A Co-evolutionary Teaching-learning-based Optimization Algorithm for Constrained Optimization Problems
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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.Abstract: In order to avoid the inconvenience of penalty factors and the tolerance amount during transforming equality constraints into inequality constraints, the self-learning process is combined with teaching-learning-based algorithm, and a co-evolutionary teaching-learning-based algorithm is thus proposed, which makes the penalty factors and tolerance amounts dynamically adjust along with the population evolution. Numerical experiments on seven common test functions verify the effectiveness of the algorithm to solve optimization problems with equality and inequality constraints.1) 本文责任编委 王占山
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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 
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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
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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
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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
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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
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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
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