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基于种群个体数自适应的多尺度量子谐振子优化算法

焦育威 王鹏

焦育威, 王鹏. 基于种群个体数自适应的多尺度量子谐振子优化算法. 自动化学报, 2020, 46(x): 1−14 doi: 10.16383/j.aas.c200247
引用本文: 焦育威, 王鹏. 基于种群个体数自适应的多尺度量子谐振子优化算法. 自动化学报, 2020, 46(x): 1−14 doi: 10.16383/j.aas.c200247
Jiao Yu-Wei, Wang Peng. Multi-scale quantum harmonic oscillator algorithm based on subpopulation number adaptive. Acta Automatica Sinica, 2020, 46(x): 1−14 doi: 10.16383/j.aas.c200247
Citation: Jiao Yu-Wei, Wang Peng. Multi-scale quantum harmonic oscillator algorithm based on subpopulation number adaptive. Acta Automatica Sinica, 2020, 46(x): 1−14 doi: 10.16383/j.aas.c200247

基于种群个体数自适应的多尺度量子谐振子优化算法

doi: 10.16383/j.aas.c200247
基金项目: 国家自然科学基金(60702075), 西南民族大学研究生创新型科研项目(CX2020SZ03) 资助
详细信息
    作者简介:

    焦育威:西南民族大学计算机科学与技术学院硕士研究生. 主要研究方向为量子启发式算法, 高性能计算. E-mail: jiaoyuwei@stu.swun.edu.cn

    王鹏:西南民族大学计算机科学与技术学院教授. 2004年获中国科学院成都计算机应用研究所计算机软件与理论专业博士学位. 主要研究方向为量子理论, 量子启发式算法, 计算智能与高性能计算. 本文通信作者. E-mail: wp002005@163.com

Multi-scale Quantum Harmonic Oscillator Algorithm Based on Subpopulation Number Adaptive

Funds: Supported by National Natural Science Foundation of China (60702075) and Innovative Research Project for Postgraduates of Southwest Minzu University (CX2020SZ03)
  • 摘要: 优化算法中多种群采样方式可转化为蒙特卡洛对当前函数积分的评估, 针对不同子种群对整体评估的差异性, 提出子种群规模 (个体数) 自适应的改进策略, 并用于多尺度量子谐振子优化算法(Multi-scale quantum harmonic oscillator algorithm, MQHOA) 的改进, 同时阐述多种群策略所具有的量子特性以及量子隧道效应与寻优性能的相关性, 已有的优化算法忽视了动态调节子种群规模对寻优能力的影响, 该策略通过动态调节子种群规模, 提高适应度差的子种群发生量子隧道效应的概率, 增强了算法的寻优能力, 将改进后的算法MQHOA-d(Multi-scale quantum harmonic oscillator algorithm based on dynamic subpopulation) 与 MQHOA 及其他优化算法在 CEC2013 测试集上进行测试, 结果表明原算法 MQHOA"早熟"问题在 MQHOA-d 中得到解决, 且 MQHOA-d 对多峰函数和复合函数优化具有显著优势, 求解误差和计算时间均小于几种经典优化算法.
  • 图  1  不同采样中心概率密度曲线

    Fig.  1  Probability density curves of different sampling centers

    图  2  MQHOA-d所生成种群在Ellipsoidal函数二维空间分布示意图

    Fig.  2  Schematic diagram of spatial distribution of subpopulations generated by MQHOA-d in Ellipsoidal function of 2D

    图  3  一维双阱函数图与隧道效应示意图

    Fig.  3  One dimensional double well function diagram and tunnel effect diagram

    图  4  MQHOA-d与MQHOA发生隧道效应点数量与迭代次数关系图

    Fig.  4  The relationship between the number of tunneling points and the number of iterations between MQHOA-d and MQHOA

    图  5  CEC2013上各算法误差等级图(维度为30,重复51次)

    Fig.  5  Rank sum test results of all algorithms repeated 51 times in the 30-dimensional case

    图  6  CEC2013测试函数在30维情况下重复51次实验箱体图

    Fig.  6  Boxplots of CEC2013 benchmark functions repeated 51 times in the 30-dimensional case

    图  7  测试函数30维情况下各算法重复51次实验平均时间消耗热力图

    Fig.  7  Rank of mean time consumption on CEC2013 repeated 51 times in the 30-dimensional case

