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求解约束优化问题的协同进化教与学优化算法

刘三阳 靳安钊

刘三阳, 靳安钊. 求解约束优化问题的协同进化教与学优化算法. 自动化学报, 2018, 44(9): 1690-1697. doi: 10.16383/j.aas.2017.c170076
引用本文: 刘三阳, 靳安钊. 求解约束优化问题的协同进化教与学优化算法. 自动化学报, 2018, 44(9): 1690-1697. doi: 10.16383/j.aas.2017.c170076
LIU San-Yang, JIN An-Zhao. A Co-evolutionary Teaching-learning-based Optimization Algorithm for Constrained Optimization Problems. ACTA AUTOMATICA SINICA, 2018, 44(9): 1690-1697. doi: 10.16383/j.aas.2017.c170076
Citation: LIU San-Yang, JIN An-Zhao. A Co-evolutionary Teaching-learning-based Optimization Algorithm for Constrained Optimization Problems. ACTA AUTOMATICA SINICA, 2018, 44(9): 1690-1697. doi: 10.16383/j.aas.2017.c170076

求解约束优化问题的协同进化教与学优化算法

doi: 10.16383/j.aas.2017.c170076
基金项目: 

国家自然科学基金 61373174

详细信息
    作者简介:

    刘三阳 西安电子科技大学教授.主要研究方向为应用数学, 最优化, 运筹学.E-mail:liusanyang@126.com

    通讯作者:

    靳安钊 西安电子科技大学硕士研究生.主要研究方向为最优化算法与应用.本文通信作者.E-mail:jinanzhao@stu.xidian.edu.cn

A Co-evolutionary Teaching-learning-based Optimization Algorithm for Constrained Optimization Problems

Funds: 

National Natural Science Foundation of China 61373174

More Information
    Author Bio:

    Professor at Xidian University. His research interest covers applied mathematics, especially optimization, and operations research

    Corresponding author: JIN An-Zhao Master student at Xidian University. His research interest covers optimization algorithms and their application. Corresponding author of this paper
  • 摘要: 对约束优化问题,为了避免罚因子和等式约束转化为不等式约束时引入的约束容忍度参数所带来的不便,本文在基本教与学优化(Teaching-learning-based optimization,TLBO)算法中加入了自我学习过程并提出了一种求解约束优化问题的协同进化教与学优化算法,使得罚因子和约束容忍度随种群的进化动态调整.对7个常见测试函数的数值实验验证了算法求解带有等式和不等式约束优化问题的有效性.
    1)  本文责任编委 王占山
  • 图  1  算法模型

    Fig.  1  Algorithm model

    图  2  算法流程

    Fig.  2  Flow chart of CTLBO

    图  3  焊接梁模型图

    Fig.  3  Model of welded beam

    图  4  压力容器模型

    Fig.  4  Model of pressure vessel

    图  5  最小化张力弦模型

    Fig.  5  Model of tension string

    表  1  不同方法求得问题1最优解的比较

    Table  1  Comparison of the best solution for Example 1 found by different methods

    变量 TLBO[12] CDE[18] UABC[24] ITLBO[25] CTLBO
    $x_1(h)$ 0.205730 0.203137 0.205730 0.205730 0.205730
    $x_2(l)$ 3.470489 3.542998 3.470489 3.470489 3.470489
    $x_3(t)$ 9.036624 9.033498 9.036624 9.036624 9.036626
    $x_4(b)$ 0.205730 0.206179 0.205730 0.205730 0.205730
    $g_1(x)$ -0.000001 -44.578568 -0.000028 -0.000000 -0.000002
    $g_2(x)$ -0.000001 -44.663534 -0.000025 -0.000000 -0.009189
    $g_3(x)$ -0.000000 -0.003042 -0.000000 -0.000000 -0.000000
    $g_4(x)$ -3.432984 -3.423726 -3.432984 -3.432984 -3.432984
    $g_5(x)$ -0.080730 -0.078137 -0.080730 -0.080730 -0.080730
    $g_6(x)$ -0.235540 -0.235557 -0.235540 -0.235540 -0.235540
    $g_7(x)$ -0.000000 -38.028268 -0.000050 -0.000000 -0.000004
    $f(x)$ 1.724852 1.733462 1.724852 1.724852 1.724852
    下载: 导出CSV

    表  2  不同方法求得问题1结果统计表

    Table  2  Statistical results of different methods for Example 1

    方法 最优解 均值 最差解 标准差
    TLBO[12] 1.7248523 1.7248525 1.7248569 8.53E-07
    CDE[8] 1.733461 1.768158 1.824105 2.2E-02
    UABC[24] 1.724852 1.724853 NA 1.70E-06
    ITLBO[25] 1.7248523 1.7248523 1.7248523 6.77E-16
    COMDE[26] 1.7248523 1.7248523 1.7248523 1.60E-12
    CTLBO 1.724852 1.724852 1.724853 0.00E-00
    下载: 导出CSV

    表  3  不同方法求得问题2最优解的比较

    Table  3  Comparison of the best solution for Example 2 found by different methods

