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基于径向空间划分的昂贵多目标进化算法

顾清华 周煜丰 李学现 阮顺领

顾清华, 周煜丰, 李学现, 阮顺领. 基于径向空间划分的昂贵多目标进化算法. 自动化学报, 2021, 47(x): 1−22 doi: 10.16383/j.aas.c200791
引用本文: 顾清华, 周煜丰, 李学现, 阮顺领. 基于径向空间划分的昂贵多目标进化算法. 自动化学报, 2021, 47(x): 1−22 doi: 10.16383/j.aas.c200791
Gu Qing-Hua, Zhou Yu-Feng, Li Xue-Xian, Ruan Shun-Ling. Expensive Many-objective Evolutionary Algorithm based on Radial Space Division. Acta Automatica Sinica, 2021, 47(x): 1−22 doi: 10.16383/j.aas.c200791
Citation: Gu Qing-Hua, Zhou Yu-Feng, Li Xue-Xian, Ruan Shun-Ling. Expensive Many-objective Evolutionary Algorithm based on Radial Space Division. Acta Automatica Sinica, 2021, 47(x): 1−22 doi: 10.16383/j.aas.c200791

基于径向空间划分的昂贵多目标进化算法

doi: 10.16383/j.aas.c200791
基金项目: 国家自然科学基金(51774228, 51864046), 陕西省自然科学基金杰青项目(2020JC-44)
详细信息
    作者简介:

    顾清华:教授, 西安建筑科技大学博士生导师. 主要研究方向为多目标优化、车辆调度和复杂系统建模和仿真. 本文通信作者. Email: qinghuagu@126.com

    周煜丰:西安建筑科技大学硕士研究生. 主要研究方向为多目标优化和车辆调度. Email: zyf18215649083@163.com

    李学现:西安建筑科技大学博士研究生. 主要研究方向为群智能优化算法在采矿系统工程中的应用. Email: lixuexian2019@163.com

    阮顺领:西安建筑科技大学副教授. 主要研究方向为矿山智能系统、深度学习. Email: ruanshunling@163.com

Expensive Many-objective Evolutionary Algorithm based on Radial Space Division

Funds: Supported by National Natural Science Foundation of China (51774228, 51864046), Outstanding Youth Project of Shaanxi Natural Science Foundation Grant(2020JC-44)
  • 摘要: 为了解决难以建立精确数学模型或者真实评估实验成本高昂的多目标优化问题, 本文提出了一种基于径向空间划分的昂贵多目标进化算法. 首先算法使用高斯回归作为代理模型逼近目标函数; 然后将目标空间的个体投影到径向空间, 结合目标空间和径向空间信息保留对种群贡献更高的个体; 之后由径向空间中个体的位置分布决定下一步应该选择哪些个体进行真实评估; 最后, 采用一种双档案管理策略维护代理模型的质量. 数值实验和现实问题上的结果表明, 与五种先进的算法相比, 本文提出的算法在解决昂贵多目标优化问题时能够提供更高质量的解.
  • 图  1  高斯过程(Kriging模型)图解

    Fig.  1  Gaussian process (Kriging model) description

    图  2  径向空间中个体的位置分布情况

    Fig.  2  Individual location distribution in radial space

    图  3  六种算法在求解3个目标DTLZ1问题过程中获得的最佳HV值对应的非支配解集

    Fig.  3  The non-dominated solution set corresponding to the best HV value obtained by the six algorithms in solving the three objective DTLZ1 problems

    图  4  六种算法在10个目标DTLZ2测试问题上最佳解集的平行坐标图和径向坐标图

    Fig.  4  Parallel coordinate diagram and radial coordinate diagram of the best solution set of the six algorithms on 10 objective DTLZ2 test problems

    图  5  K-RSEA和五种对比算法在求解3个和10个目标DTZL4问题时的IGD和HV变化. (a)3个目标的IGD变化、(b)3个目标的HV变化、(c)10个目标的IGD变化、(d)10个目标的HV变化

    Fig.  5  The IGD and HV changes of K-RSEA and five comparison algorithms when solving 3 and 10 objective DTZL4 problems. (a) IGD change for 3 objectives, (b) HV change for 3 objectives, (c) IGD change for 10 objectives, (d) HV change for 10 objectives

    图  6  K-RSEA和五种对比算法在求解3个目标DTZL5问题时的IGD和HV变化

    Fig.  6  The IGD and HV changes of K-RSEA and five comparison algorithms when solving 3 objective DTZL5 problems

    图  7  K-RSEA和五种对比算法求解DTLZ7问题过程中获得的最佳HV值对应的非支配前沿

    Fig.  7  The non-dominant frontier corresponding to the best HV value obtained in the process of solving the DTLZ7 problem by K-RSEA and five comparison algorithms

    图  8  K-RSEA和五种对比算法求解WFG2问题过程中获得的最佳HV值对应的非支配前沿

    Fig.  8  The non-dominant frontier corresponding to the best HV value obtained in the process of solving the WFG2 problem with K-RSEA and five comparison algorithms

    图  9  K-RSEA和五种对比算法在求解3个目标WFG5、WFG6和WFG8问题时的IGD变化

    Fig.  9  The IGD changes of K-RSEA and five comparison algorithms when solving three objective WFG5, WFG6 and WFG8 problems

    图  10  K-RSEA和五种对比算法在求解10个目标WFG5、WFG6和WFG8问题时的HV变化

    Fig.  10  The HV changes of K-RSEA and five comparison algorithms when solving 10 objective WFG5, WFG6 and WFG8 problems

    图  11  六种算法在10个目标WFG7测试问题上最佳解集的平行坐标图和径向坐标图

    Fig.  11  Parallel coordinate diagram and radial coordinate diagram of the best solution set of the six algorithms on 10 objective WFG7 test problems

    图  12  六种算法分别在不同问题上的运行时间比较

    Fig.  12  Comparison of the running time of the six algorithms on different problems

    图  13  算法在汽车耐撞性优化问题上的非支配解

    Fig.  13  The non-dominant solution of the algorithm in the optimization of automobile crashworthiness

    表  1  测试问题及特征

    Table  1  Test problems and their features

    问题特征
    DTLZ1,3多模态的、DTLZ1线性
    DTLZ2,4−6凹的、DTLZ4有偏好、DTLZ5退化、
    DTLZ6退化且有偏好
    DTLZ7混合、不连续、多模态的
    WFG1凸的、混合有偏好的
    WFG2凸的、不连续的
    WFG3线性的、退化的
    WFG4-9凹的、WFG4多模态、WFG5具有欺骗性、
    WFG6不可分、WFG7有偏好的、
    WFG8不可分且有偏好的、
    WFG9多模态、有偏好的
    下载: 导出CSV

    表  2  六种算法在不同维数的DTLZ测试问题上获得的IGD平均值和标准差, 最佳结果被突出显示

    Table  2  The IGD average and standard deviation obtained by the six algorithms on DTLZ test problems of different dimensions, and the best results are highlighted

