	5.38E-04	1.37E-10
	std	2.62E-03	8.19E-04	5.86E-04	2.06E-10
	best	0.00E+00	0.00E+00	0.00E+00	0.00E+00
f₉	worst	7.36E-228	7.28E-266	2.36E-263	0.00E+00
f₉	mean	1.34E-253	1.16E-277	8.47E-282	0.00E+00
	std	0.00E+00	0.00E+00	0.00E+00	0.00E+00
	best	2.37E-01	3.52E-02	9.99E-02	0.00E+00
f₁₀	worst	6.25E-01	9.34E-01	2.00E-01	0.00E+00
f₁₀	mean	5.34E-01	9.27E-02	1.20E-01	0.00E+00
	std	5.88E-01	8.04E-01	7.61E-02	0.00E+00

表4 求解1 000维测试函数的算法性能对比

Table 4 Comparison of algorithms for solving 1000D functions

Function	Index	WOA	LWOA	LXWOA	DPWOA
	best	3.06E-28	0.00E+00	0.00E+00	0.00E+00
f₁	worst	6.58E+00	8.12E-35	6.51E-49	0.00E+00
f₁	mean	9.31E-19	2.39E-44	4.39E-57	0.00E+00
	std	1.06E-18	6.29E-44	1.01E-56	0.00E+00
	best	3.73E-148	8.27E-201	4.13E-238	0.00E+00
f₂	worst	8.03E-70	1.63E-110	1.11E-115	4.91E-245
f₂	mean	1.61E-131	9.12E-163	2.22E-179	0.00E+00
	std	3.21E-131	2.68E-165	8.14E-171	0.00E+00
	best	1.64E-88	2.09E-205	6.11E-127	0.00E+00
f₃	worst	3.26E-45	5.49E-81	4.25E-107	0.00E+00
f₃	mean	1.31E-80	3.37E-86	8.50E-108	0.00E+00
	std	1.11E-80	3.14E-85	1.70E-107	0.00E+00
	best	1.12E-48	7.28E-58	9.55E-85	0.00E+00
f₄	worst	2.14E-45	6.88E-48	5.70E-37	8.83E-62
f₄	mean	4.55E-46	5.15E-52	1.20E-56	6.52E-64
	std	8.41E-46	5.21E-53	2.25E-58	7.46E-65
	best	2.01E-47	8.52E-62	3.25E-115	0.00E+00
f₅	worst	1.52E-45	3.26E-45	1.99E-37	8.29E-306
f₅	mean	3.60E-46	4.87E-46	5.35E-38	1.34E-307
	std	5.81E-46	3.51E-51	7.55E-38	0.00E+00
	best	8.88E-16	2.86E-16	8.88E-16	0.00E+00
f₆	worst	1.33E-14	2.86E-15	2.86E-15	8.88E-16
f₆	mean	5.86E-15	8.36E-16	8.88E-16	1.23E-16
	std	4.26E-15	6.38E-16	0.00E+00	3.55E-17
	best	9.00E-01	9.00E-01	9.00E-01	9.00E-01
f₇	worst	9.19E+00	3.96E+00	9.00E-01	9.00E-01
f₇	mean	5.02E+00	3.76E+00	9.00E-01	9.00E-01
	std	9.97E-01	9.03E-01	0.00E+00	0.00E+00
	best	9.19E-04	1.37E-05	7.62E-05	4.21E-12
f₈	worst	7.60E-01	9.81E-02	1.65E-02	9.88E-04
f₈	mean	4.57E-03	3.55E-03	5.38E-04	1.37E-10
	std	2.62E-03	8.19E-04	5.86E-04	2.06E-10
	best	0.00E+00	0.00E+00	0.00E+00	0.00E+00
f₉	worst	7.36E-228	7.28E-266	2.36E-263	0.00E+00
f₉	mean	1.34E-253	1.16E-277	8.47E-282	0.00E+00
	std	0.00E+00	0.00E+00	0.00E+00	0.00E+00
	best	2.37E-01	3.52E-02	9.99E-02	0.00E+00
f₁₀	worst	6.25E-01	9.34E-01	2.00E-01	0.00E+00
f₁₀	mean	5.34E-01	9.27E-02	1.20E-01	0.00E+00
	std	5.88E-01	8.04E-01	7.61E-02	0.00E+00

图6 f1收敛曲线

Fig.6 Convergence curve of f1

图7 f2收敛曲线

Fig.7 Convergence curve of f2

图8 f3收敛曲线

Fig.8 Convergence curve of f3

图9 f4收敛曲线

Fig.9 Convergence curve of f4

图10 f5收敛曲线

Fig.10 Convergence curve of f5

图11 f6收敛曲线

Fig.11 Convergence curve of f6

图12 f7收敛曲线

Fig.12 Convergence curve of f7

图13 f8收敛曲线

Fig.13 Convergence curve of f8

图14 f9收敛曲线

Fig.14 Convergence curve of f9

图15 f10收敛曲线

Fig.15 Convergence curve of f10

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求解高维优化问题的改进鲸鱼优化算法

Improved Whale Optimization Algorithm for Solving High-Dimensional Optimiza-tion Problems

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