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一类新型动态多目标鲁棒进化优化方法

陈美蓉 郭一楠 巩敦卫 杨振

陈美蓉, 郭一楠, 巩敦卫, 杨振. 一类新型动态多目标鲁棒进化优化方法. 自动化学报, 2017, 43(11): 2014-2032. doi: 10.16383/j.aas.2017.c160300
引用本文: 陈美蓉, 郭一楠, 巩敦卫, 杨振. 一类新型动态多目标鲁棒进化优化方法. 自动化学报, 2017, 43(11): 2014-2032. doi: 10.16383/j.aas.2017.c160300
CHEN Mei-Rong, GUO Yi-Nan, GONG Dun-Wei, YANG Zhen. A Novel Dynamic Multi-objective Robust Evolutionary Optimization Method. ACTA AUTOMATICA SINICA, 2017, 43(11): 2014-2032. doi: 10.16383/j.aas.2017.c160300
Citation: CHEN Mei-Rong, GUO Yi-Nan, GONG Dun-Wei, YANG Zhen. A Novel Dynamic Multi-objective Robust Evolutionary Optimization Method. ACTA AUTOMATICA SINICA, 2017, 43(11): 2014-2032. doi: 10.16383/j.aas.2017.c160300

一类新型动态多目标鲁棒进化优化方法

doi: 10.16383/j.aas.2017.c160300
基金项目: 

国家重点基础研究发展计划(973计划) 2014CB046300

国家自然科学基金 61573361

中国矿业大学创新团队 2015QN003

详细信息
    作者简介:

    陈美蓉  中国矿业大学博士研究生.2006年获得中国矿业大学理学院硕士学位.主要研究方向为进化计算.E-mail:cmrzl@126.com

    巩敦卫 中国矿业大学信息与电气工程学院教授.1999年在中国矿业大学获博士学位.主要研究方向为进化计算与应用.E-mail:dwgong@vip.163.com

    杨振 中国矿业大学硕士研究生.主要研究方向为多目标进化优化.E-mail:yangzhen.cumt@foxmail.com

    通讯作者:

    郭一楠 中国矿业大学信息与电气工程学院教授.主要研究方向为智能优化算法与控制, 数据挖掘与知识发现.本文通信作者.E-mail:nanfly@126.com

A Novel Dynamic Multi-objective Robust Evolutionary Optimization Method

Funds: 

National Basic Research Program of China (973 Program) 2014CB046300

National Natural Science Foundation of China 61573361

Innovation Team of China University of Mining and Technology 2015QN003

More Information
    Author Bio:

    Ph. D. candidate at China University of Mining and Technology. She received her master degree from China University of Mining and Technology in 2006. Her main research interest is evolutionary computation

    Professor at the School of Information and Electronic Engineering, China University of Mining and Technology. He received his Ph. D. degree from China University of Mining and Technology in 1999. His research interest covers evolutionary computation and its applications

    Master student at China University of Mining and Technology. His main research interest is multi-objective evolutionary optimization

    Corresponding author: GUO Yi-Nan Professor at the School of Information and Electronic Engineering, China University of Mining and Technology. Her research interest covers intelligence optimization and control, data mining, and knowledge discovery. Corresponding author of this paper
  • 摘要: 传统动态多目标优化问题(Dynamic multi-objective optimization problems,DMOPs)的求解方法,通常需要在新环境下,通过重新激发寻优过程,获得适应该环境的Pareto最优解.这可能导致较高的计算代价和资源成本,甚至无法在有限时间内执行该优化解.由此,提出一类寻找动态鲁棒Pareto最优解集的进化优化方法.动态鲁棒Pareto解集是指某一时刻下的Pareto较优解可以以一定稳定性阈值,逼近未来多个连续动态环境下的真实前沿,从而直接作为这些环境下的Pareto解集,以减小计算代价.为合理度量Pareto解的环境适应性,给出了时间鲁棒性和性能鲁棒性定义,并将其转化为两类鲁棒优化模型.引入基于分解的多目标进化优化方法和无惩罚约束处理方法,构建了动态多目标分解鲁棒进化优化方法.特别是基于移动平均预测模型实现了未来动态环境下适应值的多维时间序列预测.基于提出的两类新型性能评价测度,针对8个典型动态测试函数的仿真实验,结果表明该方法得到满足决策者精度要求,且具有较长平均生存时间的动态鲁棒Pareto最优解.
  • 图  1  函数FDA1在三个时刻的真实Pareto解与前沿的比较