    ProblemMNSGA-IIICPS-MOEACSEAK-RVEAMOEA/D-EGOK-RSEA
    DTLZ139.9972×101
    (2.47×101) −
    8.2935×101
    (1.74×101)=
    5.6789×101
    (1.00×101) +
    8.2001×101
    (1.84×101)=
    8.2705×101
    (1.33×101)=
    8.6600×101
    (1.91×101)
    47.3063×101
    (1.63×101) −
    6.4843×101
    (1.64×101)=
    4.3597×101
    (1.34×101) +
    5.6857×101
    (1.25×101)=
    6.9011×101
    (1.39×101)=
    5.9990×101
    (1.41×101)
    63.3806×101
    (1.24×101)=
    3.1427×101
    (7.13×100)=
    1.5953×101
    (5.45×100) +
    2.7870×101
    (1.01×101)=
    3.3203×101
    (1.04×101) −
    2.6831×101
    (7.05×100)
    86.7072×100
    (3.99×100)=
    1.1629×101
    (2.84×100) −
    3.6585×100
    (2.21×100) +
    7.8298×100
    (2.68×100)=
    1.0297×101
    (4.54×100)=
    8.0032×100
    (3.95×100)
    105.0209×10−1
    (3.59×10−1) −
    4.7694×10−1
    (1.80×10−1) −
    2.8192×10−1
    (5.67×10−2)=
    3.7187×10−1
    (8.49×10−2)=
    4.3847×10−1
    (1.13×10−1) −
    3.3052×10−1
    (1.04×10−1)
    DTLZ233.2991×10−1
    (2.63×10−2) −
    3.2931×10−1
    (2.69×10−2) −
    2.1298×10−1
    (2.81×10−2) −
    1.1972×10−1
    (1.36×10−2) −
    3.4065×10−1
    (3.28×10−2) −
    1.1075×10−1
    (7.03×10−2)
    43.5643×10−1
    (2.82×10−2) −
    3.7663×10−1
    (2.18×10−2) −
    2.9615×10−1
    (2.61×10−2) −
    2.2144×10−1
    (1.80×10−2)=
    3.6787×10−1
    (2.61×10−2) −
    2.3347×10−1
    (4.97×10−2)
    64.8763×10−1
    (2.37×10−2) −
    4.9201×10−1
    (2.47×10−2) −
    4.2782×10−1
    (4.32×10−2) −
    3.6589×10−1
    (1.98×10−2)=
    4.7191×10−1
    (2.23×10−2) −
    3.6494×10−1
    (1.93×10−2)
    85.8827×10−1
    (3.63×10−2) −
    5.8008×10−1
    (2.61×10−2) −
    5.8141×10−1
    (3.39×10−2) −
    4.1673×10−1
    (1.44×10−2) +
    5.3642×10−1
    (2.73×10−2) −
    4.2547×10−1
    (1.04×10−2)
    106.3227×10−1
    (2.04×10−2) −
    6.3801×10−1
    (1.80×10−2) −
    6.7240×10−1
    (2.38×10−2) −
    5.0316×10−1
    (1.57×10−2) −
    5.2187×10−1
    (2.24×10−2) −
    4.7549×10−1
    (7.36×10−3)
    DTLZ332.8787×102
    (6.58×101) −
    2.3395×102
    (3.82×101)=
    1.5653×102
    (3.81×101) +
    2.3708×102
    (4.75×101)=
    1.9962×102
    (2.65×101) +
    2.3648×102
    (5.58×101)
    42.0889×102
    (6.54×101)=
    1.6267×102
    (4.26×101)=
    1.2297×102
    (2.83×101) +
    1.8558×102
    (3.48×101)=
    1.5411×102
    (1.28×101)=
    1.8023×102
    (5.61×101)
    61.0529×102
    (2.46×101)=
    9.4164×101
    (1.89×101)=
    5.6044×101
    (1.64×101) +
    8.4557×101
    (2.82×101)=
    9.6519×101
    (1.60×101)=
    8.9132×101
    (3.24×101)
    82.6642×101
    (9.61×100)=
    2.8860×101
    (1.24×101)=
    1.3883×101
    (5.25×100) +
    2.2607×101
    (8.99×100) +
    3.7525×101
    (1.23×101)=
    3.0443×101
    (1.15×101)
    101.5073×100
    (4.05×10−1) −
    1.5000×100
    (3.78×10−1) −
    1.0257×100
    (2.63×10−1)=
    1.2960×100
    (3.55×10−1)=
    1.2942×100
    (3.36×10−1)=
    1.1642×100
    (2.93×10−1)
    DTLZ437.2107×10−1
    (1.19×10−1) −
    5.9002×10−1
    (3.92×10−2) −
    5.1951×10−1
    (1.51×10−1)=
    3.0267×10−1
    (7.37×10−2) +
    5.9687×10−1
    (6.57×10−2) −
    4.8903×10−1
    (1.47×10−1)
    47.0404×10−1
    (1.25×10−1) −
    6.2009×10−1
    (4.24×10−2) −
    4.6042×10−1
    (7.33×10−2) +
    4.0021×10−1
    (8.08×10−2) +
    6.8696×10−1
    (4.77×10−2) −
    5.5991×10−1
    (1.28×10−1)
    68.0377×10−1
    (7.80×10−2) −
    6.5279×10−1
    (1.84×10−2)=
    4.9747×10−1
    (5.83×10−2) +
    4.8631×10−1
    (5.15×10−2) +
    6.8610×10−1
    (2.90×10−2) −
    6.2727×10−1
    (6.36×10−2)
    87.4087×10−1
    (4.38×10−2) −
    6.3356×10−1
    (1.32×10−2) −
    5.8324×10−1
    (3.16×10−2)=
    5.5700×10−1
    (2.98×10−2)=
    6.5251×10−1
    (1.32×10−2) −
    5.7685×10−1
    (4.08×10−2)
    107.3581×10−1
    (4.34×10−2) −
    6.5510×10−1
    (1.01×10−2) −
    6.3597×10−1
    (3.28×10−2) −
    5.9740×10−1
    (2.69×10−2)=
    6.4268×10−1
    (9.34×10−3) −
    5.8861×10−1
    (2.14×10−2)
    DTLZ532.5926×10−1
    (3.51×10−2) −
    2.4959×10−1
    (2.59×10−2) −
    1.1067×10−1
    (2.85×10−2) −
    8.0805×10−2
    (2.49×10−2) −
    2.4856×10−1
    (2.24×10−2) −
    6.5513×10−2
    (4.58×10−2)
    41.9155×10−1
    (2.36×10−2) −
    2.0562×10−1
    (2.30×10−2) −
    1.2544×10−1
    (2.99×10−2) −
    5.9826×10−2
    (9.29×10−3) −
    2.1704×10−1
    (2.73×10−2) −
    2.6100×10−2
    (1.00×10−2)
    61.4568×10−1
    (2.39×10−2) −
    1.2304×10−1
    (1.85×10−2) −
    7.4651×10−2
    (2.05×10−2) −
    3.4219×10−2
    (1.06×10−2) −
    1.5500×10−1
    (1.93×10−2) −
    1.6379×10−2
    (1.12×10−2)
    88.7377×10−2
    (1.55×10−2) −
    6.5776×10−2
    (1.34×10−2) −
    3.8178×10−2
    (8.52×10−3) −
    2.0890×10−2
    (5.60×10−3) −
    8.2292×10−2
    (1.25×10−2) −
    1.2091×10−2
    (2.70×10−3)
    104.7648×10−2
    (1.45×10−2) −
    2.5322×10−2
    (4.41×10−3) −
    1.1891×10−2
    (1.17×10−3) −
    1.2745×10−2
    (2.29×10−3) −
    2.2116×10−2
    (2.45×10−3) −
    7.4301×10−3
    (1.10×10−3)
    DTLZ636.1232×100
    (2.01×10−1) −
    4.1051×100
    (4.45×10−1) −
    4.9049×100
    (6.04×10−1) −
    3.1198×100
    (3.59×10−1) −
    1.8762×100
    (5.59×10−1)=
    2.0558×100
    (4.60×10−1)
    45.4855×100
    (2.45×10−1) −
    3.4387×100
    (4.98×10−1) −
    4.9792×100
    (5.02×10−1) −
    2.4647×100
    (3.55×10−1) −
    1.6752×100
    (7.38×10−1)=
    2.0560×100
    (3.25×10−1)
    63.8995×100
    (2.21×10−1) −
    2.3140×100
    (5.21×10−1) −
    3.1061×100
    (5.07×10−1) −
    1.2890×100
    (3.22×10−1)=
    9.2648×10−1
    (3.65×10−1) +
    1.2599×100
    (3.56×10−1)
    82.1839×100
    (2.82×10−1) −
    9.0259×10−1
    (2.50×10−1) −
    1.4584×100
    (4.60×10−1) −
    5.3505×10−1
    (1.82×10−1)=
    5.1243×10−1
    (2.60×10−1)=
    5.9215×10−1
    (2.14×10−1)
    105.9508×10−1
    (2.62×10−1) −
    5.2061×10−2
    (1.55×10−2) +
    1.3300×10−1
    (9.60×10−2)=
    7.1410×10−2
    (2.06×10−2)=
    1.9708×10−1
    (7.72×10−2) −
    8.4350×10−2
    (2.65×10−2)
    DTLZ735.7028×100
    (8.46×10−1) −
    4.9515×100
    (7.40×10−1) −
    1.7040×100
    (5.07×10−1) −
    1.4314×10−1
    (4.87×10−2) −
    2.3750×10−1
    (9.67×10−2) −
    9.0290×10−2
    (6.53×10−2)
    47.1332×100
    (8.88×10−1) −
    5.3360×100
    (1.43×100) −
    2.6700×100
    (9.51×10−1) −
    3.7612×10−1
    (1.37×10−1)=
    5.3337×10−1
    (9.02×10−2) −
    3.3295×10−1
    (1.07×10−1)
    68.6385×100
    (2.00×100) −
    6.2498×100
    (1.86×100) −
    4.5051×100
    (8.80×10−1) −
    6.3454×10−1
    (8.41×10−2) +
    8.6377×10−1
    (6.08×10−2) +
    1.0134×100
    (1.85×10−1)
    81.0623×101
    (2.69×100) −
    4.5094×100
    (3.09×100) −
    6.1099×100
    (1.99×100) −
    8.7425×10−1
    (6.82×10−2) +
    1.0618×100
    (3.82×10−2) +
    2.2654×100
    (4.25×10−1)
    103.8760×100
    (1.81×100) −
    1.5793×100
    (9.05×10−2) +
    2.0827×100
    (5.27×10−1)=
    1.0910×100
    (4.35×10−2) +
    1.2142×100
    (2.27×10−2) +
    2.1206×100
    (3.77×10−1)
    +/−/=0/30/52/25/810/19/68/10/175/20/10
    下载: 导出CSV