    表  1  量子理论与优化算法的对应关系

    Table  1  The relationship between quantum theory and optimization algorithm

    量子理论 优化算法 说明
    势能评估/势能最低点 $f(x)$ / $f(x)$ 的全局最小值 量子系统下将函数优化问题转化为寻找势能最低点问题
    波函数 解的概率分布 量子系统下波函数对优化问题的概率描述
    势阱等效 优化问题目标函数的近似方法 如QPSO中用Delta势阱逼近目标函数, MQHOA以谐振子逼近目标函数
    势垒宽度 子种群位置到最优解的相对距离 势垒宽度越大, 隧道效应概率越低
    量子隧道效应 跳出局部最优能力 波函数计算隧道效应概率, 增加迭代过程中发生隧道效应的采样点数量, 可提高算法性能
    下载: 导出CSV

    表  2  CEC2013测试函数维度为30情况下各算法51重复实验的平均误差和标准差

    Table  2  Mean errors and standard deviation of each algorithm on CEC2013 repeated 51 times in the 30-dimensional

    $ f $ MQHOA MQHOA-d SPSO2011 QPSO EFWA AFWA
    MeanErr Std MeanErr Std MeanErr Std MeanErr Std MeanErr Std MeanErr Std
    1 $9.71\!\!\times\!\! 10^{\!-\!11}$ $1.44\!\!\times\!\! 10^{\!-\!11}$ $3.25\!\!\times\! \!10^{\!-\!13}$ $1.14\!\!\times\!\! 10^{\!-\!13}$ $2.36\!\!\times\!\! 10^{\!-\!13}$ $4.46\!\!\times\!\! 10^{\!-\!14}$ $2.59\!\!\times\!\! 10^{\!-\!13}$ $9.12\!\!\times \!\!10^{\!-\!14}$ $7.82\!\!\times\!\! 10^{-2}$ $1.31\!\!\times\!\! 10^{\!-\!2}$ ${\bf{0.00\!\!\times\!\! 10^{0} } }$ ${\bf{0.00\!\!\times\! \!10^{0} } }$
    2 $1.15\!\times\! 10^{6}$ $2.65\!\times \!10^{5}$ $1.76\!\times \!10^{6}$ $3.97\!\times\! 10^{5}$ ${\bf{9.68\!\times \!10^{4} } }$ ${\bf{4.82\!\times \!10^{4} } }$ $1.62\!\times\! 10^{7}$ $7.01\!\times\! 10^{6}$ $5.43\!\times\! 10^{5}$ $2.04\!\times\! 10^{5}$ $8.92\!\times\! 10^{5}$ $3.92\!\times\! 10^{5}$
    3 $3.32\!\times\! 10^{7}$ $ 3.94\times 10^{7} $ ${\bf{3.60\times 10^{6}}} $ ${\bf{4.19\times 10^{6}}} $ $ 1.07\times 10^{8} $ $ 1.58\times 10^{8} $ $ 2.28\times 10^{8} $ $ 3.23\times 10^{8} $ $ 1.26\times 10^{8} $ $ 2.15\!\times\! 10^{8} $ $ 1.26\!\times\! 10^{8} $ $ 1.54\!\times\! 10^{8} $
    4 $ 3.83\times 10^{4} $ $ 5.80\times 10^{3} $ $ 4.24\times 10^{4} $ $ 4.43\times 10^{3} $ $ 1.55\times 10^{3} $ $ 5.87\times 10^{2} $ $ 1.03\times 10^{4} $ $ 2.49\times 10^{3} $ ${\bf{1.09\times 10^{0}}} $ ${\bf{3.53\!\times\! 10^{\!-\!1}}} $ $ 1.14\times 10^{1} $ $ 6.83\!\times \!10^{0} $
    5 $ 1.92\!\!\times\!\! 10^{\!-\!3} $ $ 2.46\!\!\times\!\! 10^{\!-\!4} $ $ 2.22\!\!\times\!\! 10^{\!-\!3} $ $ 4.23\!\!\times\!\! 10^{\!-\!4} $ $ 4.03\!\!\times\!\! 10^{\!-\!4} $ $ 2.92\!\!\times\!\! 10^{\!-\!5} $ ${\bf{3.40\!\!\times\!\! 10^{\!-\!7}}} $ ${\bf{3.94\!\!\times\!\! 10^{\!-\!7}}} $ $ 7.90\!\!\times\!\! 10^{\!-\!2} $ $ 1.01\!\!\times\!\! 10^{\!-\!2} $ $ 6.04\!\!\times\!\! 10^{\!-\!4} $ $ 9.24\!\!\times\!\! 10^{\!-\!5} $