    Fig.  1  True Pareto solutions and Pareto fronts of FDA1 in three environments

    图  2  函数FDA3在三个时刻的真实Pareto解与前沿的比较

    Fig.  2  True Pareto solutions and Pareto fronts of FDA3 in three environments

    图  3  函数的平均适应度值与时刻$k=1, 2, 3$的真实Pareto前沿的比较

    Fig.  3  Average fitness and true PF($k$) with $k=1, 2, 3$

    图  4  函数的平均适应度值与鲁棒解RPS在$t=1, 2, 3$时刻的鲁棒Pareto前沿的比较

    Fig.  4  Average fitness and robust Pareto fronts with $k=1, 2, 3$

    图  5  不同变化强度$n_{ d} =5, 10, 20$下方法2和方法3的平均生存时间比较($\eta=0.4$, $T=2$, $\tau _{ d} =30$)

    Fig.  5  Average survival time of methods 2 and 3 in different change severity $n_{ d} =5, 10, 20$ ($\eta=0.4$, $T=2$, $\tau _{ d} =30$)

    图  6  不同变化频率$\tau _{ d}=10, 20, 30$下方法2和方法3的平均生存时间比较($\eta =0.4$, $T=2$, $n_{ d} =10$)

    Fig.  6  Average survival time of methods 2 and 3 in different change frequency $\tau _{ d}=10, 20, 30$ ($\eta =0.4$, $T=2$, $n_{ d} =10$)

    图  7  100个动态环境下三种方法的鲁棒生存时间比较$\eta=0.4$, $T=2$, $n_{ d} =10$, $\tau _{ d} =30$

    Fig.  7  Robust survival time of three methods in 100 environments ($\eta =0.4$, $T=2$, $n_{ d} =10$, $\tau _{ d} =30$)

    图  8  3种方法获得鲁棒Pareto解集的生存时间百分比比较($\eta =0.4$, $T=2$, $n_{ d}=10$, $\tau _{ d} =30$)

    Fig.  8  Robust survival time percentage of three methods ($\eta =0.4$, $T=2$, $n_{ d}=10$, $\tau _{ d} =30$)

    图  9  8个测试函数的前四个鲁棒Pareto解与其所适应的动态环境下的真实Pareto解的比较图\\ ($\eta=0.4$, $T=2$, $n_d =10$, $\tau _d =100$)

    Fig.  9  The first four robust Pareto fronts and true Pareto fronts in their adaptive environments ($\eta=0.4$, $T=2$, $n_d =10$, $\tau _d =100$)

    表  1  RPS(1)与相邻$k=2, 3$环境下解的相对适应度差值

    Table  1  The relative fitness errors of RPS(1) and adjacent two environments

    相对适应度差 $\Delta (1)$ $\Delta (2)$
    FDA1 0.301 1.2177
    FDA3 0.6728 1.7687
    下载: 导出CSV