    表  3  六种算法在不同维数的DTLZ测试问题上获得的HV平均值和标准差, 最佳结果被突出显示

    Table  3  The HV average and standard deviation obtained by the six algorithms on DTLZ test problems of different dimensions, and the best results are highlighted

    ProblemMNSGA-IIICPS-MOEACSEAK-RVEAMOEA/D-EGOK-RSEA
    DTLZ130.0000×100 (0.00×100)=0.0000×100 (0.00×100)=0.0000×100 (0.00×100)=0.0000×100 (0.00×100)=0.0000×100 (0.00×100)=0.0000×100 (0.00×100)
    40.0000×100 (0.00×100)=0.0000×100 (0.00×100)=0.0000×100 (0.00×100)=0.0000×100 (0.00×100)=0.0000×100 (0.00×100)=0.0000×100 (0.00×100)
    60.0000×100 (0.00×100)=0.0000×100 (0.00×100)=0.0000×100 (0.00×100)=0.0000×100 (0.00×100)=0.0000×100 (0.00×100)=0.0000×100 (0.00×100)
    80.0000×100 (0.00×100)=0.0000×100 (0.00×100)=0.0000×100 (0.00×100)=0.0000×100 (0.00×100)=0.0000×100 (0.00×100)=0.0000×100 (0.00×100)
    102.6667×10−1 (2.12×10−1) −1.9822×10−1 (1.68×10−1) −6.2982×10−1 (1.71×10−1) +3.2562×10−1 (2.12×10−1)=1.9434×10−1 (1.65×10−1) −4.0899×10−1 (2.13×10−1)
    DTLZ231.1734×10−1 (2.42×10−2) −1.2929×10−1 (2.86×10−2) −3.1461×10−1 (5.78×10−2) −4.4429×10−1 (2.12×10−2) −1.3839×10−1 (4.54×10−2) −4.6981×10−1 (1.16×10−1)
    42.1214×10−1 (3.91×10−2) −2.0737×10−1 (3.04×10−2) −3.5746×10−1 (7.24×10−2) −5.7115×10−1 (2.41×10−2)=2.2493×10−1 (4.31×10−2) −5.3573×10−1 (1.08×10−1)
    63.0640×10−1 (3.14×10−2) −3.0719×10−1 (3.32×10−2) −5.0211×10−1 (7.01×10−2) −6.8636×10−1 (3.86×10−2)=3.6893×10−1 (3.19×10−2) −6.9607×10−1 (4.86×10−2)
    84.4578×10−1 (4.27×10−2) −4.3830×10−1 (2.13×10−2) −5.6690×10−1 (4.55×10−2) −7.6366×10−1 (3.84×10−2) −5.5566×10−1 (2.98×10−2) −8.3822×10−1 (2.06×10−2)
    105.4274×10−1 (2.52×10−2) −5.9705×10−1 (2.81×10−2) −6.3060×10−1 (2.88×10−2) −8.6321×10−1 (1.23×10−2) −8.1393×10−1 (1.64×10−2) −9.1251×10−1 (7.76×10−3)
    DTLZ330.0000×100 (0.00×100)=0.0000×100 (0.00×100)=0.0000×100 (0.00×100)=0.0000×100 (0.00×100)=0.0000×100 (0.00×100)=0.0000×100 (0.00×100)
    40.0000×100 (0.00×100)=0.0000×100 (0.00×100)=0.0000×100 (0.00×100)=0.0000×100 (0.00×100)=0.0000×100 (0.00×100)=0.0000×100 (0.00×100)
    60.0000×100 (0.00×100)=0.0000×100 (0.00×100)=0.0000×100 (0.00×100)=0.0000×100 (0.00×100)=0.0000×100 (0.00×100)=0.0000×100 (0.00×100)
    80.0000×100 (0.00×100)=0.0000×100 (0.00×100)=0.0000×100 (0.00×100)=0.0000×100 (0.00×100)=0.0000×100 (0.00×100)=0.0000×100 (0.00×100)
    103.5213×10−2 (4.90×10−2) −2.8103×10−2 (4.73×10−2) −2.6473×10−1 (1.54×10−1) +9.2144×10−2 (1.22×10−1)=6.5751×10−2 (7.30×10−2)=8.5208×10−2 (8.29×10−2)
    DTLZ431.1785×10−2 (2.18×10−2) −1.5375×10−2 (2.50×10−2) −1.7626×10−1 (8.31×10−2) +1.7423×10−1 (1.08×10−1) +1.7616×10−2 (2.25×10−2) −7.5949×10−2 (7.70×10−2)
    44.9153×10−2 (3.74×10−2)=2.4608×10−2 (2.91×10−2) −3.1304×10−1 (6.63×10−2) +2.2725×10−1 (8.66×10−2) +2.8193×10−2 (2.68×10−2) −1.3719×10−1 (1.25×10−1)
    61.1881×10−1 (4.18×10−2) −1.2211×10−1 (3.42×10−2) −5.6483×10−1 (8.07×10−2) +4.1915×10−1 (1.31×10−1) +1.0424×10−1 (4.08×10−2) −2.1455×10−1 (9.54×10−2)
    83.1494×10−1 (7.82×10−2) −4.3521×10−1 (5.84×10−2) −6.8774×10−1 (3.61×10−2) +6.2693×10−1 (7.95×10−2) +3.3577×10−1 (6.00×10−2) −5.4786×10−1 (9.97×10−2)
    106.1117×10−1 (5.26×10−2) −7.4214×10−1 (2.37×10−2) −8.0539×10−1 (3.71×10−2)=8.3652×10−1 (4.00×10−2)=7.4450×10−1 (2.53×10−2) −8.2411×10−1 (4.08×10−2)
    DTLZ531.7464×10−2 (8.81×10−3) −2.3213×10−2 (1.15×10−2) −9.1632×10−2 (2.53×10−2) −1.2641×10−1 (2.94×10−2) −1.8468×10−2 (2.17×10−2) −1.5632×10−1 (4.42×10−2)
    41.8002×10−2 (8.78×10−3) −1.8297×10−2 (7.45×10−3) −5.6265×10−2 (2.37×10−2) −1.1696×10−1 (6.53×10−3) −2.6234×10−2 (2.37×10−2) −1.3582×10−1 (1.02×10−2)
    61.9682×10−2 (1.21×10−2) −2.6909×10−2 (1.97×10−2) −7.5674×10−2 (1.88×10−2) −1.0592×10−1 (3.54×10−3) −5.0133×10−2 (2.49×10−2) −1.1271×10−1 (8.05×10−3)
    85.2712×10−2 (2.30×10−2) −6.5698×10−2 (1.40×10−2) −9.3848×10−2 (4.29×10−3) −1.0261×10−1 (2.69×10−3) −8.5375×10−2 (6.49×10−3) −1.0487×10−1 (3.42×10−4)
    108.6211×10−2 (1.15×10−2) −9.7441×10−2 (9.74×10−4) −9.9494×10−2 (5.70×10−4) −9.7695×10−2 (7.46×10−4) −9.6720×10−2 (7.96×10−4) −1.0031×10−1 (2.98×10−4)
    DTLZ630.0000×100 (0.00×100)=0.0000×100 (0.00×100)=0.0000×100 (0.00×100)=0.0000×100 (0.00×100)=0.0000×100 (0.00×100)=0.0000×100 (0.00×100)
    40.0000×100 (0.00×100)=0.0000×100 (0.00×100)=0.0000×100 (0.00×100)=0.0000×100 (0.00×100)=4.7399×10−3 (2.03×10−2) +0.0000×100 (0.00×100)
    60.0000×100 (0.00×100)=0.0000×100 (0.00×100)=0.0000×100 (0.00×100)=0.0000×100 (0.00×100)=3.2073×10−2 (4.49×10−2) +4.5458×10−3 (2.03×10−2)
    80.0000×100 (0.00×100) −1.8646×10−4 (7.43×10−4)=9.2532×10−4 (4.14×10−3) −2.2110×10−2 (3.93×10−2)=6.0557×10−2 (4.40×10−2) +8.0524×10−3 (2.08×10−2)
    101.7423×10−2 (3.52×10−2) −8.0620×10−2 (2.45×10−2)=5.7994×10−2 (3.98×10−2) −9.4737×10−2 (1.76×10−3)=9.2210×10−2 (1.35×10−3) −9.2912×10−2 (1.31×10−2)
    DTLZ730.0000×100 (0.00×100) −0.0000×100 (0.00×100) −4.2044×10−2 (4.24×10−2) −2.4684×10−1 (5.96×10−3) −2.0656×10−1 (1.66×10−2) −2.7126×10−1 (8.89×10−3)
    40.0000×100 (0.00×100) −5.5343×10−7 (2.48×10−6) −3.3341×10−2 (3.99×10−2) −2.3586×10−1 (7.10×10−3) −7.9986×10−2 (5.40×10−2) −2.5760×10−1 (7.67×10−3)
    60.0000×100 (0.00×100) −0.0000×100 (0.00×100) −2.9404×10−2 (3.52×10−2) −1.9617×10−1 (8.41×10−3) −3.0127×10−2 (3.37×10−2) −2.1700×10−1 (5.99×10−3)
    80.0000×100 (0.00×100) −2.5202×10−4 (9.83×10−4) −3.0467×10−2 (3.86×10−2) −1.8919×10−1 (4.73×10−3)=5.7129×10−3 (5.65×10−3) −1.8593×10−1 (8.01×10−3)
    103.3199×10−3 (5.02×10−3) −7.5420×10−4 (8.53×10−4) −3.5750×10−2 (3.02×10−2) −1.7539×10−1 (4.01×10−3)=1.3231×10−2 (1.67×10−2) −1.7687×10−1 (4.85×10−3)
    +/−/=0/23/120/22/136/17/124/11/203/22/10
    下载: 导出CSV