    6 $ 3.16\times 10^{1} $ $ 2.65\times 10^{1} $ $ 2.41\times 10^{1} $ $ 1.60\times 10^{1} $ ${\bf{1.70\times 10^{1}}} $ ${\bf{2.02\times 10^{1}}} $ $ 3.21\times 10^{1} $ $ 2.15\times 10^{1} $ $ 3.49\times 10^{1} $ $ 2.71\times 10^{1} $ $ 2.99\times 10^{1} $ $ 2.63\times 10^{1} $
    7 ${\bf{2.07\times 10^{1}}} $ ${\bf{1.26\times 10^{1}}} $ $ 2.23\times 10^{1} $ $ 8.22\times 10^{0} $ $ 5.65\times 10^{1} $ $ 2.03\times 10^{1} $ $ 4.71\times 10^{1} $ $ 1.66\times 10^{1} $ $ 1.33\times 10^{2} $ $ 4.34\times 10^{1} $ $ 9.19\times 10^{1} $ $ 2.63\times 10^{1} $
    8 $ 2.10\!\times\! 10^{1} $ $ 4.43\!\times\! 10^{\!-\!2} $ ${\bf{2.09\!\times\! 10^{1}}} $ ${\bf{5.62\!\times\! 10^{\!-\!2}}} $ ${\bf{2.09\!\times\! 10^{1}}} $ ${\bf{6.88\!\times\! 10^{\!-\!2}}} $ ${\bf{2.09\!\times\! 10^{1}}} $ ${\bf{5.15\!\times\! 10^{\!-\!2}}} $ $ 2.10\!\times\! 10^{1} $ $ 4.82\!\times\! 10^{\!-\!2} $ ${\bf{2.09\!\times\! 10^{1}}} $ ${\bf{7.85\!\times\! 10^{\!-\!2}}} $
    9 $ 3.91\times 10^{1} $ $ 1.37\times 10^{0} $ $ 2.29\times 10^{1} $ $ 6.38\times 10^{0} $ $ 2.41\times 10^{1} $ $ 4.10\times 10^{0} $ ${\bf{2.12\times 10^{1}}} $ ${\bf{7.99\times 10^{0}}} $ $ 3.19\times 10^{1} $ $ 3.48\times 10^{0} $ $ 2.48\times 10^{1} $ $ 4.89\times 10^{0} $
    10 $ 4.75\!\times\! 10^{\!-\!1} $ $ 2.53\!\times\! 10^{\!-\!1} $ $ 5.43\!\times\! 10^{\!-\!2} $ $ 2.97\!\times\! 10^{\!-\!2} $ $ 2.13\!\times\! 10^{\!-\!1} $ $ 9.54\!\times\! 10^{\!-\!2} $ $ 1.92\!\times\! 10^{0} $ $ 1.26\!\times\! 10^{0} $ $ 8.29\!\times\! 10^{\!-\!1} $ $ 8.42\!\times\! 10^{\!-\!2} $ ${\bf{4.73\!\times\! 10^{\!-\!2}}} $ ${\bf{3.44\!\times\! 10^{\!-\!2}}} $
    11 $ 1.67\times 10^{2} $ $ 4.78\times 10^{1} $ ${\bf{3.41\times 10^{1}}} $ ${\bf{2.23\times 10^{1}}} $ $ 8.61\times 10^{1} $ $ 3.02\times 10^{1} $ $ 1.59\times 10^{2} $ $ 2.01\times 10^{1} $ $ 4.22\times 10^{2} $ $ 9.26\times 10^{1} $ $ 1.05\times 10^{2} $ $ 3.43\times 10^{1} $
    12 $ 1.71\times 10^{2} $ $ 3.93\times 10^{1} $ ${\bf{2.64\times 10^{1}}} $ ${\bf{6.96\times 10^{0}}} $ $ 7.21\times 10^{1} $ $ 2.53\times 10^{1} $ $ 2.03\times 10^{2} $ $ 1.51\times 10^{1} $ $ 6.33\times 10^{2} $ $ 1.38\times 10^{2} $ $ 1.52\times 10^{2} $ $ 4.43\times 10^{1} $