    表  2  测试函数

    Table  2  Benchmark functions

    函数 测试函数 变量维数 变量范围 说明
    Fun 1 $ f_1 (X_{Ⅰ})=x_1, \ f_2 =g\times h$
    $g(X_{Ⅱ})=1+\sum_{x_i \in X_{Ⅱ}} {(x_i -G(k))^2}$
    $ h(f_1, g)=1-\sqrt {{{f_1 }/ g}}, \ G(k)=\sin (0.5\pi k)$
    $ N=10$
    $| X_{Ⅰ} | =1$
    $| X_{Ⅱ} | $=9
    $ X_{Ⅰ} =(x_1)\in [0, 1]$
    $X_{Ⅱ} =(x_2, {\cdots}, x_N)\in [-1, 1]$
    FDA1 in [6]
    PS随环境变化
    PF不随环境变化
    Fun 2 $f_1 (X_{Ⅰ})=x_1, f_2 =g\times h$
    $g(X_{Ⅱ})=1+\sum_{x_i \in X_{Ⅱ} } {x_i^2 }$
    $h(f_1, g)=1-\left({{{f_1 }/g}} \right)^{H(k)^{-1}}$
    $H(k)=0.75+0.7\sin (0.5\pi k)$
    $ N=10$
    $| X_{Ⅰ} | =1$
    $| X_{Ⅱ} | =9$
    $X_\rm{1} =(x_1)\in [0, 1],$
    $X_\rm{2} \in [-1, 1]$
    FDA2 in [6]
    PS不随环境变化
    PF随环境变化
    Fun 3 $ f_1 (X_{Ⅰ})=\sum_{x_i \in X_\rm{1} }^n {x_i^{F(k)} }, \quad f_2 =g\times h$
    $g(X_{Ⅱ})=1+G(k)+\sum_{x_i \in X_{Ⅱ}}{(x_i -G(k))^2}$
    $h(f_1, g)=1-\sqrt {{{f_1 }/g}}, \quad G(k)=\left| {\sin (0.5\pi k)} \right|$
    $F(k)=10^{2\sin (0.5\pi k)}$
    $ N=10$
    $| X_{Ⅰ} | =1$
    $| X_{Ⅱ} |$ =9
    $X_{Ⅰ}=(x_1)\in [0, 1]$
    $X_{Ⅱ} =(x_2, {\cdots}, x_N)\in [-1, 1]$
    FDA3 in [6]
    PS随环境变化
    PF随环境变化
    Fun 4 $ f_1 (x)=(1+g)\cos (0.5\pi x_1)\cos (0.5\pi x_2)$
    $f_2 (x)=(1+g)\cos (0.5\pi x_1)\sin (0.5\pi x_2)$
    $f_3 (x)=(1+g)\sin (0.5\pi x_1)$
    $g(x)=\sum_{i=3}^n {(x_i -G(k))^2}$
    $G(k)=\left| {\sin (0.5\pi t)} \right|$
    $N=12$ $ x_i \in [0, 1]$
    $\forall\, i=1, 2, {\cdots}, N $
    FDA4 in [6]
    PS随环境变化
    PF不随环境变化
    Fun 5 $ f_1 (x)=(1+g)\cos (0.5\pi y_1)\cos (0.5\pi y_2)$
    $f_2 (x)=(1+g)\cos (0.5\pi y_1)\sin (0.5\pi y_2)$
    $f_3 (x)=(1+g)\sin (0.5\pi y_1)$
    $g(x)=G(k)+\sum_{i=3}^n {(x_i -G(k))^2}$
    $y_i =x_i^{F(k)}$
    $G(k)=\left| {\sin (0.5\pi k)} \right|$
    $F(k)=1+100\sin ^4(0.5\pi k)$
    $N=12$ $ x_i \in [0, 1]$
    $\forall\, i=1, 2, {\cdots}, N $
    FDA5 in [6]
    PS随环境变化
    PF随环境变化
    Fun 6 $ f_1 (x_1)=x_1, f_2 =g\times h$
    $g(x_2, {\cdots}, x_n, k)=1+9\sum_{i=2}^n {x_i^2 }$
    $h(f_1, g)=1-\left({{{f_1 } /g}} \right)^{H(k)}$
    $H(k)=0.75\sin (0.5\pi k)$+1.25
    $N=10$ $ x_i \in [0, 1]$
    $\forall\, i=1, 2, {\cdots}, N $
    DMOP1 in [8]
    PS不随环境变化
    PF随环境变化
    Fun 7 $ f_1 (x_1)=x_1, f_2 =g\times h$
    $g(x_2, {\cdots}, x_n, k)=1+\sum_{i=2}^n {(x_i -G(k))^2}$
    $h(f_1, g)=1-\left({{{f_1 }/g}} \right)^{H(k)}$
    $H(k)=0.75\sin (0.5\pi k)$+1.25
    $N=10$ $ x_i \in [0, 1]$
    $\forall\, i=1, 2, {\cdots}, N $
    DMOP2 in [8]
    PS随环境变化
    PF随环境变化
    Fun 8 $f_1 (x_1)=x_1, f_2 =g\times h$
    $g(x_2, {\cdots}, x_n, k)=1+9\sum_{i=2}^n {(x_i -G(k))^2}$
    $h(f_1, g)=1-\sqrt {{{f_1 } / g}}$
    $H(t)=\sin (0.5\pi k)$
    $N=10$ $ x_i \in [0, 1]$
    $\forall\, i=1, 2, {\cdots}, N $
    DMOP3 in [8]
    PS随环境变化
    PF不随环境变化
    下载: 导出CSV