    表  4  六种算法在不同维数的WFG测试问题上获得的IGD平均值和标准差, 最佳结果被突出显示

    Table  4  The IGD average and standard deviation obtained by the six algorithms on WFG test problems of different dimensions, and the best results are highlighted

    ProblemMNSGAIIICPS-MOEACSEAK-RVEAMOEA/D-EGOK-RSEA
    WFG132.3044×100 (7.04×10−2) −2.2817×100 (6.96×10−2) −1.7246×100 (7.58×10−2) +1.7654×100 (9.23×10−2) +2.1936×100 (6.64×10−2) −1.8391×100 (8.45×10−2)
    42.1642×100 (1.32×10−1)=1.9453×100 (1.71×10−2) +1.9531×100 (1.11×10−1) +2.0752×100 (1.55×10−1)=2.0130×100 (7.58×10−2) +2.0919×100 (1.31×10−1)
    62.7873×100 (5.89×10−2) −2.8059×100 (6.16×10−2) −2.4902×100 (5.09×10−2)=2.4594×100 (9.80×10−2) +2.7526×100 (5.35×10−2) −2.5304×100 (1.00×10−1)
    83.1194×100 (5.49×10−2) −3.1420×100 (4.58×10−2) −2.8827×100 (6.05×10−2)=2.8566×100 (8.07×10−2)=3.0855×100 (5.32×10−2) −2.8335×100 (2.54×10−1)
    103.4110×100 (4.80×10−2) −3.4220×100 (4.68×10−2) −3.1817×100 (8.33×10−2)=3.1589×100 (5.88×10−2)=3.3937×100 (3.38×10−2) −3.1129×100 (2.06×10−1)
    WFG238.8153×10−1 (1.09×10−1) −7.8323×10−1 (4.94×10−2) −5.3426×10−1 (4.34×10−2) −3.9180×10−1 (4.71×10−2) −6.9725×10−1 (4.98×10−2) −3.4170×10−1 (4.16×10−2)
    41.4386×100 (2.16×10−1) −1.2247×100 (1.92×10−1) −1.2237×100 (3.42×10−1) −9.9551×10−1 (1.34×10−1) −1.1293×100 (1.61×10−1) −9.1510×10−1 (1.09×10−1)
    61.9640×100 (4.29×10−1) −1.5348×100 (1.38×10−1) −1.2796×100 (4.14×10−1) −7.6122×10−1 (5.92×10−2) −1.3485×100 (1.72×10−1) −7.1461×10−1 (3.51×10−2)
    82.5026×100 (5.75×10−1) −2.0548×100 (2.06×10−1) −1.8266×100 (7.53×10−1) −1.0294×100 (4.52×10−2)=1.7364×100 (2.53×10−1) −1.0125×100 (3.41×10−2)
    103.8506×100 (7.40×10−1) −3.0779×100 (4.97×10−1) −3.1377×100 (9.98×10−1) −1.1813×100 (5.78×10−2) +2.1748×100 (3.75×10−1) −1.2223×100 (6.17×10−2)
    WFG336.0295×10−1 (4.49×10−2) −6.0035×10−1 (2.83×10−2) −4.7826×10−1 (7.08×10−2) −3.8454×10−1 (5.62×10−2)=6.3111×10−1 (3.08×10−2) −4.2141×10−1 (8.51×10−2)
    45.6516×10−1 (5.35×10−2) −5.5932×10−1 (3.92×10−2) −3.6608×10−1 (6.42×10−2) −2.2293×10−1 (3.65×10−2) +6.0164×10−1 (4.47×10−2) −2.5478×10−1 (2.73×10−2)
    61.0518×100 (1.02×10−1) −9.8651×10−1 (8.00×10−2) −7.0767×10−1 (9.63×10−2) −6.6060×10−1 (8.61×10−2) −9.7316×10−1 (4.10×10−2) −4.9911×10−1 (9.35×10−2)
    89.2359×10−1 (1.11×10−1) −7.8556×10−1 (7.35×10−2) −4.4627×10−1 (1.18×10−1) −5.3840×10−1 (7.52×10−2) −8.2687×10−1 (8.87×10−2) −3.4794×10−1 (5.25×10−2)
    101.0022×100 (8.69×10−2) −8.7818×10−1 (1.11×10−1) −5.9822×10−1 (1.14×10−1) −6.3955×10−1 (8.21×10−2) −9.2789×10−1 (7.38×10−2) −4.2756×10−1 (6.74×10−2)
    WFG436.3183×10−1 (6.30×10−2) −5.3726×10−1 (2.65×10−2) −4.4571×10−1 (3.28×10−2) +4.5254×10−1 (1.90×10−2) +5.8593×10−1 (3.46×10−2) −5.0332×10−1 (2.65×10−2)
    41.7199×100 (1.84×10−1) −1.2114×100 (8.23×10−2) −1.5483×100 (2.75×10−1) −8.1670×10−1 (8.21×10−2)=9.8678×10−1 (7.18×10−2) −7.7364×10−1 (7.65×10−2)
    63.6040×100 (3.76×10−1) −2.5352×100 (1.50×10−1) −2.9141×100 (3.36×10−1) −1.7992×100 (4.76×10−2) +2.1228×100 (1.22×10−1) −1.8370×100 (5.25×10−2)
    85.9740×100 (4.07×10−1) −4.3118×100 (2.44×10−1) −5.8308×100 (4.61×10−1) −3.2283×100 (2.39×10−1)=3.4432×100 (1.44×10−1) −3.3000×100 (2.59×10−1)
    109.1735×100 (5.27×10−1) −7.2985×100 (3.71×10−1) −8.6988×100 (9.68×10−1) −5.9483×100 (5.44×10−1) −5.0437×100 (2.82×10−1) +5.4156×100 (3.45×10−1)
    WFG536.9770×10−1 (3.25×10−2) −5.8013×10−1 (1.71×10−2) −5.2657×10−1 (3.82×10−2) −4.3283×10−1 (6.77×10−2) −5.8135×10−1 (2.95×10−2) −3.8674×10−1 (6.33×10−2)
    41.3558×100 (1.03×10−1) −1.3003×100 (4.72×10−2) −1.1067×100 (1.51×10−1) −7.8509×10−1 (6.09×10−2) −9.8506×10−1 (4.70×10−2) −7.1114×10−1 (4.18×10−2)
