融入社会心理学理论的教与学优化算法

doi:10.3778/j.issn.1673-9418.2010049

计算机科学与探索 ›› 2022, Vol. 16 ›› Issue (6): 1362-1373.DOI: 10.3778/j.issn.1673-9418.2010049

融入社会心理学理论的教与学优化算法

何佩苑(), 刘勇

上海理工大学管理学院,上海 200093

收稿日期:2020-10-19 修回日期:2021-01-07 出版日期:2022-06-01 发布日期:2021-01-25
通讯作者: + E-mail: 17805058112@163.com
作者简介:何佩苑（1997—）,女,江苏溧阳人,硕士研究生,主要研究方向为智能优化、系统工程。
刘勇（1982—）,男,江苏金湖人,博士,副教授,硕士生导师,主要研究方向为智能优化、服务网络设计与优化、系统工程。
基金资助:
上海市“科技创新行动计划”软科学研究重点项目(18692110500);上海市社会科学规划课题(2019BGL014);上海市高原科学建设项目（第2期）;上海理工大学科技发展资助项目(2020KJFZ040)

Teaching-Learning-Based Optimization Algorithm with Social Psychology Theory

HE Peiyuan(), LIU Yong

School of Management, University of Shanghai for Science and Technology, Shanghai 200093, China

Received:2020-10-19 Revised:2021-01-07 Online:2022-06-01 Published:2021-01-25
About author:HE Peiyuan, born in 1997, M.S. candidate. Her research interests include intelligent optimization and system engineering.
LIU Yong, born in 1982, Ph.D., associate profes-sor, M.S. supervisor. His research interests include intelligent optimization, design and optimization of service network and system engineering.
Supported by:
Key Project of Soft Science Research in Shanghai Science and Technology Innovation Action Plan(18692110500);Social Science Planning Project of Shanghai(2019BGL014);Plateau Science Construction Project of Shanghai (NO.2);Science and Technology Development Funding Project of University of Shanghai for Science and Technology(2020KJFZ040)

摘要/Abstract

摘要：

教与学优化算法（TLBO）是一种模拟教学过程的启发式优化算法。针对TLBO算法寻优精度低、稳定性差的特点,提出了基于社会心理学理论改进的教与学优化算法（SPTLBO）。该算法在改进中考虑了人的心理因素：在原TLBO算法的“教”阶段中结合社会心理学的“期望效应”理论,教师对高期望学生采取一对一教学策略,使得优秀学生更快向教师靠近;为了保留学生的多样性,学生依据认知风格可分为“场独立”与“场依存”两种类型,不同类型的学生将采取不同的交流方式进行学习;在“教”“学”阶段后,结合自我调节理论,学生进入学习方法调整阶段,从而增强了自我探索能力,提高学生整体水平。此外,引入自适应学生更新因子,模拟环境对学生学习效率的影响,增加算法的全局搜索能力,避免出现在初期迭代中陷入局部最优的情况。在25个标准测试函数上进行实验,结果表明SPTLBO算法相较基本TLBO算法和其他智能优化算法,在寻优精度和收敛速度方面都更具优势。

关键词: 教与学优化算法（TLBO）, 社会心理学, 期望效应, 场独立-场依存, 自我调节理论, 自适应更新因子

Abstract:

Teaching-learning-based optimization (TLBO) is a heuristic optimization algorithm that simulates the teaching process. In view of the low precision and poor stability of TLBO algorithm, an improved teaching-learning-based optimization algorithm named SPTLBO (social psychology teaching-learning-based optimization) is proposed. Human psychological factors are considered in the improvement of the algorithm. In the “teaching” stage, combining the “expectation effect” theory in social psychology, teachers adopt one-to-one teaching strategy for students with high expectations, which makes outstanding students approach teachers faster. According to cognitive style, students can be divided into two types, “field independence” and “field dependence”, so that it can preserve the diversity of students. Different types of students will adopt different communication methods to learn. After the “teaching” and “learning” stages, combined with the theory of self-regulation, students enter the stage of learning method adjustment. It can enhance the ability of self-exploration and improve the overall level of students. In addition, an adaptive student update factor is introduced to simulate the influence of environment on students’ learning efficiency, which increases the global search ability of the algorithm and avoids falling into local optimum in the initial iteration. The test of 25 test functions shows that, compared with the basic TLBO algorithm and other intelligent optimization algorithms, the SPTLBO algorithm has more advantages in the optimization accuracy and convergence speed.

Key words: teaching-learning-based optimization algorithm (TLBO), social psychology, expectation effect, field independence-field dependence, self-regulation theory, adaptive update factor

中图分类号:

TP18

何佩苑, 刘勇. 融入社会心理学理论的教与学优化算法[J]. 计算机科学与探索, 2022, 16(6): 1362-1373.