    表  3  不同稳定性阈值下两种模型的平均生存时间比较($T=2$, $n_{ d} =10$, $\tau_{ d} =30$)

    Table  3  Average survival time of methods 2 and 3 in different stability thresholds ($T=2$, $n_{ d} =10$, $\tau_{ d} =30$)

    方法 参数 函数1 函数2 函数3 函数4 函数5 函数6 函数7 函数8
    方法2 $\eta =0.2$ 2.4480±0.0314 7.2233±0.0082 2.3433±0.0342 2.6867±0.0049 1.6120±0.0041 9.9680±0.0041 2.2813±0.0106 1.2407±0.0110
    $\eta=0.4$ 3.4640±0.0099 21.658±0.0086 3.9428±0.0208 4.2027±0.0096 2.4093±0.0026 30.197±0.5440 3.6000±9.1E-16 1.7000±7.3E-15
    $\eta=0.6$ ${\bf 4.5720\pm0.0077}$ ${\bf 49.51\pm0} $ $ {\bf 5.9933\pm0.0049} $ ${\bf 6.1293\pm0.0080}$ $ {\bf 4.3727\pm0.0059} $ $ {\bf 49.510\pm0} $ $ {\bf 4.8247\pm0.0210} $ $ {\bf 2.2867\pm0.0105}$
    方法3 $\eta=0.2$ 2.5067±0.0315 7.4120±0.2402 2.5500±0.0746 2.7800±9.1E-16 1.7200±2.2E-16 9.9453±0.0074 2.3347±0.0106 1.2600±2.2E-16
    $\eta=0.4$ 3.5133±0.0082 20.941±0.0083 4.3147±0.0280 4.3187±0.0106 2.5500±0.0318 30.007±0.4951 3.6520±0.0068 1.7800±4.6E-16
    $\eta=0.6$ ${\bf 4.5940\pm0.0019}$ $ {\bf 49.510\pm0} $ $ {\bf 6.1227\pm0.0046} $ $ {\bf 6.2727\pm0.0110} $ $ {\bf 4.4787\pm0.0035} $ $ {\bf 49.510\pm0} $ $ {\bf 4.8607\pm0.0026} $ $ {\bf 2.4033\pm0.0082}$
    下载: 导出CSV

    表  4  不同时间窗下两种模型的平均生存时间比较($\eta =0.4$, $n_{ d} =10$, $\tau_{ d} =30$)

    Table  4  Average survival time of methods 2 and 3 in different time windows ($\eta =0.4$, $n_{ d} =10$, $\tau_{ d} =30$)

