基于双学习策略的金鹰优化算法

一、理论基础

1、金鹰优化算法

请参考这里。

2、基于双学习策略的金鹰优化算法

原始的GEO算法无法在探索阶段和开发阶段之间实现很好的平衡。为了提高GEO的搜索性能，该算法通过引入个体示例学习策略和镜面反射学习策略来改善这种平衡。

（1）个体示例学习

为了使GEO的搜索朝着一个好的方向发展，同时提高GEO的搜索能力，减少GEO陷入局部最优的可能性，将个体示例学习应用于GEO。首先，种群中的个体根据其适合度按升序进行排序。然后，个体$X_i$可以从其示例库中学习，个体$X_i$的示例库由排序在其前的个体和自身组成。对于已排序的种群，个体$X_i$的示例可以是个体$X_b(1\le b\le i)$ 。从由个体最佳位置组成的示例池中选择第$m$个个体的示例值，该示例值用于更新攻击向量，如下式所示：$$\overline{X_p^{\ast }}=X_p^{\ast }(m)\text{\hspace{1em}(1)}$$$$m=ceil(i\ast rand)\text{\hspace{1em}(2)}$$$$A_i=\overline{X_p^{\ast }}-X_i\text{\hspace{1em}(3)}$$其中，$X_p^{\ast }(m)$是由个体最佳位置组成的示例库中第$m$个个体的示例值，$ceil$是向上取整操作符，$X_i$表示第$i$个个体的位置。

（2）镜面反射学习

镜面反射学习的灵感来自光线的反射。根据反射定律，入射角$\alpha$应等于反射角$\beta$。假设实数的范围是$[Z_L,Z_U]$，$o=(x_0,0)$是$Z_L$和$Z_U$的中点，$x(a,0)$是$[Z_L,Z_U]$中的一个随机变量。$x$的反向点是$\breve{x}=(b,0)$。可以推导出以下公式：$$tan(\alpha )=\frac{x_0-a}{A_0}\text{\hspace{1em}(4)}$$$$tan(\beta )=\frac{b-x_0}{B_0}\text{\hspace{1em}(5)}$$由于$\alpha =\beta$ $$\frac{x_0-a}{A_0}=\frac{b-x_0}{B_0}\to b=\frac{B_0(x_0-a)}{A_0}+x_0\text{\hspace{1em}(6)}$$假设$B_0=\lambda A_0$，其中$\lambda$是实数且$\lambda >0$，则式(5)变为：$$b=(\lambda +1)x_0-\lambda a=(0.5\lambda +0.5)\times (Z_L+Z_U)-\lambda a\text{\hspace{1em}(7)}$$$b$的值如下所示：$$b=\{\begin{array}{l}{b}_1,0<\lambda <1\\ 2x_0-a,\lambda =1\\ b_2,\lambda >1\end{array}\text{\hspace{1em}(8)}$$其中，$b_1\in (x_0,2x_0-a),b_2\in (2x0-a,Z_U]$。当$\lambda =1$时，镜面反射学习变成了反向学习。在镜面反射学习中，$\lambda$由式(9)计算所得。$$\lambda =\{\begin{array}{l}1+\mu Q,if{r}_1>r_2\\ 1-\mu Q,otherwise\end{array}\text{\hspace{1em}(9)}$$其中，$r_1\in [0,1],r_2\in [0,1]$，$Q$为邻域半径且$Q\in [0,1]$，$\mu$为弹性因子且$\mu \in [0,1]$。
根据镜面反射学习的原理，$x$的反向点$\breve{x}$可通过式(10)计算。$$\breve{x}=(0.5\lambda +0.5)\times (Z_L+Z_U)-\lambda x\text{\hspace{1em}(10)}$$扩展到$D$维空间，$\breve{x}$可进一步写成：$${\breve{x}}_j=(0.5\lambda +0.5)\times (Z_{L,j}+Z_{U,j})-\lambda x_j,j=1,2,\cdots ,D\text{\hspace{1em}(11)}$$当镜面反射学习与元启发式算法相结合时，式(11)变为：$${\breve{x}}_j=(0.5\lambda +0.5)\times (L_j+U_j)-\lambda x_j,j=1,2,\cdots ,D\text{\hspace{1em}(12)}$$其中$L$和$U$分别表示当前种群的下限和上限。

（3）GEO-DLS算法伪代码

基于双学习策略的金鹰优化算法(Golden eagle optimizer with double learning strategies, GEO-DLS)的伪代码如图1所示。

图1 GEO-DLS算法伪代码

二、仿真实验与结果分析

将GEO-DLS与GEO、SCA、WOA和GWO进行对比，以CEC2013测试函数中的F3、F4(单峰函数/50维)、F14、F15(基本多峰函数/50维)、F21、F22(组合函数/50维)为例，实验设置种群规模为100，最大迭代次数为1000，每种算法独立运算1次，结果显示如下：