    13 $ 1.87\times 10^{2} $ $ 1.58\times 10^{1} $ ${\bf{6.83\times 10^{1}}} $ ${\bf{1.40\times 10^{1}}} $ $ 1.39\times 10^{2} $ $ 3.02\times 10^{1} $ $ 2.06\times 10^{2} $ $ 1.51\times 10^{1} $ $ 4.51\times 10^{2} $ $ 7.45\times 10^{1} $ $ 2.36\times 10^{2} $ $ 6.06\times 10^{1} $
    14 $ 7.13\times 10^{3} $ $ 2.03\times 10^{2} $ $ 3.16\times 10^{3} $ $ 1.07\times 10^{3} $ $ 4.54\times 10^{3} $ $ 8.04\times 10^{2} $ $ 6.17\times 10^{3} $ $ 5.54\times 10^{2} $ $ 4.16\times 10^{3} $ $ 6.16\times 10^{2} $ ${\bf{2.97\times 10^{3}}} $ ${\bf{5.70\times 10^{2}}} $
    15 $ 7.32\times 10^{3} $ $ 2.59\times 10^{2} $ ${\bf{2.80\times 10^{3}}} $ ${\bf{6.30\times 10^{2}}} $ $ 4.45\times 10^{3} $ $ 6.60\times 10^{2} $ $ 7.25\times 10^{3} $ $ 3.79\times 10^{2} $ $ 4.13\times 10^{3} $ $ 5.61\times 10^{2} $ $ 3.81\times 10^{3} $ $ 5.03\times 10^{2} $
    16 $ 2.48\times 10^{0} $ $2.67\!\times\! 10^{\!-\!1}$ ${\bf{3.23\!\times\! 10^{\!-\!1} } }$ ${\bf{2.85\!\times\! 10^{\!-\!1} } }$ $ 1.88\times 10^{0} $ $ 3.94\!\times\! 10^{\!-\!1} $ $ 2.50\times 10^{0} $ $ 2.67\!\times\! 10^{\!-\!1} $ $ 5.92\!\times\! 10^{\!-\!1} $ $ 2.30\!\times\! 10^{\!-\!1} $ $ 4.97\!\times\! 10^{\!-\!1} $ $ 2.56\!\times\! 10^{\!-\!1} $
    17 $ 2.10\times 10^{2} $ $ 1.33\times 10^{1} $ ${\bf{5.94\times 10^{1}}} $ ${\bf{1.33\times 10^{1}}} $ $ 1.34\times 10^{2} $ $ 3.06\times 10^{1} $ $ 2.32\times 10^{2} $ $ 1.44\times 10^{1} $ $ 3.10\times 10^{2} $ $ 6.52\times 10^{1} $ $ 1.45\times 10^{2} $ $ 2.55\times 10^{1} $
    18 $ 2.09\times 10^{2} $ $ 1.21\times 10^{1} $ ${\bf{6.02\times 10^{1}}} $ ${\bf{2.00\times 10^{1}}} $ $ 1.38\times 10^{2} $ $ 2.48\times 10^{1} $ $ 2.38\times 10^{2} $ $ 1.64\times 10^{1} $ $ 1.75\times 10^{2} $ $ 3.81\times 10^{1} $ $ 1.75\times 10^{2} $ $ 4.92\times 10^{1} $
    19 $ 1.58\times 10^{1} $ $ 1.58\times 10^{0} $ ${\bf{3.01\times 10^{0}}} $ ${\bf{1.46\times 10^{0}}} $ $ 7.91\times 10^{0} $ $ 3.37\times 10^{0} $ $ 1.73\times 10^{1} $ $ 1.75\times 10^{0} $ $ 1.23\times 10^{1} $ $ 3.68\times 10^{0} $ $ 6.92\times 10^{0} $ $ 2.37\times 10^{0} $
    20 ${\bf{1.20\times 10^{1}}} $ ${\bf{4.32\!\times\! 10^{\!-\!1} } }$ $ 1.46\times 10^{1} $ $2.32\!\times\! 10^{\!-\!1}$ $ 1.31\times 10^{1} $ $ 1.91\times 10^{0} $ $ 1.25\times 10^{1} $ $2.55\!\times\! 10^{\!-\!1}$ $ 1.46\times 10^{1} $ $1.73\!\times\! 10^{\!-\!1}$ $ 1.30\times 10^{1} $ $9.72\!\times\! 10^{\!-\!1}$