    方法 参数 函数1 函数2 函数3 函数4 函数5 函数6 函数7 函数8
    方法2 $T=2 $ $ {\bf 3.4640\pm0.0099} $ $ {\bf 21.658\pm0.0086} $ 3.9428±0.0208 $ {\bf 4.2027\pm0.0096}$ 2.4093±0.0026 $ {\bf 30.197\pm0.5440} $ $ {\bf 3.6000\pm9.1}$E-${\bf 16}$ $ {\bf 1.7000\pm7.3}$E-${\bf 15}$
    $T=4 $ 3.1460±0.0051 21.606±0.1788 3.6447±0.0168 3.9440±0.0203 $ {\bf 2.4500\pm0.0038} $ 29.448±0.7324 3.1707±0.0406 1.5220±0.0056
    $T=6 $ $ 3.1180\pm0.0115 $ $ 21.650\pm3.6$E-15 $ {\bf 4.0373\pm0.0059} $ $ 3.9007\pm0.0167$ $ {{\bf 2.4500\pm0}} $ 29.085±0.6836 3.2027±0.0448 1.5220±0.0056
    方法3 $T=2 $ $ {\bf 3.5133\pm0.0082} $ $ {\bf 20.941\pm0.0083} $ $ {\bf 4.3147\pm0.0280} $ $ {\bf 4.3187\pm0.0106}$ 2.5500±0.0318 $ {\bf 30.007\pm0.4951} $ $ {\bf 3.6520\pm0.0068} $ $ {\bf 1.7800\pm4.6}$E$-{\bf 16}$
    $T=4 $ 3.4033±0.0082 20.900±0 3.9533±0.0082 4.1893±0.0110 2.7100±4.5E-16 28.068±0.2995 3.3927±0.0546 1.5600±2.2E-16
    $T=6 $ 3.4120±0.0108 20.644±0.3445 4.1467±0.0374 4.2313±0.0125 $ {\bf 2.8813\pm0.0376} $ 28.340±0.4003 3.4820±0.0441 1.5940±0.0356
    下载: 导出CSV

    表  5  不同时间窗下两种模型的RIGD测度比较($\eta =0.4$, $n_{ d} =10$, $\tau_{ d} =30$)

    Table  5  RIGD of methods 2 and 3 in different time windows ($\eta =0.4$, $n_{ d} =10$, $\tau_{ d} =30$)

    方法 参数 函数1 函数2 函数3 函数4 函数5 函数6 函数7 函数8
    方法2 $T=2$ $ 0.2267\pm0.0092 $ $ {\bf 0.0816\pm0.0555} $ $ {\bf 0.3815\pm0.0230} $ $ 0.4009\pm0.0178$ $ {\bf 0.4638\pm0.0218} $ $ 0.1009\pm0.0754 $ $ 0.3476\pm0.0132 $ $ 2.5449\pm0.1079$
    $T=4$ $ {\bf 0.1944\pm0.0081} $ $ 0.1364\pm0.0911 $ $ 0.4142\pm0.0296 $ $ 0.4009\pm0.0177$ $ 0.5424\pm0.0250 $ $ {\bf 0.0707\pm0.0323} $ $ {\bf 0.3306\pm0.0171} $ $ 2.1396\pm0.0972$
    $T=6$ $ 0.2270\pm0.0077 $ $ 0.1552\pm0.1226 $ $ 0.8548\pm0.1841 $ $ {\bf 0.4008\pm0.0200} $ $ 0.7334\pm0.0571 $ $ 0.0928\pm0.0457 $ $ 0.3476\pm0.0181 $ $ {\bf 2.0528\pm0.0880}$
    方法3 $T=2$ $ 0.2276\pm0.0063 $ $ {\bf 0.0849\pm0.0559} $ $ {\bf 0.3842\pm0.0215} $ $ 0.4520\pm0.0169$ $ {\bf 0.5907\pm0.0444} $ $ {\bf 0.0811\pm0.0543} $ $ 0.3477\pm0.0128 $ $ {\bf 2.5629\pm0.0999}$
    $T=4$ $ {\bf 0.1976\pm0.0066} $ $ 0.2238\pm0.1382 $ $ 0.5184\pm0.1738 $ $ {\bf 0.4057\pm0.0070}$ $ 0.7655\pm0.0278 $ $ 0.1168\pm0.1083 $ $ {\bf 0.3328\pm0.0089} $ $ 3.4348\pm0.1043$
    $T=6$ $ 0.5411\pm0.1200 $ $ 0.1432\pm0.0457 $ $ 1.2358\pm0.0851 $ $ 0.6615\pm0.0048 $ $ 1.2572\pm0.1641 $ $ 0.1812\pm0.0857 $ $ 0.9319\pm0.0289 $ $ 11.416\pm0.2446 $
    下载: 导出CSV

    表  6  三种方法获得鲁棒Pareto解集的平均生存时间比较($\eta =0.4$, $T=2$, $n_{ d}=10$, $\tau _{ d} =30$)