    62.7646×100 (1.73×10−1) −2.5373×100 (1.03×10−1) −2.4254×100 (2.43×10−1) −1.7916×100 (8.92×10−2)=2.1839×100 (1.45×10−1) −1.8386×100 (8.92×10−2)
    84.7298×100 (2.08×10−1) −4.5985×100 (1.45×10−1) −4.7238×100 (4.32×10−1) −3.0908×100 (7.65×10−2) +4.4346×100 (2.48×10−1) −3.1917×100 (1.30×10−1)
    107.3037×100 (2.91×10−1) −7.0171×100 (3.14×10−1) −7.0938×100 (3.27×10−1) −4.8049×100 (3.18×10−1) +6.6075×100 (4.97×10−1) −5.0163×100 (2.42×10−1)
    WFG638.0656×10−1 (3.30×10−2) −7.8806×10−1 (2.19×10−2) −7.1317×10−1 (3.92×10−2)=7.1713×10−1 (4.87×10−2)=8.0570×10−1 (4.64×10−2) −7.2868×10−1 (3.97×10−2)
    41.3827×100 (8.35×10−2) −1.2283×100 (5.92×10−2) −1.0173×100 (8.09×10−2)=1.0307×100 (9.56×10−2)=1.1020×100 (4.08×10−2) −1.0481×100 (4.41×10−2)
    62.8461×100 (2.15×10−1) −2.6368×100 (1.36×10−1) −2.3941×100 (1.92×10−1) −2.2878×100 (1.10×10−1) −2.1672×100 (5.45×10−2) −2.1051×100 (7.48×10−2)
    84.9875×100 (3.24×10−1) −4.6116×100 (2.17×10−1) −4.7585×100 (5.04×10−1) −3.6354×100 (8.70×10−2) −3.7239×100 (1.43×10−1) −3.4682×100 (1.18×10−1)
    107.4853×100 (4.37×10−1) −6.9814×100 (4.94×10−1) −7.2251×100 (6.41×10−1) −5.1438×100 (1.55×10−1)=5.3090×100 (3.81×10−1)=5.0901×100 (1.35×10−1)
    WFG736.6448×10−1 (4.60×10−2)=6.3768×10−1 (3.13×10−2)=5.8351×10−1 (3.07×10−2) +6.0448×10−1 (2.89×10−2) +6.6027×10−1 (3.26×10−2)=6.5385×10−1 (4.34×10−2)
    41.5156×100 (1.31×10−1) −1.4000×100 (1.10×10−1) −1.3798×100 (1.35×10−1) −8.9610×10−1 (6.80×10−2)=1.2373×100 (1.22×10−1) −8.6343×10−1 (6.67×10−2)
    63.0239×100 (2.27×10−1) −2.6972×100 (1.90×10−1) −2.5951×100 (2.70×10−1) −1.9468×100 (4.95×10−2) +2.4804×100 (1.69×10−1) −2.0185×100 (5.45×10−2)
    85.2874×100 (2.64×10−1) −4.9740×100 (4.19×10−1) −5.4691×100 (4.66×10−1) −3.4310×100 (1.05×10−1)=5.1211×100 (3.92×10−1) −3.4353×100 (7.80×10−2)
    108.0948×100 (4.84×10−1) −7.6933×100 (4.38×10−1) −8.1050×100 (5.92×10−1) −5.1689×100 (1.77×10−1)=6.9773×100 (6.25×10−1) −5.2380×100 (3.09×10−1)
    WFG838.7226×10−1 (3.67×10−2) −8.4186×10−1 (2.79×10−2) −7.3788×10−1 (5.34×10−2) −7.2196×10−1 (3.90×10−2) −8.6180×10−1 (2.30×10−2) −6.7398×10−1 (4.58×10−2)
    41.7785×100 (1.02×10−1) −1.7130×100 (8.86×10−2) −1.7122×100 (1.64×10−1) −1.3654×100 (5.98×10−2) −1.3509×100 (3.48×10−2) −1.2069×100 (5.72×10−2)
    63.2270×100 (2.38×10−1) −2.8330×100 (1.62×10−1) −3.0250×100 (2.19×10−1) −2.3476×100 (9.36×10−2) −2.4530×100 (5.19×10−2) −2.2095×100 (4.49×10−2)
    85.2767×100 (2.91×10−1) −5.1048×100 (3.42×10−1) −5.4616×100 (3.15×10−1) −3.5830×100 (1.18×10−1)=4.2845×100 (2.58×10−1) −3.5916×100 (1.02×10−1)
    107.8537×100 (3.65×10−1) −7.3927×100 (4.54×10−1) −7.9521×100 (4.13×10−1) −5.0690×100 (1.21×10−1) +5.6966×100 (2.55×10−1) −5.1969×100 (1.70×10−1)
    WFG938.1617×10−1 (4.19×10−2) −7.8612×10−1 (4.07×10−2) −6.2832×10−1 (6.58×10−2)=6.6725×10−1 (4.52×10−2)=7.8107×10−1 (5.79×10−2) −6.6782×10−1 (5.50×10−2)
    41.3187×100 (8.63×10−2) −1.3147×100 (7.88×10−2) −1.1780×100 (1.25×10−1) −1.1332×100 (2.03×10−1)=1.3423×100 (1.28×10−1) −1.0305×100 (1.72×10−1)
    63.0388×100 (2.54×10−1) −3.0806×100 (1.69×10−1) −2.9442×100 (2.94×10−1) −2.1042×100 (1.22×10−1) +2.7797×100 (3.25×10−1) −2.4146×100 (1.84×10−1)
    85.1537×100 (3.04×10−1) −5.1279×100 (3.30×10−1) −5.2404×100 (4.41×10−1) −3.9706×100 (6.27×10−1)=4.8820×100 (4.82×10−1) −4.0674×100 (5.76×10−1)
    107.6142×100 (4.20×10−1) −7.6831×100 (3.48×10−1) −7.5833×100 (5.26×10−1) −6.2079×100 (5.79×10−1)=7.2823×100 (5.75×10−1) −6.0729×100 (6.67×10−1)
    +/−/=0/43/21/43/14/35/612/14/192/41/2
    下载: 导出CSV

