图共 5个 表共 7
    • 图  1  冰晶连续优化模型收敛示意图

      Figure 1.  Convergence diagram of ice crystal continuous optimization model

    • 图  2  基于角度惩罚距离的偏差策略示意图

      Figure 2.  Deviation strategy based on angle penalty distance

    • 图  3  5种算法在8个30维基准函数上的演化曲线

      Figure 3.  Evolution curves of the five algorithms on eight benchmark functions with 30 dimensions as examples

    • 图  4  各算法的Friedman排名

      Figure 4.  Friedman ranks of each optimization algorithms

    • 图  5  加入概率更新策略前后算法的演化曲线

      Figure 5.  Evolution curves of the algorithm before and after joining probabilistic update atrategy

    • BenchmarkFunctionDRangeMinimum
      Sphere${f_1}(x) = \sum\limits_{i = 1}^D {x_i^2} $30,50,100[−100, 100]0
      SumSquares${f_{\rm{2}}}(x) = \sum\limits_{i = 1}^D {ix_i^2} $30,50,100[−10, 10]0
      Rosenbrock${f_3}(x) = \sum\limits_{i = 1}^{D - 1} {(100{{(x_i^2 - {x_{i + 1}})}^2} + {{(1 - {x_i})}^2})} $30,50,100[−30, 30]0
      Schwefel 2.22${f_{\rm{4}}}(x) = \sum\limits_{i = 1}^D {\left| {{x_i}} \right|} + \prod\limits_{i = 1}^D {\left| {{x_i}} \right|} $30,50,100[−10,10]0
      Rastrigin${f_5}(x) = \sum\limits_{i = 1}^D {(x_i^2 - 10\cos (2{\text{π}} {x_i}) + 10)}$30,50,100[−10,10]0
      Schwefel${f_6}(x) = \sum\limits_{i = 1}^n { - {x_i}\sin (\sqrt {\left| {{x_i}} \right|} )} $30,50,100[−500, 500]$ - 418.9829 \times D$
      Ackley$\begin{array}{l} {f_7}(x) = 20 + \exp - 20\exp ( - 0.2\sqrt {\frac{1}{D}\sum\limits_{i = 1}^D {x_i^2} } ) \\ - \exp (\frac{1}{D}\sum\limits_{i = 1}^D {\cos (2{\text{π}} {x_i})} ) \\ \end{array}$30,50,100[−32, 32]0
      Griewank${f_8}(x) = \frac{1}{{4\;000}}(\sum\limits_{i = 1}^D {x_i^2} ) - (\prod\limits_{i = 1}^D {\cos (\frac{{{x_i}}}{{\sqrt i }})} ) + 1$30,50,100[−600, 600]0
      F4${f_9} = 418.9829n + \sum\limits_{i = 1}^D { - {x_i}\sin (\sqrt {\left| {{x_i}} \right|} )} $30,50,100[−100, 100]0
      Step${f_{10}} = \sum\limits_{i = 1}^n {{{(\left\lfloor {{x_i} + 0.5} \right\rfloor )}^2}} $30,50,100[−100, 100]0
      Zakharov${f_{11}}(x) = \sum\limits_{i = 1}^n {x_i^2} + {(\sum\limits_{i = 1}^n {0.5i{x_i}} )^2} + {(\sum\limits_{i = 1}^n {0.5i{x_i}} )^4}$30,50,100[−5, 10]0
      Salomon${f_{ {\rm{12} } } } = - \cos (2{\text{π}} \sqrt {\sum\limits_{i = 1}^D {x_i^2} } ) + 0.1\sqrt {\sum\limits_{i = 1}^D {x_i^2 + 1} }$30,50,100[−100, 100]0

