Natural Computing Method Based on Nonlinear Dimension Reduction
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摘要:
随着人工智能的发展，许多优化问题发展为高维的大规模优化问题。在自然计算方法中，针对高维问题虽然能避免算法陷入局部最优，但是在收敛速度和时间可行性上却不占优势。该文在传统自然计算方法的基础上，提出了非线性降维的自然计算方法(NDR)，该策略不依赖具体的算法，具有普适性。该方法将初始化的N个个体看做一个N行D列的矩阵，然后对矩阵的列向量求最大线性无关组，从而减少矩阵的冗余度，达到降低维度的目的。在此过程中，由于剩余的任意列向量组均可由最大线性无关组表示，所以通过对最大线性无关组施加一个随机系数来维持种群的多样性和完整性。将该文所提策略分别应用到标准遗传算法(GA)和粒子群优化算法(PSO)中，并与标准粒子群算法、遗传算法以及目前主流的对维数进行优化的4个算法对比，实验证明，改进的算法对大部分标准测试函数都具有很强的全局收敛能力，其寻优能力超过了上述6个算法，同时改进后的算法在运行时间上远优于对比算法。
Abstract:Many optimization problems develop into high-dimensional large-scale optimization problems in the process of the development of artificial intelligence. Although the high-dimensional problem can avoid the algorithm falling into local optimum, it has no advantage in convergence speed and time feasibility. Therefore, the natural computing method for Nonlinear Dimension Reduction (NDR) is proposed. This strategy does not depend on specific algorithm and has universality. In this method, the initialized N individuals are regarded as a matrix of N rows and D columns, and then the maximum linear independent group is calculated for the column vector of the matrix, so as to reduce the redundancy of the matrix and reduce the dimension. In this process, since any remaining column vector group can be represented by the maximum linearly independent group, a random coefficient is applied to the maximum linearly independent group to maintain the diversity and integrity of the population. The standard genetic algorithm and particle swarm optimization using NDR strategy compare with Particle Swarm Optimization (PSO), Genetic Algorithm (GA) and the four mainstream algorithms for dimension optimization. Experiments show that the improved algorithm has strong global convergence ability and better time complexity for most standard test functions.
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Key words:
- Natural computing method /
- Optimization /
- Dimension reduction /
- Nonlinearity
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表 1 非线性降维的自然计算方法(NDR)
种群规模为N，终止进化代数为G，测试次数为T (1) 初始化N, G, T 等参数，随机产生第1代种群$\bf{pop}$； (2) 将生成的种群$\bf{pop}$做线性变换，求得列向量的最大线性无关
组，即为新的种群$\bf{newpop}$；(3) 对新种群$\bf{newpop}$的各维乘以随机系数${r_i}$，更新$\bf{newpop}$； (4) 将得到的$\bf{newpop}$使用基于种群的自然计算方法进化； (5) 结束。 
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表 2 标准测试函数
测试函数 维数 可行解空间 ${F_1} = \displaystyle\sum\nolimits_{i = 1}^D { {x_i}^2} $ 1000 [–100, 100]n ${F_2} = {\left( { {x_1} - 1} \right)^2} + {\displaystyle\sum\nolimits_{i = 1}^D {i\cdot \left( {2{x_i}^2 - {x_{i - 1} } } \right)} ^2}$ 1000 [–10, 10]n ${F_3} = \displaystyle\sum\nolimits_{i = 1}^D { { {\left| { {x_i} } \right|}^{i + 1} } }$ 1000 [–1, 1]n $\begin{array}{l}{F_4} = - a\cdot\exp \left( { - b\sqrt {\dfrac{1}{D}\displaystyle\sum\nolimits_{i = 1}^D { {x_i}^2} } } \right) - \exp \left( {\dfrac{1}{D}\displaystyle\sum\nolimits_{i = 1}^D {\cos \left( {c{x_i} } \right)} } \right) + a + \exp \left( 1 \right), a = 20,b = 0.2,c = 2\pi \end{array}$ 1000 [–32.768, 32.768]n ${F_5} = \displaystyle\sum\nolimits_{i = 1}^D {({x_i}^2 - 10 \cos (2 \pi {x_i})} + 10)$ 1000 [–5.12, 5.12]n ${F_6} = \displaystyle\sum\limits_{i = 1}^D {\frac{ { {x_i}^2} }{ {4000} } - \prod\limits_{i = 1}^D {\cos \left( {\frac{ { {x_i} } }{ {\sqrt i } } } \right)} } + 1$ 1000 [–600,600]n $\begin{array}{l}{F_7} = {\sin ^2}\left( {\pi {\omega _1} } \right) + {\displaystyle\sum\nolimits_{i = 1}^{D - 1} {\left( { {\omega _i} - 1} \right)} ^2}\left[ {1 + 10{ {\sin }^2}\left( {\pi {\omega _i} + 1} \right)} \right]\\ \quad\ \ + {\left( { {\omega _D} - 1} \right)^2}\left[ {1 + { {\sin }^2}\left( {2\pi {\omega _D} } \right)} \right]{\omega _i} = 1 + \dfrac{ { {\omega _i} - 1} }{4}\end{array}$ 1000 [–10, 10]n ${F_8} = 418.9829D - \displaystyle\sum\nolimits_{i = 1}^D { {x_i}\sin \left( {\sqrt {\left| { {x_i} } \right|} } \right)}$ 1000 [–500, 500]n ${F_9} = \displaystyle\sum\nolimits_{i = 1}^{D - 1} {\left[ {100{ {\left( { {x_{i + 1} } - x_i^2} \right)}^2} + { {\left( { {x_i} - 1} \right)}^2} } \right]} $ 1000 [–5, 10]n ${F_{10} } = \displaystyle\sum\nolimits_{i = 1}^{D/4} {\left[ { { {\left( { {x_{4i - 3} } + 10{x_{4i - 2} } } \right)}^2} + 5{ {\left( { {x_{4i - 1} } - {x_{4i} } } \right)}^2} + { {\left( { {x_{4i - 2} } - 2{x_{4i - 1} } } \right)}^4} + 10{ {\left( { {x_{4i - 3} } - {x_{4i} } } \right)}^4} } \right]} $ 1000 [–4, 5]n ${F_{11} }{\rm{ = } }\displaystyle\sum\nolimits_{i = 1}^D {ix_i^2} $ 1000 [–10, 10]n ${F_{12} } = \displaystyle\sum\nolimits_{i = 1}^D {x_i^2 + { {\left( {\displaystyle\sum\nolimits_{i = 1}^D {0.5i{x_i} } } \right)}^2} + { {\left( {\displaystyle\sum\nolimits_{i = 1}^D {0.5i{x_i} } } \right)}^4} } $ 1000 [–5, 10]n
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表 3 各算法对12个标准测试函数进行优化结果
函数 平均结果及标准方差 PSO NDRPSO GA NDRGA F1 4.92e+05±5.09e+04 345.2211±96.7891 1.78e+05±6.12e+03 12.7059±3.2681 F2 1.18e+08±2.47e+07 301.8188±114.0539 3.65e+07±1.85e+06 3.8379±2.8421 F3 6.71e-08±7.10e-08 6.48e-09±8.88e-09 1.11e-07±1.84e-07 3.17e-08±4.51e-08 F4 17.1198±0.5185 7.0752±0.7417 13.2492±0.1323 5.2058±0.5977 F5 9.72e+03±464.6314 115.6225±21.7166 9.68e+03±86.5390 104.5187±18.5246 F6 4.34e+03±366.1741 3.6991±0.7714 1.61e+03±64.0915 1.4019±0.1196 F7 1.66e+04±1.45e+03 13.0269±5.8130 8.62e+03±366.1671 0.4960±0.1577 F8 3.47e+05±7.50e+03 9.58e+03±866.1230 3.70e+05±2.42e+03 9.92e+03±619.0421 F9 9.20e+06±2.18e+06 694.6303±588.9850 3.48e+06±2.40e+05 103.1690±45.9190 F10 1.16e+05±1.50e+04 14.2205±5.1053 2.48e+04±1.50e+03 0.3623±0.2182 F11 2.82e+06±2.62e+05 80.1161±31.8298 7.96e+05±2.70e+04 2.8521±1.2240 F12 1.31e+04±3.41e+03 53.57±12.84 2.18e+19±3.03e+09 27.9986±12.1954
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表 4 各算法的实验对比结果
${F_1}$ ${F_3}$ ${F_4}$ ${F_5}$ ${F_6}$ ${F_8}$ DMS-CC best 13.2 5.88e+05 1.60e+14 2.30e+06 2.20e+04 2.00e+11 SLPSO best 4.78e-14 3.82e+06 1.90e+14 5.15e+09 9.42e+06 – DMS-PSO best 1.66e+01 1.14e+08 4.36e+07 1.68e+09 9.23e+04 4.51e+10 CSO best 2.43e-09 3.71e+06 4.61e+14 3.25e+09 9.68e+06 2.13e+11 NDRPSO best 6.54 1.11e-08 2.89 12.91 0.88 2.02e+03
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