    Table  6  Average survival time of robust Pareto solutions on three methods ($\eta =0.4$, $T=2$, $n_{ d}=10$, $\tau _{ d} =30$)

    方法 函数1 函数2 函数3 函数4 函数5 函数6 函数7 函数8
    方法1 2.5800±0.0086 10.090±0.1376 2.6200±0.3880 2.7300±0.6077 2.9900±0.0560 17.940±0.8734 2.2400±0.3525 1.3500±3.6E-15
    方法2 3.4640±0.0099 $ {\bf 21.658\pm0.0086} $ 3.9428±0.0208 4.2027±0.0096 $2.4093\pm0.0026 $ $ {\bf 30.197\pm0.5440} $ 3.6000±9.1E-16 1.7000±7.3E-15
    方法3 ${\bf 4.5073\pm0.0133} $ 20.941±0.0083 $ {\bf 4.3147\pm0.0280} $ $ {\bf 4.3187\pm0.0106}$ ${\bf 2.5500\pm0.0318} $ 30.007±0.4951 $ {\bf 3.6520\pm0.0068} $ $ {\bf 1.7800\pm4.6}$E-${\bf 16}$
    下载: 导出CSV

    表  7  三种方法获得鲁棒Pareto解集的RIGD测度比较($\eta =0.4$, $T=2$, $n_{ d}=10$, $\tau _{ d} =30$)

    Table  7  RIGD of robust Pareto solutions on three methods ($\eta =0.4$, $T=2$, $n_{ d}=10$, $\tau _{ d} =30$)

    方法 函数1 函数2 函数3 函数4 函数5 函数6 函数7 函数8
    方法1 ${\bf 0.0095\pm0.0021} $ 0.2438±0.0316 $ {\bf 0.1544\pm0.0579} $ $ {\bf 0.0543\pm0.0043}$ ${\bf 0.3619\pm0.1683} $ 0.1564±1.04E-7 $ {\bf 0.1335\pm0.0743} $ $ {\bf 0.1866\pm1.40}$E-${\bf 6}$
    方法2 0.2267±0.0092 $ {\bf 0.0816\pm0.0555} $ 0.3815±0.0230 0.4009±0.0178 0.4638±0.0218 0.1009±0.0754 0.3476±0.0132 2.5449±0.1079
    方法3 0.2276±0.0063 0.0849±0.0559 0.3842±0.0215 0.4520±0.0169 0.5907±0.0444 $ {\bf 0.0811\pm0.0543} $ 0.3477±0.0128 2.5629±0.0999
    下载: 导出CSV

    表  8  三种方法求解鲁棒Pareto最优解的平均运行时间(s)比较($\eta=0.4$, $T=2$, $n_{ d} =10$, $\tau _{ d} =30$)

    Table  8  Average elapsed time of robust Pareto solutions on three methods ($\eta=0.4$, $T=2$, $n_{ d} =10$, $\tau _{ d} =30$)

    方法 函数1 函数2 函数3 函数4 函数5 函数6 函数7 函数8
    方法1 51.3681±3.4762 13.2158±4.0439 70.8508±1.2940 185.9928±10.9027 710.6224±70.2617 1.7005±0.0077 54.8422±2.6081 128.4283±2.3889
    方法2 ${\bf 46.0068\pm1.4818} $ $ {\bf 7.0358\pm4.2498} $ 59.0582±1.1731 $ {\bf 172.3075\pm7.6934}$ ${\bf 315.2275\pm8.4846} $ $ {\bf 0.8673\pm0.0140} $ $ {\bf 45.7206\pm2.3813} $ 89.7115±0.9206
    方法3 49.3816±1.8679 10.2509±3.2404 $ {\bf 55.8101\pm2.2671} $ 190.0556±19.8364 340.7134±21.4715 1.0038±0.0046 47.8325±2.0709 $ {\bf 84.1739\pm0.8742}$
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出版历程
  • 收稿日期:  2016-03-31
  • 录用日期:  2016-11-17
  • 刊出日期:  2017-11-20

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