HE Peiyuan, LIU Yong. Teaching-Learning-Based Optimization Algorithm with Social Psychology Theory[J]. Journal of Frontiers of Computer Science and Technology, 2022, 16(6): 1362-1373.

图/表 7

图1 学生的多样性对比图

Fig.1 Comparison chart of students’ diversity

表1 低维测试函数

Table 1 Low-dimensional test functions

函数编号	函数名	表达式
$f 1$	Camel-Three Hump	$f 1 (x) = 2 x 12 - 1.05 x 14 + 16 x 16 + x 1 x 2 + x 22$
$f 2$	Rotated Ellipse 2	$f 2 (x) = x 12 - x 1 x 2 + x 22$
$f 3$	Scahffer 1	$f 3 (x) = 0.5 + s i n 2 (x 12 + x 22) 2 - 0.5 (1 + 0.001 (x 12 + x 22)) 2$
$f 4$	Rotated Ellipse	$f 4 (x) = 7 x 12 - 63 x 1 x 2 + 13 x 22$
$f 5$	Schumer Steiglitz	$f 5 (x) = ∑ i = 1 2 x i 4$
$f 6$	Price 4	$f 6 (x) = (2 x 13 x 2 - x 23) 2 + (6 x 1 - x 22 + x 2) 2$
$f 7$	Freudenstein Roth	$f 7 (x) = (x 1 - 13 + ((5 - x 2) x 2 - 2) x 2) 2 + (x 1 - 29 + ((x 2 + 1) x 2 - 14) x 2) 2$
$f 8$	Treccani	$f 8 (x) = x 14 - 4 x 13 + 4 x 12 + x 22$
$f 9$	Egg Crate	$f 9 (x) = x 12 + x 22 + 25 (s i n 2 x 1 + s i n 2 x 2)$
$f 10$	Matyas	$f 10 (x) = 0.26 (x 12 + x 22) - 0.48 x 1 x 2$

表1 低维测试函数

Table 1 Low-dimensional test functions

函数编号	函数名	表达式
$f 1$	Camel-Three Hump	$f 1 (x) = 2 x 12 - 1.05 x 14 + 16 x 16 + x 1 x 2 + x 22$
$f 2$	Rotated Ellipse 2	$f 2 (x) = x 12 - x 1 x 2 + x 22$
$f 3$	Scahffer 1	$f 3 (x) = 0.5 + s i n 2 (x 12 + x 22) 2 - 0.5 (1 + 0.001 (x 12 + x 22)) 2$
$f 4$	Rotated Ellipse	$f 4 (x) = 7 x 12 - 63 x 1 x 2 + 13 x 22$
$f 5$	Schumer Steiglitz	$f 5 (x) = ∑ i = 1 2 x i 4$
$f 6$	Price 4	$f 6 (x) = (2 x 13 x 2 - x 23) 2 + (6 x 1 - x 22 + x 2) 2$
$f 7$	Freudenstein Roth	$f 7 (x) = (x 1 - 13 + ((5 - x 2) x 2 - 2) x 2) 2 + (x 1 - 29 + ((x 2 + 1) x 2 - 14) x 2) 2$
$f 8$	Treccani	$f 8 (x) = x 14 - 4 x 13 + 4 x 12 + x 22$
$f 9$	Egg Crate	$f 9 (x) = x 12 + x 22 + 25 (s i n 2 x 1 + s i n 2 x 2)$
$f 10$	Matyas	$f 10 (x) = 0.26 (x 12 + x 22) - 0.48 x 1 x 2$

表2 高维测试函数

Table 2 High-dimensional test functions

函数编号	函数名	表达式
$f 11$	Chung Reynolds	$f 11 = ∑ i = 1 D x i 2 2$
$f 12$	Sphere	$f 12 (x) = ∑ i = 1 D x i 2$
$f 13$	Schwefel2.22	$f 13 (x) = ∑ i = 1 D x i + ∏ i = 1 D \| x i \|$
$f 14$	Schwefel1.2	$f 14 (x) = ∑ i = 1 D (∑ j = 1 i x j) 2$
$f 15$	Schwefel2.21	$f 15 (x) = m a x (\| x \|)$
$f 16$	Powell Sum	$f 16 (x) = ∑ i = 1 D \| x i \| i + 1$
$f 17$	W/Wavy	$f 17 (x) = 1 - g D, g = ∑ i = 1 D c o s (10 x i) e - x i 2 2$
$f 18$	Quartic	$f 18 (x) = ∑ i = 1 D i x i 4 + r a n d [0,1)$
$f 19$	Zakharov	$f 19 (x) = ∑ i = 1 D x i 2 + ∑ i = 1 D 12 i x i 2 + ∑ i = 1 D 12 i x i 4$
$f 20$	Alpine	$f 20 (x) = ∑ i = 1 D \| x i s i n x i + 0.1 x \|$
$f 21$	Cigar	$f 21 (x) = x 12 + 106 g, g = ∑ i = 1 D x i 2$
$f 22$	Schwefel2.23	$f 22 (x) = ∑ i = 1 D x i 10$
$f 23$	Csendes	$f 23 (x) = ∑ i = 1 D x i 6 2 + s i n 1 x i, ∏ i = 1 D x i ≠ 0 0, ∏ i = 1 D x i = 0$
$f 24$	Salomon	$f 24 (x) = 1 - c o s (2 π g) + 0.1 g, g = ∑ i = 1 D x i 2$
$f 25$	Rastrigin	$f 25 (x) = 10 D + g, g = ∑ i = 1 D x i 2 - 10 c o s (2 π x i)$