    表  5  六种算法在不同维数的WFG测试问题上获得的HV平均值和标准差, 最佳结果被突出显示

    Table  5  The HV average and standard deviation obtained by the six algorithms on WFG test problems of different dimensions, and the best results are highlighted

    ProblemMNSGA-IIICPS-MOEACSEAK-RVEAMOEA/D-EGOK-RSEA
    WFG13 0.0000 ×100 (0.00×100) − 2.3754 ×10−3 (5.39×10−3) − 1.5551 ×10−1 (4.56×10−2)= 1.6255 ×10−1 (2.88×10−2)= 5.6758 ×10−3 (1.33×10−2) − 1.4541 ×10−1 (3.90×10−2)
    4 1.9475 ×10−1 (3.30×10−2)= 2.9110 ×10−1 (6.17×10−3) + 2.7323 ×10−1 (3.27×10−2) + 2.3182 ×10−1 (4.43×10−2)= 2.7638 ×10−1 (1.47×10−2) + 2.1878 ×10−1 (6.03×10−2)
    6 3.0659 ×10−2 (2.10×10−2) − 9.9187 ×10−2 (1.20×10−2) − 2.0714 ×10−1 (3.65×10−2)= 2.0241 ×10−1 (5.23×10−2)= 1.2496 ×10−1 (2.73×10−2) − 1.8319 ×10−1 (6.40×10−2)
    8 1.0632 ×10−1 (2.34×10−2) − 1.7126 ×10−1 (9.62×10−3)= 2.0631 ×10−1 (2.29×10−2)= 2.0651 ×10−1 (3.16×10−2)= 1.8449 ×10−1 (1.97×10−2)= 1.9834 ×10−1 (1.06×10−1)
    10 1.0646 ×10−1 (2.91×10−2) − 1.9083 ×10−1 (6.61×10−3)= 2.1349 ×10−1 (2.85×10−2)= 2.1469 ×10−1 (7.69×10−3)= 1.8981 ×10−1 (1.13×10−2)= 1.8994 ×10−1 (6.85×10−2)
    WFG23 5.7155 ×10−1 (3.23×10−2) − 6.0180 ×10−1 (1.47×10−2) − 7.0816 ×10−1 (2.55×10−2) − 7.6969 ×10−1 (2.32×10−2) − 6.3021 ×10−1 (2.06×10−2) − 7.9608 ×10−1 (3.12×10−2)
    4 6.0584 ×10−1 (3.12×10−2) − 6.5097 ×10−1 (2.87×10−2) − 6.8373 ×10−1 (5.07×10−2) − 7.5823 ×10−1 (3.40×10−2) − 6.5436 ×10−1 (3.73×10−2) − 7.8405 ×10−1 (2.80×10−2)
    6 6.4455 ×10−1 (3.97×10−2) − 6.8126 ×10−1 (2.19×10−2) − 7.9691 ×10−1 (6.05×10−2) − 8.6678 ×10−1 (3.48×10−2) − 6.9228 ×10−1 (2.95×10−2) − 9.3463 ×10−1 (1.69×10−2)
    8 8.1555 ×10−1 (6.07×10−2) − 8.6795 ×10−1 (1.99×10−2) − 9.1947 ×10−1 (4.73×10−2) − 9.6566 ×10−1 (1.18×10−2) − 8.5448 ×10−1 (3.00×10−2) − 9.9044 ×10−1 (4.99×10−3)
    10 7.6764 ×10−1 (4.56×10−2) − 8.3857 ×10−1 (3.36×10−2) − 8.7953 ×10−1 (5.73×10−2) − 9.6830 ×10−1 (1.01×10−2) − 8.4508 ×10−1 (4.68×10−2) − 9.9466 ×10−1 (2.01×10−3)
    WFG33 1.5521 ×10−1 (1.05×10−2) − 1.6033 ×10−1 (9.18×10−3) − 1.9966 ×10−1 (2.70×10−2)= 2.4291 ×10−1 (2.18×10−2) + 1.4825 ×10−1 (9.09×10−3) − 2.0974 ×10−1 (3.45×10−2)
    4 8.3428 ×10−2 (2.38×10−2) − 8.8702 ×10−2 (2.16×10−2) − 1.6360 ×10−1 (3.01×10−2) − 2.2785 ×10−1 (2.52×10−2) + 8.1881 ×10−2 (1.96×10−2) − 2.0271 ×10−1 (2.29×10−2)
    6 0.0000 ×100 (0.00×100) − 0.0000 ×100 (0.00×100) − 6.2531 ×10−3 (1.43×10−2) − 7.5288 ×10−3 (1.24×10−2) − 1.2130 ×10−3 (5.42×10−3) − 2.7253 ×10−2 (2.58×10−2)
    8 4.5931 ×10−3 (2.01×10−2) − 2.6674 ×10−4 (9.23×10−4) − 3.3423 ×10−2 (4.46×10−2)= 1.4620 ×10−2 (2.21×10−2) − 9.6695 ×10−3 (2.28×10−2) − 4.9333 ×10−2 (3.21×10−2)
    10 0.0000 ×100 (0.00×100) − 8.4506 ×10−5 (3.78×10−4)= 2.1320 ×10−3 (8.52×10−3)= 2.5236 ×10−3 (1.13×10−2)= 0.0000 ×100 (0.00×100) − 5.1517 ×10−3 (1.19×10−2)
    WFG43 3.1526 ×10−1 (1.26×10−2) − 3.3905 ×10−1 (1.08×10−2)= 3.8760 ×10−1 (2.00×10−2) + 3.6710 ×10−1 (1.16×10−2) + 3.3783 ×10−1 (1.33×10−2) − 3.4727 ×10−1 (1.35×10−2)
    4 3.1231 ×10−1 (1.20×10−2) − 3.8630 ×10−1 (1.57×10−2) − 3.6772 ×10−1 (3.32×10−2) − 4.7669 ×10−1 (2.16×10−2) − 4.3451 ×10−1 (2.48×10−2) − 4.8908 ×10−1 (1.89×10−2)
    6 3.6756 ×10−1 (2.66×10−2) − 4.6632 ×10−1 (1.82×10−2) − 4.6846 ×10−1 (2.89×10−2) − 5.8503 ×10−1 (3.32×10−2) − 5.1160 ×10−1 (2.69×10−2) − 6.3545 ×10−1 (2.31×10−2)
    8 4.2581 ×10−1 (1.93×10−2) − 5.5373 ×10−1 (2.95×10−2) − 5.0393 ×10−1 (4.44×10−2) − 7.0749 ×10−1 (3.11×10−2) − 6.5901 ×10−1 (2.55×10−2) − 7.7127 ×10−1 (3.11×10−2)
    10 4.1946 ×10−1 (2.30×10−2) − 5.4379 ×10−1 (2.16×10−2) − 4.9750 ×10−1 (5.18×10−2) − 6.5396 ×10−1 (3.22×10−2) − 6.5974 ×10−1 (3.52×10−2) − 7.6087 ×10−1 (2.71×10−2)
    WFG53 2.4395 ×10−1 (1.12×10−2) − 2.9918 ×10−1 (7.98×10−3) − 3.4805 ×10−1 (2.40×10−2) − 3.9021 ×10−1 (3.64×10−2) − 3.6004 ×10−1 (1.89×10−2) − 4.2110 ×10−1 (3.46×10−2)
    4 2.7954 ×10−1 (1.14×10−2) − 3.3688 ×10−1 (1.26×10−2) − 3.9549 ×10−1 (3.14×10−2) − 4.7832 ×10−1 (2.30×10−2) − 4.0185 ×10−1 (1.44×10−2) − 5.0051 ×10−1 (2.40×10−2)