      表 1  基准测试函数表

      Table 1.  Benchmark function table

    • AlgorithmParameter setting
      RGAThe initial number of root tips is 1, and the number of branches is 3, the parameter $\alpha $ is a random number ranges from 0 to 1 and $\beta $ is a random number ranges from 5 to 10.
      GSOThe initial angle is ${\text{π}} /4$, the parameter $a$ is a random number ranges from 0 to $\sqrt {n + 1} $ and the maximum angle of pursuit is ${\text{π}} /{a^2}$.
      SSOThe threshold parameter $PF$ is 0.7.
      LSAThe channel time is set as 10.
      APD-CEOThe number of precipitated molecules is 80, parameter $\alpha ,\beta $ and $\gamma $ are random numbers range from 0 to 1. At the stage of energy change, the parameter ${\lambda _1}$=100 and ${\lambda _{\rm{2}}}$=0. At the regrowth stage, the parameter ${\lambda _1}$=0,${\lambda _{\rm{2}}}$=-100 and ${\lambda _{\rm{3}}}$=-50.

      表 2  各个算法的参数设置

      Table 2.  Parameter settings of each algorithms

    • FunctionIndexAPD-CEORGAGSOSSOLSA
      ${f_1}$ AB 8.537 0×10−99 5.774 4×10−8 4.468 5×10−5 1.965 3×10−3 3.213 8×10−30
      SD 2.103 9×10−98 6.487 1×10−9 4.260 9×10−2 9.960 1×10−4 4.628 5×10−31
      ${f_2}$ AB 6.140 2×10−100 3.709 5×10−5 0 4.446 5×10−4 2.471 5×10−20
      SD 1.020 3×10−99 1.942 6×10−6 0 2.901 9×10−4 5.815 3×10−21
      ${f_3}$ AB 1.898 8×10 2.307 2×10 8.004 9×10 1.148 3×102 1.164 8
      SD 1.387 1×10−2 1.339 6 2.428 8×10 3.942 8×10 2.381 8×10−1
      ${f_4}$ AB 2.261 8×10−78 5.417 7×10−3 5.608 7×10−1 1.371 0×10−2 4.395 1×10−20
      SD 1.590 6×10−78 1.775 3×10−4 7.340 9×10−2 3.117 9×10−3 3.781 9×10−21
      ${f_5}$ AB 1.117 7×102 1.694 2×10 2.113 6×10 8.594 8 2.167 8×10
      SD 9.919 2×10 8.539 8 3.233 8 1.114 3 4.032 6
      ${f_6}$ AB −3.099 1×103 −6.942 3×103 −7.224 ×102 −9.364 3×102 −9.211 5×103
      SD 4.732 0×102 8.684 9×102 1.343 5×10 1.614 9×10 4.085 6×102
      ${f_7}$ AB 4.440 2×10−16 1.864 9×10−8 8.660 6×10−4 1.360 5×10−2 2.614 8×10−10
      SD 4.440 8×10−16 2.200 8×10−9 3.669 0×10−5 2.361 2×10−3 4.258 4×10−11
      ${f_8}$ AB 0 7.394 7×10−6 6.422 9×10−1 3.294 0×10−3 2.806 5×10−16
      SD 0 1.493 2×10−7 7.920 5×10−2 5.490 1×10−4 1.398 6×10−17
      ${f_9}$ AB 1.207 5×104 7.112 9×102 1.184 7×104 6.789 3×103 1.892 5×102
      SD 1.429 0×102 1.259 8×102 1.535 8×10 1.013 2×103 3.948 6×10
      ${f_{10}}$ AB 0 5.263 7×10−42 0 2.689 2×10−3 4.754 5×10−16
      SD 0 2.253 2×10−43 0 6.059 8×10−4 9.843 5×10−17
      ${f_{11}}$ AB 5.188 2×10−98 6.542 3×10−5 2.322 9×10−6 6.810 3×10 8.549 8×10−7
      SD 1.275 4×10−97 2.168 3×10−5 1.532 5×10−6 3.008 4×10 7.924 8×10−8
      ${f_{12}}$ AB 1.873 6×101 3.326 3×10−1 5.800 2×10−1 2.740 9×10−1 3.405 5×10−1
      SD 1.490 9×10−2 1.965 3×10−2 6.310 2×10−2 5.177 5×10−2 4.540 5×10−2

