﻿ 基于非线性收敛因子和标杆管理的改进教与学优化算法
 上海理工大学学报  2022, Vol. 44 Issue (5): 508-518 PDF

Modified teaching-learning-based optimization algorithm based on the nonlinear convergence factor and benchmarking management
CHEN Xuefen, YE Chunming
Business School, University of Shanghai for Science and Technology, Shanghai 200093, China
Abstract: A modified teaching-learning-based optimization algorithm (MTLBO) was proposed to solve the shortcomings of standard teaching-learning-based optimization algorithm (TLBO), such as low optimization accuracy, slow convergence speed and weak avoidance of local optimization. In the teaching and learning stages of the TLBO, the nonlinear convergence factor adjustment and benchmarking management strategies were introduced respectively. Based on the random combination of the two strategies, three different MTLBOs were constructed. Subsequently, the experimental results show that the three MTLBOs are better than the TLBO. Among them, the MTLBO3 with the two modified strategies achieves the best numerical results, which is much better than the original TLBO. In order to further verify the effectiveness of the proposed algorithm, numerical experiments are carried out with other well-known swarm intelligence optimization algorithms. Numerical results and convergence curves show that the optimization performance of MTLBO3 is significantly better than other comparison methods, with higher solution accuracy, faster convergence speed and better local optimization avoidance ability. Finally, the effectiveness of the proposed algorithm is further verified using constrained engineering optimization problems.
Key words: swarm intelligence optimization algorithm     teaching-learning-based optimization algorithm     convergence factor     benchmarking management

2种改进策略分别是非线性收敛因子调整策略以及标杆管理策略。对于收敛因子，在标准TLBO中，收敛因子是随机取值为1或2，这种策略显然不符合实际，因为，随着教学经验的累积，教师的教学水平也应该得以提升。从算法角度来看，为加快收敛，算法的收敛因子应该随种群所获得的信息量进行动态调整，而不是笼统地设为1或2。此外，标杆管理是一种先进的管理学思想，针对学习阶段随机学习所导致的“无效学习”等问题，在班级中推选精英组，进而建立班级学习标杆，可以更好地帮助班级成员学习到更多“有用”的知识。从算法角度来看，标杆管理策略进一步主导了种群朝着正确的方向进化，从而加快了算法的收敛速度，使得算法有更多的精力致力于全局最优解的局部开发。为验证改进策略的有效性，基于2种策略的部署组合提出了3种改进教与学优化算法（modified teaching-learning-based optimization algorithm，MTLBO）的变体，并通过11个基准测试函数进行了实验对比。随后，为进一步验证提出算法的性能，使用性能最优的MTLBO与其他著名的优化算法进行了实验验证与分析。

1 教与学优化算法

TLBO的灵感来源主要是课堂上师生之间的教与学行为，其设计思想主要是基于一个典型的优化系统（即教学活动）的构建。在该系统中，TLBO将一个自然班级视为动态演化的种群，老师和学生都可作为种群中的搜索代理，他们的知识水平或考试成绩作为群智能优化算法的适应值。

 $f\left(x\right)=\frac{1}{\sigma \sqrt{2\text{π} }}{{\rm{e}}}^{-\frac{{(x-\mu )}^{2}}{2{\sigma }^{2}}}$ (1)

 图 1 不同教师的教学效果分布图 Fig. 1 Distribution of grades obtained by students taught by two different teachers

 图 2 教学活动前后班级学生成绩分布图 Fig. 2 Distribution of grades obtained by students taught by a teacher

 ${\boldsymbol{P}}=\left[\begin{array}{c}{{\boldsymbol{X}}}_{1}\\ {{\boldsymbol{X}}}_{2}\\ {{\boldsymbol{X}}}_{3}\\ \vdots \\ {{\boldsymbol{X}}}_{N}\end{array}\right]=\left[\begin{array}{c}{x}_{\mathrm{1,1}}\quad{x}_{\mathrm{1,2}}\quad\dots \quad{x}_{1,n}\\ {x}_{\mathrm{2,1}}\quad{x}_{\mathrm{2,2}}\quad\dots \quad{x}_{2,n}\\ {x}_{\mathrm{3,1}}\quad{x}_{\mathrm{3,2}}\quad\dots\quad {x}_{3,n}\\ \vdots\;\qquad\vdots\quad\quad\quad\;\; \quad\vdots \\ {x}_{N,1}\quad{x}_{N,2}\quad\dots\quad {x}_{N,n}\end{array}\right]$ (2)

1.1 教学阶段

 ${\boldsymbol{D}}={r}_{k} ({{\boldsymbol{T}}}_{{{k}}}-{{{T}}}_{{\rm{F}}}{{\boldsymbol{M}}}_{k})$ (3)
 ${T}_{{\rm{F}}}=\left\{\begin{array}{l}1,\quad r\leqslant 0.5\\ 2,\quad r > 0.5\end{array}\right.$ (4)