    6 3.2385 ×10−1 (1.38×10−2) − 4.0172 ×10−1 (1.59×10−2) − 4.9501 ×10−1 (3.25×10−2) − 5.6239 ×10−1 (4.28×10−2) − 5.0007 ×10−1 (1.55×10−2) − 5.9915 ×10−1 (3.21×10−2)
    8 3.6783 ×10−1 (2.95×10−2) − 4.6548 ×10−1 (1.24×10−2) − 5.4239 ×10−1 (3.56×10−2) − 6.6693 ×10−1 (4.28×10−2) − 5.3718 ×10−1 (2.14×10−2) − 7.2794 ×10−1 (2.07×10−2)
    10 3.7342 ×10−1 (2.07×10−2) − 4.6464 ×10−1 (1.42×10−2) − 5.2257 ×10−1 (2.90×10−2) − 6.2569 ×10−1 (4.07×10−2) − 5.5522 ×10−1 (3.41×10−2) − 7.1028 ×10−1 (3.10×10−2)
    WFG63 2.0832 ×10−1 (1.01×10−2) − 2.0973 ×10−1 (7.97×10−3) − 2.5872 ×10−1 (1.90×10−2) + 2.5438 ×10−1 (1.91×10−2) + 2.5210 ×10−1 (1.88×10−2) + 2.2621 ×10−1 (1.93×10−2)
    4 2.6812 ×10−1 (1.80×10−2) − 2.9358 ×10−1 (1.92×10−2) − 3.5227 ×10−1 (2.85×10−2) + 3.7527 ×10−1 (4.14×10−2) + 2.9643 ×10−1 (1.56×10−2) − 3.3339 ×10−1 (3.07×10−2)
    6 3.0489 ×10−1 (1.75×10−2) − 3.4169 ×10−1 (1.88×10−2) − 4.0660 ×10−1 (3.70×10−2)= 3.9532 ×10−1 (4.83×10−2)= 4.1998 ×10−1 (1.48×10−2)= 4.0315 ×10−1 (3.75×10−2)
    8 3.9079 ×10−1 (4.10×10−2) − 4.6339 ×10−1 (2.22×10−2) − 5.1016 ×10−1 (4.92×10−2) − 6.6623 ×10−1 (2.77×10−2) − 5.1183 ×10−1 (2.19×10−2) − 7.3524 ×10−1 (4.15×10−2)
    10 3.9579 ×10−1 (2.55×10−2) − 4.7448 ×10−1 (2.34×10−2) − 4.9997 ×10−1 (4.80×10−2) − 6.7299 ×10−1 (3.78×10−2) − 5.3339 ×10−1 (2.18×10−2) − 7.9146 ×10−1 (4.70×10−2)
    WFG73 2.7294 ×10−1 (1.31×10−2)= 2.7768 ×10−1 (1.21×10−2)= 3.2790 ×10−1 (2.16×10−2) + 2.9637 ×10−1 (1.66×10−2) + 2.8180 ×10−1 (1.13×10−2)= 2.7832 ×10−1 (1.80×10−2)
    4 3.0770 ×10−1 (1.53×10−2) − 3.3113 ×10−1 (1.54×10−2) − 3.6574 ×10−1 (2.60×10−2) − 4.5508 ×10−1 (2.66×10−2)= 3.5393 ×10−1 (1.72×10−2) − 4.6052 ×10−1 (2.59×10−2)
    6 3.5607 ×10−1 (1.68×10−2) − 4.0546 ×10−1 (1.42×10−2) − 4.6728 ×10−1 (2.93×10−2) − 5.1981 ×10−1 (3.50×10−2) − 4.1921 ×10−1 (1.60×10−2) − 5.5770 ×10−1 (4.09×10−2)
    8 4.1954 ×10−1 (3.00×10−2) − 4.7747 ×10−1 (2.38×10−2) − 5.3422 ×10−1 (4.07×10−2) − 6.6874 ×10−1 (3.33×10−2) − 4.9354 ×10−1 (2.52×10−2) − 7.9566 ×10−1 (1.82×10−2)
    10 4.3978 ×10−1 (2.68×10−2) − 4.9236 ×10−1 (1.55×10−2) − 5.3589 ×10−1 (3.53×10−2) − 6.4650 ×10−1 (4.91×10−2) − 5.2947 ×10−1 (3.36×10−2) − 8.2131 ×10−1 (3.53×10−2)
    WFG83 2.0563 ×10−1 (9.02×10−3) − 2.1117 ×10−1 (9.38×10−3) − 2.5649 ×10−1 (1.96×10−2)= 2.7987 ×10−1 (1.16×10−2) + 2.0196 ×10−1 (9.26×10−3) − 2.5619 ×10−1 (2.12×10−2)
    4 2.2504 ×10−1 (1.58×10−2) − 2.4391 ×10−1 (1.41×10−2) − 2.7342 ×10−1 (2.59×10−2)= 3.0350 ×10−1 (2.26×10−2)= 2.6327 ×10−1 (1.60×10−2) − 2.9119 ×10−1 (2.71×10−2)
    6 2.9374 ×10−1 (2.14×10−2) − 3.1166 ×10−1 (1.44×10−2) − 3.5815 ×10−1 (3.05×10−2) − 3.4101 ×10−1 (1.70×10−2) − 3.4154 ×10−1 (1.45×10−2) − 3.8034 ×10−1 (1.95×10−2)
    8 3.3827 ×10−1 (2.25×10−2) − 3.9489 ×10−1 (2.24×10−2) − 4.3731 ×10−1 (2.62×10−2) − 4.9176 ×10−1 (3.96×10−2) − 4.3251 ×10−1 (3.33×10−2) − 5.4014 ×10−1 (4.21×10−2)
    10 3.7424 ×10−1 (1.66×10−2) − 4.1948 ×10−1 (1.39×10−2) − 4.3470 ×10−1 (2.64×10−2) − 5.0197 ×10−1 (4.74×10−2) − 4.7105 ×10−1 (2.48×10−2) − 6.1406 ×10−1 (5.41×10−2)
    WFG93 2.1195 ×10−1 (1.55×10−2) − 2.2346 ×10−1 (2.06×10−2) − 2.8163 ×10−1 (3.02×10−2)= 2.6005 ×10−1 (1.97×10−2)= 2.2416 ×10−1 (2.21×10−2) − 2.6152 ×10−1 (3.23×10−2)
    4 3.1377 ×10−1 (2.84×10−2) − 3.3327 ×10−1 (1.77×10−2) − 3.5582 ×10−1 (4.06×10−2) − 3.8240 ×10−1 (6.37×10−2)= 3.0577 ×10−1 (1.98×10−2) − 4.1389 ×10−1 (6.77×10−2)
    6 3.1971 ×10−1 (2.97×10−2) − 3.3719 ×10−1 (1.64×10−2) − 3.8869 ×10−1 (4.85×10−2) − 4.8276 ×10−1 (5.15×10−2)= 3.4419 ×10−1 (2.18×10−2) − 4.7707 ×10−1 (6.45×10−2)
    8 4.5419 ×10−1 (3.01×10−2) − 4.7014 ×10−1 (2.00×10−2) − 5.4358 ×10−1 (3.44×10−2) − 6.3781 ×10−1 (6.24×10−2)= 4.9095 ×10−1 (3.53×10−2) − 6.5968 ×10−1 (6.23×10−2)
    10 4.8221 ×10−1 (2.31×10−2) − 4.7502 ×10−1 (1.84×10−2) − 5.6346 ×10−1 (4.24×10−2) − 6.1737 ×10−1 (3.46×10−2) − 5.0326 ×10−1 (3.14×10−2) − 6.7554 ×10−1 (4.40×10−2)
    +/−/=0/43/21/39/55/29/117/25/132/39/4
    下载: 导出CSV