      表 3  APD-CEO、RGA、GSO、SSO和LSA在基准函数上的实验结果(D=30)

      Table 3.  Experimental results of APD-CEO, RGA, GSO, SSO and LSA on benchmark functions (D=30)

    • FunctionIndexAPD-CEORGAGSOSSOLSA
      ${f_1}$ AB 9.789 0×10−99 2.907 4×10−6 1.451 3×10−4 1.642 4×10−1 5.671 5×10−16
      SD 2.274 5×10−98 4.790 8×10−7 8.946 4×10−5 3.218 4×10−2 2.114 8×10−17
      ${f_2}$ AB 9.781 3×10−99 3.213 2×10−3 3.801 3×10−3 4.768 4 8.484 3×10−9
      SD 1.964 0×10−98 2.998 6×10−3 1.167 8×10−2 1.445 6 3.481 6×10−10
      ${f_3}$ AB 4.898 5×10 1.945 2×102 1.723 7×102 7.349 5×10 5.538 4×10
      SD 1.836 3×10−2 1.567 8×10 1.057 4×10 2.011 9×10 2.859 9×10
      ${f_4}$ AB 1.038 5×10−77 1.515 3×10−2 2.357 4×10−1 2.034 6 3.915 8×10−11
      SD 7.593 2×10−78 1.205 4×10−3 6.328 4×10−2 1.722 4×10−1 2.191 5×10−12
      ${f_5}$ AB 9.054 4×10 5.224 6×10 1.008 5×102 7.945 6×10 5.804 5×10
      SD 9.782 3×10 6.951 3 1.531 5×10 1.435 9 4.439 5
      ${f_6}$ AB −4.130 6×103 −9.596 1×103 −1.204 1×103 −1.456 2×104 −1.195 3×104
      SD 7.035 4×102 4.265 4×103 8.132 4×10−6 1.624 0×103 2.178 5×103
      ${f_7}$ AB 4.235 2×10−14 2.137 9×10−4 5.187 4×10−2 3.384 6×10−1 1.486 2×10−5
      SD 1.846 3×10−15 6.644 5×10−5 7.165 8×10−3 3.023 4×10−2 2.018 9×10−6
      ${f_8}$ AB 0 8.214 4×10−5 6.226 1 2.488 3×10−2 7.415 8×10−8
      SD 0 2.237 8×10−6 4.502 9×10−2 7.085 4×10−3 2.784 6×10−9
      ${f_9}$ AB 2.029 6×104 1.895 6×104 1.974 5×104 6.794 5×103 1.075 4×103
      SD 1.562 9×102 2.349 5×103 3.001 8×102 1.332 4×103 2.008 1×102
      ${f_{10}}$ AB 0 3.387 9×10−26 1.764 1×10−7 0 4.675 1×10−13
      SD 0 3.246 4×10−27 5.161 5×10−8 0 1.456 8×10−14
      ${f_{11}}$ AB 5.149 1×10−98 3.913 4×10−3 1.387 6×10−2 6.214 7 9.198 7×10−4
      SD 1.950 8×10−97 3.240 0×10−4 1.135 4×10−3 3.684 5 4.528 3×10−4
      ${f_{12}}$ AB 1.772 1×10−1 6.791 3×10−1 9.299 9×10−1 4.802 5×10−1 5.219 8×10−1
      SD 4.023 2×10−2 4.362 5×10−2 3.154 6×10−2 4.004 6×10−2 4.159 8×10−2

      表 4  APD-CEO、RGA、GSO、SSO和LSA在基准函数上的实验结果(D=50)

      Table 4.  Experimental results of APD-CEO, RGA, GSO, SSO and LSA on benchmark functions (D=50)