 ${{\boldsymbol{X}}}_{{\rm{new}},i}^{k}={{\boldsymbol{X}}}_{i}^{k}+{\boldsymbol{D}}$ (5)

1.2 学习阶段

 ${{\boldsymbol{X}}}_{i}^{k+1}={{{\boldsymbol{X}}}_{i}^{k}}^{\mathrm{{'}}}+{r}_{i}({{{\boldsymbol{X}}}_{i}^{k}}^{\mathrm{{'}}}-{{{{\boldsymbol{X}}}_{j}}^{k}}^{\mathrm{{'}}})$ (6)
 ${{\boldsymbol{X}}}_{i}^{k+1}={{{\boldsymbol{X}}}_{i}^{k}}^{\mathrm{{'}}}+{r}_{i}({{{\boldsymbol{X}}}_{j}^{k}}^{\mathrm{{'}}}-{{{\boldsymbol{X}}}_{i}^{k}}^{\mathrm{{'}}})$ (7)

2 改进的教与学优化算法 2.1 非线性收敛因子

 ${{{T}}}_{{\rm{F}}}=2+{(k/T)}^{1/2}$ (8)

2.2 标杆管理策略

 ${{\boldsymbol{G}}}_{l}=[{{\boldsymbol{X}}}_{{\rm{s}}},{{\boldsymbol{X}}}_{{\rm{c}}}]$ (9)

 ${{\boldsymbol{X}}}_{i}^{k+1}={{{\boldsymbol{X}}}_{i}^{k}}^{\mathrm{{'}}}+{r}_{i}({{\boldsymbol{X}}}_{{\rm{e}}}-{{{\boldsymbol{X}}}_{i}^{k}}^{\mathrm{{'}}})$ (10)

2.3 MTLBO算法

a. 参数初始化。设定种群规模N，总迭代次数T

b. 待解问题参数空间范围内进行班级种群初始化，并计算每个个体的适应值。

c. $t = 1$ ，迭代开始。

d. 教学阶段开始，计算 ${{{\boldsymbol{M}}}}_{{k}}$ ，依据式(8)计算 ${{{{T}}}}_{{{\rm{F}}}}$

e. 计算D，班级个体依据D更新位置。

f. 更新班级所有个体适应值，教学阶段结束。

g. 学习阶段开始，推选标杆个体Xe

h. 班级个体以Xe为榜样根据式(10)更新位置。

i. 更新班级所有个体适应值，学习阶段结束。

j. 判断是否满足循环终止条件，若是，返回全局最佳解；否则，令 $t = t + 1$ ，返回步骤d。

3 实验与分析

3.1 MTLBOs与标准TLBO的比较

3.2 与其他群智能优化算法的比较

HS，PSO，GA，MFO，TLBO以及MTLBO3这6种算法在11个测试函数上30次独立运行的总时间列于表8。由表8可知，TLBO和MTLBO的运行时间明显优于其他算法。具体地，TLBO在函数F1，F2，F3，F6以及F8上取得最短的运行时间，而MTLBO3则在函数F4，F5，F7，F9，F10以及F11具有最高的计算效率。总体上讲，MTLBO3和TLBO计算效率差异不大，MTLBO3略胜一筹，同时前者在计算精度上优于后者。

 图 3 6种比较算法在部分测试函数上的收敛曲线 Fig. 3 Convergence curves of 6 comparison algorithms on some test functions
3.3 约束优化问题

 图 4 拉伸/压缩弹簧设计优化问题 Fig. 4 Tension/compression spring design problem

 $x = [{x_1},{x_2},{x_3}] = [d,D',N']$

 $f(x) = ({x_3} + 2){x_2}x_1^2$

 ${g_1}(x) = 1 - \dfrac{{x_2^3{x_3}}}{{71\;785x_1^4}} \leqslant 0$
 ${g_2}(x) = \frac{{4x_2^2 - {x_1}{x_2}}}{{12\;566({x_2}x_1^3 - x_1^4)}} + \frac{1}{{5\;108x_1^2}} - 1 \leqslant 0$
 ${g_{\text{3}}}(x) = {\text{1}} - \frac{{140.45{x_1}}}{{x_2^2{x_3}}} \leqslant 0$
 ${g_{\text{4}}}(x) = \frac{{{x_1} + {x_2}}}{{1.5}} - 1 \leqslant 0$

 $0.05 \leqslant {x_1} \leqslant 2.00,\;\;0.25 \leqslant {x_2} \leqslant 1.30,\;\;2.00 \leqslant {x_3} \leqslant 15.0$

4 结束语

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