    表  6  汽车碰撞优化设计问题上获得的IGD和HV的平均值, 最佳结果被突出显示

    Table  6  The average value of IGD and HV obtained on the car crash optimization design problem, the best result is highlighted

    IGDHV
    NSGA-III2.49750.0308
    CPS-MOEA2.36650.0340
    CSEA1.31440.0368
    K-RVEA0.71420.0374
    MOEA/D-EGO0.67930.0384
    K-RSEA0.49200.0389
    下载: 导出CSV
  • [1] Coello Coello C A, González Brambila S, Figueroa Gamboa J, Castillo Tapia M G, Hernández Gómez R. Evolutionary multiobjective optimization: open research areas and some challenges lying ahead. Complex & Intelligent Systems, 2020, 6(2): 221−236
    [2] Koziel S, Leifsson L. Multi-level CFD-based airfoil shape optimization with automated low-fidelity model selection. Procedia Computer Science, 2013, 18(1): 889−898
    [3] Lim D, Jin Y, Ong Y, Sendhoff B. Generalizing surrogate-assisted evolutionary computation. IEEE Transactions on Evolutionary Computation, 2010, 14(3): 329−355 doi: 10.1109/TEVC.2009.2027359
    [4] Jin Y. Surrogate-assisted evolutionary computation: Recent advances and future challenges. Swarm and Evolutionary Computation, 2011, 1(2): 61−70 doi: 10.1016/j.swevo.2011.05.001
    [5] Chugh T, Sindhya K, Hakanen J, Miettinen K. A survey on handling computationally expensive multiobjective optimization problems with evolutionary algorithms. Soft Computing, 2019, 23(9): 3137−3166 doi: 10.1007/s00500-017-2965-0
    [6] 刘晓路, 陈盈果, 贺仁杰, 陈英武. Kriging代理模型在对地观测卫星系统优化中的应用. 自动化学报, 2012, 38(01): 120−127 doi: 10.3724/SP.J.1004.2012.00120

    Liu Xiao-Lu, Chen Ying-Guo, He Ren-Jie, Chen Ying-Wu. Application of kriging proxy model in the optimization of earth observation satellite system. Acta Automatica Sinica, 2012, 38(01): 120−127 doi: 10.3724/SP.J.1004.2012.00120
    [7] Knowles J. ParEGO: A hybrid algorithm with on-line landscape approximation for expensive multiobjective optimization problems. IEEE Transactions on Evolutionary Computation, 2006, 10(1): 50−66 doi: 10.1109/TEVC.2005.851274
    [8] Hochstrate N, Naujoks B, Emmerich M. SMS-EMOA: Multiobjective selection based on dominated hypervolume. European Journal of Operational Research, 2007, 181(3): 1653−1669 doi: 10.1016/j.ejor.2006.08.008
    [9] Zhang Qing-Fu, Liu Wu-Dong, Tsang E, Virginas B. Expensive multiobjective optimization by MOEA/D With gaussian process model. IEEE Transactions on Evolutionary Computation, 2010, 14(3): 456−474 doi: 10.1109/TEVC.2009.2033671
    [10] Chugh T, Jin Y, Miettinen K, Hakanen J, Sindhya K. A surrogate-assisted reference vector guided evolutionary algorithm for computationally expensive many-objective optimization. IEEE Transactions on Evolutionary Computation, 2018, 22(1): 129−142 doi: 10.1109/TEVC.2016.2622301
    [11] Zhang Jin-Yuan, Zhou Ai-Min, Zhang Gui-Xu. A classification and pareto domination based multiobjective evolutionary algorithm. In: Proceedings of 2015 IEEE Congress on Evolutionary Computation (CEC). Sendai, Japan: IEEE, 2015. 2883−2890
    [12] Pan Lin-Qiang, He Cheng, Tian Ye, Wang Han-Ding, Zhang Xing-Yi, Jin Y. A classification-based surrogate-assisted evolutionary algorithm for expensive many-objective optimization. IEEE Transactions on Evolutionary Computation, 2019, 23(1): 74−88 doi: 10.1109/TEVC.2018.2802784
    [13] Shahriari B, Swersky K, Wang Z, Adams R P, De Freitas N. Taking the human out of the loop: A review of bayesian optimization. Proceedings of the IEEE, 2015, 104(1): 148−175
    [14] Rasmussen C E. Gaussian Processes in Machine Learning. In: Proceedings of Summer School on Machine Learning. Berlin, Heidelberg: Springer, 2003. 63−71
    [15] Hoffman P, Grinstein G, Marx K, Grosse I, Stanley E. DNA visual and analytic data mining. In: Proceedings of Visualization '97 (Cat No 97CB36155). Phoenix, USA: IEEE, 1997. 437−441
    [16] Walker D J, Everson R, Fieldsend J E. Visualizing mutually nondominating solution sets in many-objective optimization. IEEE Transactions on Evolutionary Computation, 2013, 17(2): 165−184 doi: 10.1109/TEVC.2012.2225064
    [17] Ibrahim A, Rahnamayan S, Martin M V, Deb K. 3D-RadVis: visualization of pareto front in many-objective optimization. In: Proceedings of 2016 IEEE Congress on Evolutionary Computation (CEC). Vancouver, Canada: IEEE, 2016. 736−745
    [18] He Cheng, Tian Ye, Jin Y, Zhang Xing-Yi, Pan Lin-Qiang. A radial space division based evolutionary algorithm for many-objective optimization. Applied Soft Computing, 2017, 61: 603−621 doi: 10.1016/j.asoc.2017.08.024
    [19] McKay M D, Beckman R J, Conover W J. A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics, 2000, 42(1): 55−61 doi: 10.1080/00401706.2000.10485979
    [20] Deb K, Jain H. An evolutionary many-objective optimization algorithm using reference-point-based nondominated sorting approach, part I: solving problems with box constraints. IEEE Transactions on Evolutionary Computation, 2014, 18(4): 577−601 doi: 10.1109/TEVC.2013.2281535
    [21] Tian Ye, Cheng Ran, Zhang Xing-Yi, Jin Y. PlatEMO: A MATLAB platform for evolutionary multi-objective optimization [Educational Forum]. IEEE Computational Intelligence Magazine, 2017, 12(4): 73−87 doi: 10.1109/MCI.2017.2742868
    [22] Deb K, Thiele L, Laumanns M, Zitzler E. Evolutionary Multiobjective Optimization: Theoretical Advances and Applications. London; Springer, 2005. 105−145
    [23] Huband S, Hingston P, Barone L, While L. A review of multiobjective test problems and a scalable test problem toolkit. IEEE Transactions on Evolutionary Computation, 2006, 10(5): 477−506 doi: 10.1109/TEVC.2005.861417
    [24] Zhou Ai-Min, Jin Y, Zhang Qing-Fu, Sendhoff B, Tsang E. Combining model-based and genetics-based offspring generation for multi-objective optimization using a convergence criterion. In: Proceedings of 2006 IEEE International Conference on Evolutionary Computation. Vancouver, Canada: IEEE, 2006. 892−899
    [25] While L, Hingston P, Barone L, Huband S. A faster algorithm for calculating hypervolume. IEEE Transactions on Evolutionary Computation, 2006, 10(1): 29−38 doi: 10.1109/TEVC.2005.851275
    [26] Ishibuchi H, Imada R, Masuyama N, Nojima Y. Comparison of hypervolume, IGD and IGD+ from the viewpoint of optimal distributions of solutions. In: Proceedings of Evolutionary Multi-Criterion Optimization. Cham: Springer International Publishing, 2019. 332−345
    [27] Liao Xing-Tao, Li Qing, Yang Xu-Jing, Zhang Wei-Gang, Li Wei. Multiobjective optimization for crash safety design of vehicles using stepwise regression model. Structural and Multidisciplinary Optimization, 2008, 35(6): 561−569 doi: 10.1007/s00158-007-0163-x
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