    • FunctionIndexAPD-CEORGAGSOSSOLSA
      ${f_1}$ AB 3.474 3×10−98 1.974 2×10−1 3.093 2×10−1 2.244 2 5.238 4×10−6
      SD 1.058 6×10−97 2.463 2×10−2 3.161 8×10−2 2.943 2 4.187 8×10−6
      ${f_2}$ AB 2.412 1×10−99 5.427 9 4.606 4×10 1.932 4×10 2.018 6×10−6
      SD 7.604 7×10−99 2.379 2 1.851 7 1.103 7 1.268 7×10−7
      ${f_3}$ AB 9.898 7×10 3.715 3×102 7.898 4×102 3.352 3×102 1.608 9×102
      SD 1.105 7×10−2 4.174 2×10 3.681 6 1.574 8×102 2.168 6×10
      ${f_4}$ AB 3.924 3×10−77 4.224 3 5.610 5 8.843 2 2.678 2×10−4
      SD 2.343 3×10−77 2.197 8×10−1 3.167 5 7.231 5×10−1 1.925 8×10−5
      ${f_5}$ AB 1.233 6×102 1.398 8×102 4.231 2×102 2.385 7×102 9.108 6×10
      SD 1.014 4×102 1.747 9×10 3.123 1×10 8.544 2×10 3.371 7
      ${f_6}$ AB −5.779 8×103 −1.668 3×104 −2.407 9×103 −2.895 3×102 −2.256 7×104
      SD 8.555 7×102 4.324 6×103 3.131 8 4.084 6×103 1.861 7×103
      ${f_7}$ AB 7.440 8×10−13 2.479 8 4.920 5 1.571 2 2.946 8×10−1
      SD 1.845 3×10−14 4.254 2×10−1 8.921 6×10−1 8.271 2×10−1 2.061 8×10−2
      ${f_8}$ AB 0 1.717 6×10−3 1.236 4×102 2.184 7×10−2 4.821 6×10−3
      SD 0 2.839 7×10−4 0.598 4×10−2 4.290 3×10−3 1.815 8×10−4
      ${f_9}$ AB 4.098 8×104 8.214 3×104 3.949 1×104 1.618 2×104 9.149 5×103
      SD 1.787 3×102 4.512 3×103 1.468 4 1.347 8×103 1.128 4×103
      ${f_{10}}$ AB 0 5.263 4×10 1.715 6×10 2.108 3 1.042 6
      SD 0 2.637 4×10 3.187 9×10−3 1.589 3 2.361 8
      ${f_{11}}$ AB 3.292 8×10−99 8.324 5×10 3.384 8×10 6.199 5×10 5.124 8
      SD 1.119 7×10−98 4.386 6 6.165 7 7.104 7×10 1.801 6×10−4
      ${f_{12}}$ AB 1.708 5×10−1 9.180 6×10−1 2.299 9 1.081 0 8.317 6×10−1
      SD 4.498 7×10−2 2.479 2×10−2 1.264 8×10−1 1.479 3×10−1 1.286 4×10−2

      表 5  APD-CEO、RGA、GSO、SSO和LSA在基准函数上的实验结果(D=100)

      Table 5.  Experimental results of APD-CEO, RGA, GSO, SSO and LSA on benchmark functions (D=100)

    • DAPD-CEORGAGSOSSOLSA
      302.1252.923.753.712.25
      501.8753.174.423.462.08
      1001.503.504.423.751.83

      表 6  各优化算法在30、50和100维下的平均排名

      Table 6.  Average ranking of each optimization algorithms

    • FunctionIndexBeforeAfter
      Sphere AB 9.435 3×10−29 5.353 5×10−83
      SD 1.469 7×10−44 1.666 1×10−84
      SumSquares AB 1.736 2×10−28 5.840 1×10−82
      SD 1.617 9×10−28 1.639 2×10−81
      Rastrigin AB 7.194 3×102 7.117 7×102
      SD 4.239 53×10 1.919 2×10
      Ackley AB 1.234 9×10−5 7.372 3×10−14
      SD 5.047 8×10−9 9.832 7×10−15

      表 7  角度惩罚距离策略的有效性

      Table 7.  Effectiveness of angle penalty distance strategy