表2 高维测试函数

Table 2 High-dimensional test functions

函数编号	函数名	表达式
$f 11$	Chung Reynolds	$f 11 = ∑ i = 1 D x i 2 2$
$f 12$	Sphere	$f 12 (x) = ∑ i = 1 D x i 2$
$f 13$	Schwefel2.22	$f 13 (x) = ∑ i = 1 D x i + ∏ i = 1 D \| x i \|$
$f 14$	Schwefel1.2	$f 14 (x) = ∑ i = 1 D (∑ j = 1 i x j) 2$
$f 15$	Schwefel2.21	$f 15 (x) = m a x (\| x \|)$
$f 16$	Powell Sum	$f 16 (x) = ∑ i = 1 D \| x i \| i + 1$
$f 17$	W/Wavy	$f 17 (x) = 1 - g D, g = ∑ i = 1 D c o s (10 x i) e - x i 2 2$
$f 18$	Quartic	$f 18 (x) = ∑ i = 1 D i x i 4 + r a n d [0,1)$
$f 19$	Zakharov	$f 19 (x) = ∑ i = 1 D x i 2 + ∑ i = 1 D 12 i x i 2 + ∑ i = 1 D 12 i x i 4$
$f 20$	Alpine	$f 20 (x) = ∑ i = 1 D \| x i s i n x i + 0.1 x \|$
$f 21$	Cigar	$f 21 (x) = x 12 + 106 g, g = ∑ i = 1 D x i 2$
$f 22$	Schwefel2.23	$f 22 (x) = ∑ i = 1 D x i 10$
$f 23$	Csendes	$f 23 (x) = ∑ i = 1 D x i 6 2 + s i n 1 x i, ∏ i = 1 D x i ≠ 0 0, ∏ i = 1 D x i = 0$
$f 24$	Salomon	$f 24 (x) = 1 - c o s (2 π g) + 0.1 g, g = ∑ i = 1 D x i 2$
$f 25$	Rastrigin	$f 25 (x) = 10 D + g, g = ∑ i = 1 D x i 2 - 10 c o s (2 π x i)$

表3 低维测数函数结果对比

Table 3 Comparison of experimental results of low-dimensional benchmark functions