    21 $ 3.38\times 10^{2} $ $ 7.88\times 10^{1} $ ${\bf{2.86\times 10^{2}}} $ ${\bf{7.01\times 10^{1}}} $ $ 3.46\times 10^{2} $ $ 8.31\times 10^{1} $ $ 3.00\times 10^{2} $ $ 6.99\times 10^{1} $ $ 3.24\times 10^{2} $ $ 9.67\times 10^{1} $ $ 3.16\times 10^{2} $ $ 9.33\times 10^{1} $
    22 $ 7.64\times 10^{3} $ $ 3.03\times 10^{2} $ ${\bf{3.12\times 10^{3}}} $ ${\bf{7.19\times 10^{2}}} $ $ 4.16\times 10^{3} $ $ 7.19\times 10^{2} $ $ 6.16\times 10^{3} $ $ 5.17\times 10^{2} $ $ 5.75\times 10^{3} $ $ 1.08\times 10^{3} $ $ 3.45\times 10^{3} $ $ 7.44\times 10^{2} $
    23 $ 7.51\times 10^{3} $ $ 3.35\times 10^{2} $ ${\bf{3.50\times 10^{3}}} $ ${\bf{5.83\times 10^{2}}} $ $ 4.52\times 10^{3} $ $ 8.56\times 10^{2} $ $ 7.30\times 10^{3} $ $ 2.83\times 10^{2} $ $ 5.74\times 10^{3} $ $ 7.59\times 10^{2} $ $ 4.70\times 10^{3} $ $ 8.98\times 10^{2} $
    24 $ 2.40\times 10^{2} $ $ 7.21\times 10^{0} $ ${\bf{2.20\times 10^{2}}} $ ${\bf{4.94\times 10^{0}}} $ $ 2.53\times 10^{2} $ $ 9.33\times 10^{0} $ $ 2.46\times 10^{2} $ $ 7.35\times 10^{0} $ $ 3.37\times 10^{2} $ $ 7.33\times 10^{1} $ $ 2.70\times 10^{2} $ $ 1.31\times 10^{1} $
    25 $ 3.14\times 10^{2} $ $ 3.34\times 10^{0} $ $ 2.73\times 10^{2} $ $ 8.64\times 10^{0} $ $ 2.81\times 10^{2} $ $ 6.78\times 10^{0} $ ${\bf{2.60\times 10^{2}}} $ ${\bf{6.08\times 10^{0}}} $ $ 3.56\times 10^{2} $ $ 2.80\times 10^{1} $ $ 2.99\times 10^{2} $ $ 1.24\times 10^{1} $
    26 $ 2.22\times 10^{2} $ $ 4.67\times 10^{1} $ ${\bf{2.03\times 10^{2}}} $ ${\bf{1.74\times 10^{1}}} $ $ 2.67\times 10^{2} $ $ 7.25\times 10^{1} $ $ 2.91\times 10^{2} $ $ 6.83\times 10^{1} $ $ 3.21\times 10^{2} $ $ 9.04\times 10^{1} $ $ 2.73\times 10^{2} $ $ 8.51\times 10^{1} $
    27 $ 7.73\times 10^{2} $ $ 2.03\times 10^{2} $ ${\bf{5.42\times 10^{2}}} $ ${\bf{3.92\times 10^{1}}} $ $ 8.10\times 10^{2} $ $ 1.11\times 10^{2} $ $ 7.47\times 10^{2} $ $ 1.26\times 10^{2} $ $ 1.28\times 10^{3} $ $ 1.10\times 10^{2} $ $ 9.72\times 10^{2} $ $ 1.33\times 10^{2} $
    28 $ 3.39\times 10^{2} $ $ 2.76\times 10^{2} $ ${\bf{3.00\times 10^{2}}} $ ${\bf{6.97\!\times\! 10^{\!-\!11} } }$ $ 4.29\times 10^{2} $ $ 5.27\times 10^{2} $ $ 3.64\times 10^{2} $ $ 2.59\times 10^{2} $ $ 4.34\times 10^{3} $ $ 2.08\times 10^{3} $ $ 4.37\times 10^{2} $ $ 4.67\times 10^{2} $
    下载: 导出CSV
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  • 收稿日期:  2020-04-24
  • 录用日期:  2020-09-22

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