Multi-scale Quantum Harmonic Oscillator Algorithm Based on Subpopulation Number Adaptive
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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.Abstract: The optimization algorithm of multiple subpopulations of sampling can be converted Monte Carlo evaluation function definite integral, because of the differences between different subpopulations in the overall evaluation, an adaptive strategy of subpopulation size (number of individuals) is proposed, which is used to improve multi-scale quantum harmonic oscillator optimization algorithm (MQHOA). The quantum properties of multi-population strategy and the correlation between quantum tunneling effect and optimization performance are discussed. The performance of existing optimization algorithm ignores the dynamic adjustment of subpopulation size influence on optimization ability, this strategy by dynamically adjusting population size, improve fitness poor child population incidence of quantum tunnel effect, increase the optimization ability of the algorithm. Compared the improved algorithm multi-scale quantum harmonic oscillator algorithm based on dynamic subpopulation (MQHOA-d) with MQHOA and other optimization algorithms on CEC2013 standard test functions. The experimental results show that the improved algorithm is more efficient in the optimization of basic multimodal functions and composition functions and the experimental results are better than several classical optimization algorithms and premature convergence problem has been solved.
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Key words:
- Optimization algorithm /
- quantum tunneling effect /
- dynamic population /
- multi-populations /
- Monte Carlo
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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测试集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 on CEC2013 test set (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 CEC2013测试函数在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 
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表 2 CEC2013测试函数维度为30情况下各算法51次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 $9.71\times 10^{-11}$ $1.44\times 10^{-11}$ $3.25\times10^{-13}$ $1.14\times 10^{-13}$ $2.36\times 10^{-13}$ $4.46\times 10^{-14}$ $2.59\times 10^{-13}$ $9.12\times 10^{-14}$ $7.82\times 10^{-2}$ $1.31\times 10^{-2}$ ${{{\boldsymbol{0.00}}{\boldsymbol{\times}} {\boldsymbol{10}}^{{\boldsymbol{0}}} } }$ ${{{\boldsymbol{0.00}}{\boldsymbol{\times}}{\boldsymbol{10}}^{{\boldsymbol{0}}} } }$ 2 $1.15\times 10^{6}$ $2.65\times 10^{5}$ $1.76\times 10^{6}$ $3.97\times 10^{5}$ ${{{\boldsymbol{9.68}}{\boldsymbol{\times}} {\boldsymbol{10}}^{{\boldsymbol{4}}} } }$ ${{{\boldsymbol{4.82}}\times {\boldsymbol{10}}^{{\boldsymbol{4}}} } }$ $1.62\times 10^{7}$ $7.01\times 10^{6}$ $5.43\times 10^{5}$ $2.04\times 10^{5}$ $8.92\times 10^{5}$ $3.92\times 10^{5}$ 3 $3.32\times 10^{7}$ $ 3.94\times 10^{7} $ ${{{\boldsymbol{3.60}}\times {\boldsymbol{10}}^{{\boldsymbol{6}}} } }$ ${{{\boldsymbol{4.19}}\times {\boldsymbol{10}}^{{\boldsymbol{6}}} } }$ $ 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} $ ${{{\boldsymbol{1.09}}{\boldsymbol{\times}} {\boldsymbol{10^{0}}}}} $ ${{{\boldsymbol{3.53}}{\boldsymbol{\times}} {\boldsymbol{10}}^{{\boldsymbol{-}}{\boldsymbol{1}}} } }$ $ 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} $ ${{{\boldsymbol{3.40}}{\boldsymbol{\times}} {\boldsymbol{10}}^{{\boldsymbol{-}}{\boldsymbol{7}}} } }$ ${{{\boldsymbol{3.94}}{\boldsymbol{\times}} {\boldsymbol{10}}^{{\boldsymbol{-}}{\boldsymbol{7}}} } }$ $ 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} $ ${{{\boldsymbol{1.70}}{\boldsymbol{\times}} {\boldsymbol{10}}^{{\boldsymbol{1}}} } }$ ${{{\boldsymbol{2.02}}{\boldsymbol{\times}} {\boldsymbol{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 ${{{\boldsymbol{2.07}}{\boldsymbol{\times}} {\boldsymbol{10^{1}}} } }$ ${{{\boldsymbol{1.26}}{\boldsymbol{\times}} {\boldsymbol{10^{1}}} } }$ $ 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} $ ${{{\boldsymbol{2.09}}{\boldsymbol{\times}} {\boldsymbol{10}}^{{\boldsymbol{1}}} } }$ ${{{\boldsymbol{5.62}}{\boldsymbol{\times}} {\boldsymbol{10}}^{{\boldsymbol{-}}{\boldsymbol{2}}} } }$ ${{{\boldsymbol{2.09}}{\boldsymbol{\times}} {\boldsymbol{10}}^{{\boldsymbol{1}}} } }$ ${{{\boldsymbol{6.88}}{\boldsymbol{\times}} {\boldsymbol{10}}^{{\boldsymbol{-}}{\boldsymbol{2}}} } }$ ${{{\boldsymbol{2.09}}{\boldsymbol{\times}} {\boldsymbol{10}}^{{\boldsymbol{1}}} } }$ ${{{\boldsymbol{5.15}}{\boldsymbol{\times}} {\boldsymbol{10}}^{{\boldsymbol{-}}{\boldsymbol{2}}} } }$ $ 2.10\times 10^{1} $ $ 4.82\times 10^{-2} $ ${{{\boldsymbol{2.09}}{\boldsymbol{\times}} {\boldsymbol{10}}^{{\boldsymbol{1}}} } }$ ${{{\boldsymbol{7.85}}{\boldsymbol{\times}} {\boldsymbol{10}}^{{\boldsymbol{-}}{\boldsymbol{2}}} } }$ 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} $ ${{{\boldsymbol{2.12}}{\boldsymbol{\times}} {\boldsymbol{10^{1}}} } }$ ${{{\boldsymbol{7.99}}{\boldsymbol{\times}} {\boldsymbol{10^{0}}} } }$ $ 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} $ ${{{\boldsymbol{4.73}}{\boldsymbol{\times}} {\boldsymbol{10}}^{{\boldsymbol{-}}{\boldsymbol{2}}} } }$ ${{{\boldsymbol{3.44}}{\boldsymbol{\times}} {\boldsymbol{10}}^{{\boldsymbol{-}}{\boldsymbol{2}}} } }$ 11 $ 1.67\times 10^{2} $ $ 4.78\times 10^{1} $ ${{{\boldsymbol{3.41}}{\boldsymbol{\times}} {\boldsymbol{10^{1}}} } }$ ${{{\boldsymbol{2.23}}{\boldsymbol{\times}} {\boldsymbol{10^{1}}} } }$ $ 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} $ ${{{\boldsymbol{2.64}}{\boldsymbol{\times}} {\boldsymbol{10^{1}}} } }$ ${{{\boldsymbol{6.96}}{\boldsymbol{\times}} {\boldsymbol{10^{0}}} } }$ $ 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} $ ${{{\boldsymbol{6.83}}{\boldsymbol{\times}} {\boldsymbol{10^{1}}} } }$ ${{{\boldsymbol{1.40}}{\boldsymbol{\times}} {\boldsymbol{10^{1}}} } }$ $ 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} $ ${{{\boldsymbol{2.97}}{\boldsymbol{\times}} {\boldsymbol{10^{3}}} } }$ ${{{\boldsymbol{5.70}}{\boldsymbol{\times}}{\boldsymbol{ 10^{2}}} } }$ 