函数	算法	平均值	标准差	最劣值	最优值
$f 1$	TLBO	1.512 6E-112	6.458 6E-112	2.894 4E-111	4.090 9E-118
	PSO	1.302 6E-20	1.933 7E-20	7.307 6E-20	5.395 1E-23
	IA	6.657 7E-10	7.446 9E-10	2.787 7E-09	9.070 7E-12
	GA	1.560 7E-12	5.808 1E-12	2.600 9E-11	0
	SPTLBO	1.628 6E-214	0	3.250 7E-213	0
$f 2$	TLBO	1.056 1E-112	2.719 4E-112	1.155 5E-111	1.288 2E-117
	PSO	1.020 7E-20	3.217 7E-20	1.456 8E-19	4.594 1E-23
	IA	3.824 2E-10	3.938 7E-10	1.780 3E-09	4.599 2E-12
	GA	4.112 9E-10	1.777 5E-09	7.959 9E-09	8.370 8E-19
	SPTLBO	1.610 5E-211	0	1.400 5E-210	3.400 4E-220
$f 3$	TLBO	2.155 9E-04	7.115 1E-04	3.189 3E-03	1.498 8E-14
	PSO	2.915 2E-03	4.567 9E-03	9.715 9E-03	0
	IA	4.372 2E-03	4.959 2E-03	9.715 9E-03	4.118 4E-10
	GA	9.809 7E-04	3.019 5E-03	9.903 5E-03	0
	SPTLBO	1.901 6E-09	7.703 1E-09	3.452 9E-08	0
$f 4$	TLBO	9.655 4E-110	3.400 5E-109	1.520 8E-108	1.460 6E-115
	PSO	7.982 6E-19	2.972 7E-18	1.330 5E-17	1.825 3E-22
	IA	5.226 4E-09	6.151 1E-09	2.764 9E-08	2.662 5E-11
	GA	5.343 2E-08	2.170 8E-07	9.737 5E-07	3.654 1E-17
	SPTLBO	7.761 0E-209	0	7.794 1E-208	1.081 0E-218
$f 5$	TLBO	2.393 1E-224	0	4.715 2E-223	6.769 0E-238
	PSO	7.593 2E-39	2.154 6E-38	9.311 5E-38	6.585 8E-49
	IA	2.091 4E-18	2.775 6E-18	1.222 9E-17	1.285 3E-21
	GA	1.080 7E-23	4.786 5E-23	2.141 6E-22	2.974 2E-31
	SPTLBO	0	0	0	0
$f 6$	TLBO	2.161 4E-11	5.273 3E-11	2.291 7E-10	7.999 7E-15
	PSO	1.442 7E-15	1.828 3E-15	6.864 7E-15	1.469 7E-19
	IA	9.009 1E-10	1.169 8E-09	3.641 5E-09	1.764 4E-11
	GA	3.219 7E-02	9.219 6E-02	3.995 9E-01	0
	SPTLBO	1.412 2E-14	2.668 1E-14	1.024 9E-13	0
$f 7$	TLBO	3.427 9E-176	0	6.850 8E-175	3.071 6E-190
	PSO	1.836 2E-19	5.306 3E-19	2.116 2E-18	2.905 8E-25
	IA	1.067 6E-08	1.073 5E-08	3.676 8E-08	4.156 2E-11
	GA	1.111 5E-07	4.970 8E-07	2.223 4E-06	0
	SPTLBO	6.924 5E-310	0	1.372 2E-308	0
$f 8$	TLBO	5.771 4E-223	0	1.154 2E-221	3.984 7E-237
	PSO	6.535 1E-40	2.623 2E-39	1.178 4E-38	1.064 0E-45
	IA	2.094 3E-19	3.927 9E-19	1.310 9E-18	3.685 9E-24
	GA	1.930 7E-18	8.438 4E-18	3.777 3E-17	8.797 8E-32
	SPTLBO	0	0	0	0
$f 9$	TLBO	2.861 5E-110	1.132 5E-109	5.077 5E-109	3.366 4E-116
	PSO	1.921 2E-19	3.848 4E-19	1.206 2E-18	1.228 8E-22
	IA	6.763 3E-09	7.252 5E-09	2.793 9E-08	7.046 8E-10
	GA	7.111 1E-09	2.390 5E-08	1.054 3E-07	1.951 7E-18
	SPTLBO	2.418 8E-213	0	4.100 5E-212	0
$f 10$	TLBO	2.190 9E-81	7.402 1E-81	3.278 3E-80	3.535 1E-85
	PSO	4.075 1E-20	1.780 0E-19	7.969 4E-19	1.105 8E-26
	IA	5.164 1E-10	5.077 0E-10	1.988 6E-09	1.900 2E-11
	GA	4.275 8E-03	5.692 9E-03	1.709 7E-02	0
	SPTLBO	5.109 4E-168	0	7.329 3E-167	0

表3 低维测数函数结果对比

Table 3 Comparison of experimental results of low-dimensional benchmark functions