    变量 TLBO[12] CDE[18] UABC[24] ITLBO[25] CTLBO
    $x_1(T_s)$ 0.8125 0.812500 0.8125 0.8125 0.81250
    $x_2(T_h)$ 0.4375 0.437500 0.4375 0.4375 0.437500
    $x_3(R)$ 42.098446 42.09841 42.098446 42.098446 42.098400
    $x_4(L)$ 176.636596 176.63769 176.636596 176.636596 176.636596
    $g_1(x)$ -0.000000 -6.677E-7 -0.000000 -0.000000 -0.000000
    $g_2(x)$ -0.035881 -0.035881 -0.035881 -0.035881 -0.035880
    $g_3(x)$ -0.000000 -3.6831 -0.000000 -0.000000 -7.0E-10
    $g_4(x)$ -63.363404 -63.3623 -63.363404 -63.363404 -63.363404
    $f(x)$ 6 059.714335 6 059.7340 6 059.714335 6 059.714335 6 059.714335
    下载: 导出CSV

    表  4  不同方法求得问题2结果统计表

    Table  4  Statistical results of different methods for Example 2

    方法 最优解 均值 最差解 标准差
    TLBO[12] 6 059.714335 6 059.714335 6 059.714335 1.85E-12
    CDE[18] 6 059.7340 6 085.2303 6 371.0455 4.3E+02
    UABC[24] 6 059.714335 6 192.116211 NA 2.04E+02
    ITLBO[25] 6 059.714335 6 059.714335 6 059.714335 1.85E-12
    COMDE[26] 6 059.714335 6 059.714335 6 059.714335 3.62E-10
    CTLBO 6 059.714335 6 059.714335 6 059.714335 0.00E-00
    下载: 导出CSV

    表  5  不同方法求得问题3最优解的比较

    Table  5  Comparison of the best solution for Example 3 found by different methods

    变量 TLBO[12] ETLBO[13] FETLBO[15] UABC[24] ITLBO[25] CTLBO
    $x_1(d)$ 0.051506 0.051565 0.051691 0.051691 0.051698 0.051664
    $x_2(D)$ 0.35327 0.353713 0.356758 0.356769 0.356723 0.356112
    $x_3(P)$ 11.555900 11.468954 11.286578 11.285988 11.288662 11.313513
    $g_1(x)$ -0.000388 -0.001029 0.000953 -0.000000 -0.000000 -1.50E-08
    $g_2(x)$ -0.000020 -0.000062 -0.000014 -0.000000 -0.000000 -5.60E-08
    $g_3(x)$ -4.042961 -4.047205 -4.053903 -4.053886 -4.053796 -4.057519
    $g_4(x)$ -0.730778 -0.729815 -0.727701 -0.727694 -0.727725 -0.728149
    $f(x)$ 0.012671 0.0126674 0.0126652 0.012665 0.012665 0.0126547
    下载: 导出CSV

    表  6  不同方法求得问题3结果统计表

    Table  6  Statistical results of different methods for Example 3

    方法 最优解 均值 最差解 标准差
    TLBO[12] 0.0126717 0.0127407 0.0127977 2.88E-05
    ETLBO[13] 0.0126674 NA NA NA
    FETLBO[15] 0.0126652 NA NA NA
    CDE[18] 0.0126702 0.012703 0.012790 2.7E-05
    UABC[24] 0.012665 0.012683 NA 3.31E-05
    ITLBO[25] 0.0126652 0.0126662 0.0126735 2.12E-06
    CTLBO 0.0126547 0.01265493 0.01265693 5.50E-07
    下载: 导出CSV

    表  7  不同方法求得问题$g05, g06, g11$ 和$g13$ 最优解的比较

    Table  7  Comparison of the best solution for example $g05, g06, g11$ and $g13$ found by different methods

    函数/最优值 统计项 方法
    TLBO[12] ETLBO[13] HTS[27] Jaya[28] CTLBO
    $g05$ /5 126.498 最好解 5 126.486 5 126.484 5 126.486 5 126.486 5 126.517
    均值 5 126.6184 5 168.7149 5 126.6831 5 126.635 5 126.605
    最差解 5 127.714 5 261.826 5 126.5152 5 126.5061 5 126.759
    $g06/-$ 6 961.814 最好解 -6 961.814 -6 961.814 -6 961.814 -6 961.814 -6 961.814
    均值 -6 961.814 -6 961.814 -6 961.814 -6 961.814 -6 961.814
    最差解 -6 961.814 -6 961.814 -6 961.814 -6 961.814 -6 961.814
    $g11$ /0.750 最好解 0.7499 0.750 0.7499 0.7499 0.750
    均值 0.7499 0.750 0.7499 0.7499 0.750
    最差解 0.7499 0.750 0.7499 0.7499 0.750
    $g13$ /0.0539498 最好解 0.44015 0.13314 0.37319 0.003625 0.053991
    均值 0.69055 0.83851 0.79751 0.003631 0.058190
    最差解 0.95605 0.99979 0.66948 0.003627 0.069077
    下载: 导出CSV
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  • 收稿日期:  2017-02-14
  • 录用日期:  2017-06-22
  • 刊出日期:  2018-09-20

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