15 $ 7.32\times 10^{3} $ $ 2.59\times 10^{2} $ ${{{\boldsymbol{2.80}}{\boldsymbol{\times}} {\boldsymbol{10^{3}}} } }$ ${{{\boldsymbol{6.30}}{\boldsymbol{\times}} {\boldsymbol{10^{2}}} } }$ $ 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} $ $2.67\times 10^{-1}$ ${{{\boldsymbol{3.23}}{\boldsymbol{\times}} {\boldsymbol{10}}^{{\boldsymbol{-}}{\boldsymbol{1}}} } }$ ${{{\boldsymbol{2.85}}{\boldsymbol{\times}} {\boldsymbol{10}}^{{\boldsymbol{-}}{\boldsymbol{1}}} } }$ $ 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} $ ${{{\boldsymbol{5.94}}{\boldsymbol{\times}}{\boldsymbol{ 10^{1}}} } }$ ${{{\boldsymbol{1.33}}{\boldsymbol{\times}} {\boldsymbol{10^{1} }}} }$ $ 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} $ ${{{\boldsymbol{6.02}}{\boldsymbol{\times}} {\boldsymbol{10^{1}}} } }$ ${{{\boldsymbol{2.00}}{\boldsymbol{\times}}{\boldsymbol{ 10^{1}}} } }$ $ 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} $ ${{{\boldsymbol{3.01}}{\boldsymbol{\times}} {\boldsymbol{10^{0} }}} }$ ${{{\boldsymbol{1.46}}{\boldsymbol{\times}} {\boldsymbol{10^{0}}} } }$ $ 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 ${{{\boldsymbol{1.20}}{\boldsymbol{\times}} {\boldsymbol{10^{1}}} } }$ ${{{\boldsymbol{4.32}}{\boldsymbol{\times}} {\boldsymbol{10}}^{{\boldsymbol{-}}{\boldsymbol{1}}} } }$ $ 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} $ ${{{\boldsymbol{2.86}}{\boldsymbol{\times}} {\boldsymbol{10^{2}}} } }$ ${{{\boldsymbol{7.01}}{\boldsymbol{\times}}{\boldsymbol{ 10^{1}}} } }$ $ 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} $ ${{{\boldsymbol{3.12}}{\boldsymbol{\times}} {\boldsymbol{10^{3} }}} }$ ${{{\boldsymbol{7.19}}{\boldsymbol{\times}} {\boldsymbol{10^{2}}} } }$ $ 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} $ ${{{\boldsymbol{3.50}}{\boldsymbol{\times}} {\boldsymbol{10^{3}}} } }$ ${{{\boldsymbol{5.83}}{\boldsymbol{\times}} {\boldsymbol{10^{2}}} } }$ $ 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} $ ${{{\boldsymbol{2.20}}{\boldsymbol{\times}} {\boldsymbol{10^{2}}} } }$ ${{{\boldsymbol{4.94}}{\boldsymbol{\times}} {\boldsymbol{10^{0}}} } }$ $ 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} $ ${{{\boldsymbol{2.60}}{\boldsymbol{\times}} {\boldsymbol{10^{2}}} } }$ ${{{\boldsymbol{6.08}}{\boldsymbol{\times}} {\boldsymbol{10^{0}}} } }$ $ 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} $ ${{{\boldsymbol{2.03}}{\boldsymbol{\times }}{\boldsymbol{10^{2}}} } }$ ${{{\boldsymbol{1.74}}{\boldsymbol{\times}} {\boldsymbol{10^{1}}} } }$ $ 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} $ ${{{\boldsymbol{5.42}}{\boldsymbol{\times}} {\boldsymbol{10^{2}}} } }$ ${{{\boldsymbol{3.92}}{\boldsymbol{\times}} {\boldsymbol{10^{1}}} } }$ $ 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} $ ${{{\boldsymbol{3.00}}{\boldsymbol{\times}} {\boldsymbol{10^{2}}} } }$ ${{{\boldsymbol{6.97}}{\boldsymbol{\times}} {\boldsymbol{10}}^{{\boldsymbol{-}}{\boldsymbol{11}}} } }$ $ 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} $
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