函数	算法	平均值	标准差	最劣值	最优值
$f 1$	TLBO	1.512 6E-112	6.458 6E-112	2.894 4E-111	4.090 9E-118
	PSO	1.302 6E-20	1.933 7E-20	7.307 6E-20	5.395 1E-23
	IA	6.657 7E-10	7.446 9E-10	2.787 7E-09	9.070 7E-12
	GA	1.560 7E-12	5.808 1E-12	2.600 9E-11	0
	SPTLBO	1.628 6E-214	0	3.250 7E-213	0
$f 2$	TLBO	1.056 1E-112	2.719 4E-112	1.155 5E-111	1.288 2E-117
	PSO	1.020 7E-20	3.217 7E-20	1.456 8E-19	4.594 1E-23
	IA	3.824 2E-10	3.938 7E-10	1.780 3E-09	4.599 2E-12
	GA	4.112 9E-10	1.777 5E-09	7.959 9E-09	8.370 8E-19
	SPTLBO	1.610 5E-211	0	1.400 5E-210	3.400 4E-220
$f 3$	TLBO	2.155 9E-04	7.115 1E-04	3.189 3E-03	1.498 8E-14
	PSO	2.915 2E-03	4.567 9E-03	9.715 9E-03	0
	IA	4.372 2E-03	4.959 2E-03	9.715 9E-03	4.118 4E-10
	GA	9.809 7E-04	3.019 5E-03	9.903 5E-03	0
	SPTLBO	1.901 6E-09	7.703 1E-09	3.452 9E-08	0
$f 4$	TLBO	9.655 4E-110	3.400 5E-109	1.520 8E-108	1.460 6E-115
	PSO	7.982 6E-19	2.972 7E-18	1.330 5E-17	1.825 3E-22
	IA	5.226 4E-09	6.151 1E-09	2.764 9E-08	2.662 5E-11
	GA	5.343 2E-08	2.170 8E-07	9.737 5E-07	3.654 1E-17
	SPTLBO	7.761 0E-209	0	7.794 1E-208	1.081 0E-218
$f 5$	TLBO	2.393 1E-224	0	4.715 2E-223	6.769 0E-238
	PSO	7.593 2E-39	2.154 6E-38	9.311 5E-38	6.585 8E-49
	IA	2.091 4E-18	2.775 6E-18	1.222 9E-17	1.285 3E-21
	GA	1.080 7E-23	4.786 5E-23	2.141 6E-22	2.974 2E-31
	SPTLBO	0	0	0	0
$f 6$	TLBO	2.161 4E-11	5.273 3E-11	2.291 7E-10	7.999 7E-15
	PSO	1.442 7E-15	1.828 3E-15	6.864 7E-15	1.469 7E-19
	IA	9.009 1E-10	1.169 8E-09	3.641 5E-09	1.764 4E-11
	GA	3.219 7E-02	9.219 6E-02	3.995 9E-01	0
	SPTLBO	1.412 2E-14	2.668 1E-14	1.024 9E-13	0
$f 7$	TLBO	3.427 9E-176	0	6.850 8E-175	3.071 6E-190
	PSO	1.836 2E-19	5.306 3E-19	2.116 2E-18	2.905 8E-25
	IA	1.067 6E-08	1.073 5E-08	3.676 8E-08	4.156 2E-11
	GA	1.111 5E-07	4.970 8E-07	2.223 4E-06	0
	SPTLBO	6.924 5E-310	0	1.372 2E-308	0
$f 8$	TLBO	5.771 4E-223	0	1.154 2E-221	3.984 7E-237
	PSO	6.535 1E-40	2.623 2E-39	1.178 4E-38	1.064 0E-45
	IA	2.094 3E-19	3.927 9E-19	1.310 9E-18	3.685 9E-24
	GA	1.930 7E-18	8.438 4E-18	3.777 3E-17	8.797 8E-32
	SPTLBO	0	0	0	0
$f 9$	TLBO	2.861 5E-110	1.132 5E-109	5.077 5E-109	3.366 4E-116
	PSO	1.921 2E-19	3.848 4E-19	1.206 2E-18	1.228 8E-22
	IA	6.763 3E-09	7.252 5E-09	2.793 9E-08	7.046 8E-10
	GA	7.111 1E-09	2.390 5E-08	1.054 3E-07	1.951 7E-18
	SPTLBO	2.418 8E-213	0	4.100 5E-212	0
$f 10$	TLBO	2.190 9E-81	7.402 1E-81	3.278 3E-80	3.535 1E-85
	PSO	4.075 1E-20	1.780 0E-19	7.969 4E-19	1.105 8E-26
	IA	5.164 1E-10	5.077 0E-10	1.988 6E-09	1.900 2E-11
	GA	4.275 8E-03	5.692 9E-03	1.709 7E-02	0
	SPTLBO	5.109 4E-168	0	7.329 3E-167	0

表4 高维测试函数结果对比

Table 4 Comparison of experimental results of high-dimensional benchmark functions

函数	算法	平均值	标准差	最劣值	最优值
$f 11$	TLBO	1.972 4E-92	4.370 7E-92	1.996 9E-91	1.696 6E-95
	PSO	4.744 0E+04	1.543 1E+04	7.997 7E+04	2.079 8E+04
	IA	5.107 2E+05	6.601 6E+05	6.440 1E+05	3.462 6E+05
	GA	5.025 8E+02	4.270 5E+02	1.705 9E+02	1.382 8E+02
	SPTLBO	1.991 5E-196	0	2.690 7E-195	0
$f 12$	TLBO	4.987 2E-47	4.125 9E-47	1.528 9E-46	1.100 8E-47
	PSO	1.930 2E+02	2.928 0E+01	2.652 5E+02	1.530 2E+02
	IA	6.909 6E+02	4.517 1E+01	7.592 7E+02	6.129 4E+02
	GA	2.212 2E+01	1.011 3E+02	5.552 8E+02	1.095 1E+01
	SPTLBO	8.184 0E-104	1.241 0E-103	4.326 4E-103	0
$f 13$	TLBO	8.259 2E-23	4.310 6E-23	2.157 4E-22	3.647 1E-23
	PSO	1.133 2E+02	1.796 7E+02	1.762 4E+02	9.473 2E+01
	IA	7.383 4E+08	3.301 5E+09	1.476 5E+10	3.053 7E+02
	GA	6.070 5E+01	2.216 4E+01	9.080 5E+01	2.551 2E+01
	SPTLBO	2.656 8E-53	4.009 8E-53	1.359 7E-52	9.473 3E-55
$f 14$	TLBO	1.095 9E-03	1.192 3E-03	5.210 8E-03	6.265 2E-05
	PSO	5.466 4E+02	1.069 5E+02	7.502 4E+02	3.242 6E+02
	IA	5.838 1E-13	9.582 4E-13	3.758 1E-12	5.186 2E-16
	GA	7.471 5E+02	1.473 9E+02	1.086 8E+03	5.003 2E+02
	SPTLBO	9.173 0E-58	3.389 2E-57	1.522 6E-56	3.479 6E-62
$f 15$	TLBO	3.427 6E-19	1.397 4E-19	6.257 0E-19	1.382 9E-19
	PSO	4.539 6E-00	5.405 1E-01	5.665 1E-00	3.531 3E-00
	IA	6.743 6E-00	1.487 3E-01	6.965 4E-00	6.332 7E-00
	GA	3.869 5E-00	5.852 4E-01	5.213 1E-00	2.588 1E-00
	SPTLBO	5.060 6E-46	4.956 8E-46	1.913 8E-45	1.196 5E-47
$f 16$	TLBO	3.498 3E-111	1.413 2E-110	6.339 4E-110	7.358 1E-117
	PSO	5.331 3E+38	1.632 2E+39	5.954 1E+39	3.673 2E+27
	IA	9.341 4E-13	1.930 6E-12	7.041 3E-12	1.218 1E-15
	GA	7.742 0E+48	3.411 8E+49	1.526 8E+50	2.385 5E+26
	SPTLBO	5.540 5E-210	0	4.912 8E-209	0
$f 17$	TLBO	8.573 7E-01	1.658 3E-02	8.868 8E-01	8.271 3E-01
	PSO	7.645 3E-01	1.903 9E-02	8.100 8E-01	7.299 8E-01
	IA	1.160 7E-11	1.588 2E-11	6.437 3E-11	2.101 7E-13
	GA	6.381 7E-01	4.672 4E-02	7.147 6E-01	5.547 1E-01
	SPTLBO	2.401 5E-10	9.335 4E-10	4.193 9E-09	0
$f 18$	TLBO	8.059 8E-83	8.753 3E-83	2.864 8E-82	3.003 0E-84
	PSO	9.721 1E+09	3.768 2E+09	1.519 8E+10	3.853 7E+09
	IA	6.301 0E-20	1.740 6E-19	6.073 9E-19	1.620 7E-27
	GA	3.216 6E+08	2.418 8E+08	1.131 4E+09	1.403 2E+08
	SPTLBO	3.628 8E-179	0	4.636 1E-178	1.628 2E-183
$f 19$	TLBO	7.547 9E+02	1.776 6E+02	1.076 7E+03	4.076 1E+02
	PSO	6.936 6E+02	1.281 9E+02	9.561 3E+02	4.552 9E+02
	IA	7.779 1E-13	1.771 2E-12	7.977 8E-12	2.557 6E-16
	GA	2.183 0E+03	5.843 3E+02	3.494 3E+03	1.292 0E+03
	SPTLBO	3.211 4E-07	1.410 2E-06	6.311 5E-06	0
$f 20$	TLBO	1.007 5E-23	3.711 2E-24	1.791 7E-23	5.063 7E-24
	PSO	6.519 2E+01	6.349 6E-00	7.340 1E+01	5.095 1E+01
	IA	4.929 2E+01	4.107 2E+00	5.772 0E+01	4.031 8E+01
	GA	1.378 9E+01	1.819 3E+00	1.793 7E+01	1.139 9E+01
	SPTLBO	2.142 9E-54	2.690 6E-54	1.104 4E-53	0
函数	算法	平均值	标准差	最劣值	最优值
$f 21$	TLBO	1.249 1E-40	1.209 1E-40	3.885 7E-40	1.775 3E-41
	PSO	2.076 1E+08	2.832 2E+07	2.603 1E+08	1.359 8E+08
	IA	7.482 2E-13	1.499 5E-12	5.304 2E-12	2.420 7E-16
	GA	1.669 7E+07	5.665 2E+06	3.528 1E+07	1.012 2E+07
	SPTLBO	2.199 9E-97	4.073 2E-97	1.517 8E-96	2.772 3E-100
$f 22$	TLBO	5.901 4E-201	0	5.807 8E-200	4.946 7E-207
	PSO	1.860 8E+06	1.147 9E+06	4.062 5E+06	3.991 8E+06
	IA	1.458 3E+08	3.891 2E+08	2.203 8E+08	8.836 9E+08
	GA	3.765 2E+03	3.027 1E+03	1.016 1E+04	4.084 1E+02
	SPTLBO	0	0	0	0
$f 23$	TLBO	2.187 3E-124	4.547 6E-124	2.044 6E-123	9.101 1E-127
	PSO	3.429 1E+04	1.319 9E+04	6.479 9E+04	1.054 2E+04
	IA	4.333 5E+05	7.887 6E+04	5.710 3E+05	2.766 5E+05
	GA	5.824 2E+02	8.617 2E+02	3.723 2E+03	3.864 4E+01
	SPTLBO	2.149 9E-249	0	3.229 2E-248	6.968 4E-256
$f 24$	TLBO	1.998 7E-01	4.723 4E-07	1.998 8E-01	1.998 7E-01
	PSO	2.094 9E-00	1.234 4E-01	2.399 9E-00	1.899 9E-00
	IA	9.486 7E-08	9.731 1E-08	3.177 6E-07	1.193 2E-09
	GA	2.609 9E+00	3.582 1E-01	3.299 9E+00	1.999 9E+00
	SPTLBO	8.795 7E-02	3.003 2E-02	9.987 3E-02	0
$f 25$	TLBO	8.410 8E-00	3.761 4E+01	1.682 1E+02	0
	PSO	1.000 5E+03	7.620 1E+03	1.147 3E+03	8.488 1E+03
	IA	2.220 7E-10	3.770 1E-10	1.457 1E-09	1.659 1E-12
	GA	4.110 5E+02	4.757 3E+01	5.280 3E+02	3.345 0E+02
	SPTLBO	0	0	0	0

表4 高维测试函数结果对比

Table 4 Comparison of experimental results of high-dimensional benchmark functions

函数	算法	平均值	标准差	最劣值	最优值
$f 11$	TLBO	1.972 4E-92	4.370 7E-92	1.996 9E-91	1.696 6E-95
	PSO	4.744 0E+04	1.543 1E+04	7.997 7E+04	2.079 8E+04
	IA	5.107 2E+05	6.601 6E+05	6.440 1E+05	3.462 6E+05
	GA	5.025 8E+02	4.270 5E+02	1.705 9E+02	1.382 8E+02
	SPTLBO	1.991 5E-196	0	2.690 7E-195	0
$f 12$	TLBO	4.987 2E-47	4.125 9E-47	1.528 9E-46	1.100 8E-47
	PSO	1.930 2E+02	2.928 0E+01	2.652 5E+02	1.530 2E+02
	IA	6.909 6E+02	4.517 1E+01	7.592 7E+02	6.129 4E+02
	GA	2.212 2E+01	1.011 3E+02	5.552 8E+02	1.095 1E+01
	SPTLBO	8.184 0E-104	1.241 0E-103	4.326 4E-103	0
$f 13$	TLBO	8.259 2E-23	4.310 6E-23	2.157 4E-22	3.647 1E-23
	PSO	1.133 2E+02	1.796 7E+02	1.762 4E+02	9.473 2E+01
	IA	7.383 4E+08	3.301 5E+09	1.476 5E+10	3.053 7E+02
	GA	6.070 5E+01	2.216 4E+01	9.080 5E+01	2.551 2E+01
	SPTLBO	2.656 8E-53	4.009 8E-53	1.359 7E-52	9.473 3E-55
$f 14$	TLBO	1.095 9E-03	1.192 3E-03	5.210 8E-03	6.265 2E-05
	PSO	5.466 4E+02	1.069 5E+02	7.502 4E+02	3.242 6E+02
	IA	5.838 1E-13	9.582 4E-13	3.758 1E-12	5.186 2E-16
	GA	7.471 5E+02	1.473 9E+02	1.086 8E+03	5.003 2E+02
	SPTLBO	9.173 0E-58	3.389 2E-57	1.522 6E-56	3.479 6E-62
$f 15$	TLBO	3.427 6E-19	1.397 4E-19	6.257 0E-19	1.382 9E-19
	PSO	4.539 6E-00	5.405 1E-01	5.665 1E-00	3.531 3E-00
	IA	6.743 6E-00	1.487 3E-01	6.965 4E-00	6.332 7E-00
	GA	3.869 5E-00	5.852 4E-01	5.213 1E-00	2.588 1E-00
	SPTLBO	5.060 6E-46	4.956 8E-46	1.913 8E-45	1.196 5E-47
$f 16$	TLBO	3.498 3E-111	1.413 2E-110	6.339 4E-110	7.358 1E-117
	PSO	5.331 3E+38	1.632 2E+39	5.954 1E+39	3.673 2E+27
	IA	9.341 4E-13	1.930 6E-12	7.041 3E-12	1.218 1E-15
	GA	7.742 0E+48	3.411 8E+49	1.526 8E+50	2.385 5E+26
	SPTLBO	5.540 5E-210	0	4.912 8E-209	0
$f 17$	TLBO	8.573 7E-01	1.658 3E-02	8.868 8E-01	8.271 3E-01
	PSO	7.645 3E-01	1.903 9E-02	8.100 8E-01	7.299 8E-01
	IA	1.160 7E-11	1.588 2E-11	6.437 3E-11	2.101 7E-13
	GA	6.381 7E-01	4.672 4E-02	7.147 6E-01	5.547 1E-01
	SPTLBO	2.401 5E-10	9.335 4E-10	4.193 9E-09	0
$f 18$	TLBO	8.059 8E-83	8.753 3E-83	2.864 8E-82	3.003 0E-84
	PSO	9.721 1E+09	3.768 2E+09	1.519 8E+10	3.853 7E+09
	IA	6.301 0E-20	1.740 6E-19	6.073 9E-19	1.620 7E-27
	GA	3.216 6E+08	2.418 8E+08	1.131 4E+09	1.403 2E+08
	SPTLBO	3.628 8E-179	0	4.636 1E-178	1.628 2E-183
$f 19$	TLBO	7.547 9E+02	1.776 6E+02	1.076 7E+03	4.076 1E+02
	PSO	6.936 6E+02	1.281 9E+02	9.561 3E+02	4.552 9E+02
	IA	7.779 1E-13	1.771 2E-12	7.977 8E-12	2.557 6E-16
	GA	2.183 0E+03	5.843 3E+02	3.494 3E+03	1.292 0E+03
	SPTLBO	3.211 4E-07	1.410 2E-06	6.311 5E-06	0
$f 20$	TLBO	1.007 5E-23	3.711 2E-24	1.791 7E-23	5.063 7E-24
	PSO	6.519 2E+01	6.349 6E-00	7.340 1E+01	5.095 1E+01
	IA	4.929 2E+01	4.107 2E+00	5.772 0E+01	4.031 8E+01
	GA	1.378 9E+01	1.819 3E+00	1.793 7E+01	1.139 9E+01
	SPTLBO	2.142 9E-54	2.690 6E-54	1.104 4E-53	0
函数	算法	平均值	标准差	最劣值	最优值
$f 21$	TLBO	1.249 1E-40	1.209 1E-40	3.885 7E-40	1.775 3E-41
	PSO	2.076 1E+08	2.832 2E+07	2.603 1E+08	1.359 8E+08
	IA	7.482 2E-13	1.499 5E-12	5.304 2E-12	2.420 7E-16
	GA	1.669 7E+07	5.665 2E+06	3.528 1E+07	1.012 2E+07
	SPTLBO	2.199 9E-97	4.073 2E-97	1.517 8E-96	2.772 3E-100
$f 22$	TLBO	5.901 4E-201	0	5.807 8E-200	4.946 7E-207
	PSO	1.860 8E+06	1.147 9E+06	4.062 5E+06	3.991 8E+06
	IA	1.458 3E+08	3.891 2E+08	2.203 8E+08	8.836 9E+08
	GA	3.765 2E+03	3.027 1E+03	1.016 1E+04	4.084 1E+02
	SPTLBO	0	0	0	0
$f 23$	TLBO	2.187 3E-124	4.547 6E-124	2.044 6E-123	9.101 1E-127
	PSO	3.429 1E+04	1.319 9E+04	6.479 9E+04	1.054 2E+04
	IA	4.333 5E+05	7.887 6E+04	5.710 3E+05	2.766 5E+05
	GA	5.824 2E+02	8.617 2E+02	3.723 2E+03	3.864 4E+01
	SPTLBO	2.149 9E-249	0	3.229 2E-248	6.968 4E-256
$f 24$	TLBO	1.998 7E-01	4.723 4E-07	1.998 8E-01	1.998 7E-01
	PSO	2.094 9E-00	1.234 4E-01	2.399 9E-00	1.899 9E-00
	IA	9.486 7E-08	9.731 1E-08	3.177 6E-07	1.193 2E-09
	GA	2.609 9E+00	3.582 1E-01	3.299 9E+00	1.999 9E+00
	SPTLBO	8.795 7E-02	3.003 2E-02	9.987 3E-02	0
$f 25$	TLBO	8.410 8E-00	3.761 4E+01	1.682 1E+02	0
	PSO	1.000 5E+03	7.620 1E+03	1.147 3E+03	8.488 1E+03
	IA	2.220 7E-10	3.770 1E-10	1.457 1E-09	1.659 1E-12
	GA	4.110 5E+02	4.757 3E+01	5.280 3E+02	3.345 0E+02
	SPTLBO	0	0	0	0

图2 低维收敛曲线对比

Fig.2 Comparison of low-dimensional convergence curves

图3 高维收敛曲线对比

Fig.3 Comparison of high-dimensional convergence curves

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融入社会心理学理论的教与学优化算法

Teaching-Learning-Based Optimization Algorithm with Social Psychology Theory

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