函数：F3
GEO-LDS：
最优位置: -24.9654      12.6448     -35.7234       53.244     -37.5686     -66.4059      61.3932      13.0567      68.2205     -23.9786      29.0946     -61.5048      -33.157      58.6511     -32.9821      72.3984       11.961       22.525      16.1373      28.5763     -48.2454      59.3854     -34.7854      -17.736      35.7556     -59.5842      39.8611     -20.2013     0.917959      -37.772      4.12143     -26.9626      15.1894     -13.6431     -39.8347      -57.266      23.0536     -70.9291      26.6816     -33.8137     -7.72825      13.3373      39.0934     -22.8424      14.6808     -26.0806      53.6524      12.6754      48.4223      6.46129
最优值: 105504531.1876
GEO：
最优位置: -21.2541      -16.844     -34.3166      71.3886     -37.0839      -46.769      50.7249      34.1737      9.96732     -16.5573      24.0602     -52.5262     -45.9411      17.3793     -39.3226      63.1496      12.0786      7.16327      13.4801      20.1288     -43.4937      23.9916     -43.2928     -22.4838      53.4138     -57.9946      53.3422     -30.5392     -15.0249     -28.8239      4.40009     -30.4774      19.1188     -11.7891     -34.8427     -37.9174      14.7838     -72.4515      25.6093     -36.5018     -3.18557      24.8825      13.1796     -23.1552      17.4279     -19.6362      47.3099      10.5145      44.4658     -2.96466
最优值: 1445338616.2294
SCA：
最优位置: -9.01011     -5.14092     -15.5439    -0.143003     -33.8611     -91.9869      12.8054      53.9635     -2.84506      -2.5386      13.3276       -55.09      -85.487     -97.8608      53.7554     -1.68432      58.8446     -70.1761     -20.8393      15.6037   -0.0631428     -6.79828     -12.9812     -46.7962      10.4439     -0.41067     -4.83858     -91.5175      21.8579      87.4853     -39.8564       -39.83      99.9868       22.419      78.5117      1.47608     -36.9591     0.838661      62.9112      12.1197      38.9095      37.3307     -37.7313     -19.0909      57.6209     -14.7255      28.3358      1.06669     -28.7306     0.184916
最优值: 146973082742.3995
WOA：
最优位置: -23.9319       -44.55     -60.5019      46.7112      18.4595     -28.0935     -6.19558      45.9326     -15.8318     -6.76327     -9.90721      1.59307       3.2329      15.4912     -16.9968      24.8921      57.6987     -6.84154      40.9176    0.0181429      2.93346      53.4505     -14.8396      9.98855     -6.99573     -17.2847      1.85275     -64.7607      22.8639      80.9881     -5.10822    -0.253368      27.5616    0.0409732     -11.6378      3.02627      69.7633      2.85353      34.3147     -5.17079      46.3308     -7.96881    0.0729811      10.4416     -12.6185     -11.9212      26.7366      40.1911     -2.75978     -52.6067
最优值: 49831000846.282
GWO：
最优位置: 0.111781     -1.33574      -31.724      73.6177      -9.4719     -63.1807      58.5847    -0.480348      67.5049     0.326354     -1.82868     -77.3955     -48.6386  -0.00840821     -27.3721      71.1694      37.7549     -22.9648      13.6048     0.704124     -51.3088     0.101907     -51.3873      -31.493      54.4823     -72.6782     -1.74251     -34.4263      3.01756      2.12714    -0.197568     -35.7089      98.4433     -4.10652      -26.762      -61.797     -1.25839      1.15323      6.74409     -25.9864    -0.230747      11.3864    -0.583174     -22.7812     -1.52263     -11.9283      39.6554     0.558738      2.95716       1.6103
最优值: 12957625876.334
函数：F4
GEO-LDS：
最优位置: -16.9871     -12.6528     -33.5142      28.3142     -15.3142     -35.3627      50.9148      16.4033     -3.10312      15.6761      14.7388     -24.8143     -39.9226      33.2401     -45.6268       39.758     -14.9935      28.8142     -12.1929      13.1928     -46.3572      36.6443     -41.7771      -18.444      12.0819     -62.1039      52.3059     -5.13592     -7.41616     -13.7184      15.7428     -11.8604     0.814566      6.81229      -8.9405     -23.9778      50.3683        -43.3      20.4801     -50.1701      13.5922     -19.0492      14.8015     -26.3194      12.8147     -19.8972      42.9233      -11.182      32.2174     -10.7173
最优值: 26260.0241
GEO：
最优位置: 17.9344     -5.73736     -38.2929      28.2461    -0.217143     -20.7725     -2.64616     -12.6552      27.8482      4.44801     -9.21133     -15.1313     -36.3787      28.8448     -23.1501       29.943      -10.804      38.8576     -4.68706      7.95805      5.67935      30.9357     -12.2972     -32.3154      31.8692     -26.0175       34.077       30.093     -22.3606     -6.72367     -1.36788     -24.1913      10.5783     -6.13032     -30.2361     -31.9242      3.74647     -66.0667      7.37713      8.81946     -8.64628       28.873      18.2899     -35.0934      13.9509     -1.21826     -22.9428      22.0084      19.1501      4.69819
最优值: 47615.148
SCA：
最优位置: 32.4145     0.135176     -1.91518     0.152868     -13.9303      -33.789       94.992    -0.170845      9.75761     -67.1922      79.9475     -59.0665      27.8023      13.3718     -8.66212      48.6082      2.23302     -29.2243    0.0193175     -8.13689     -14.3242      54.9899     -6.73332    0.0529294      99.6174     -72.2007      6.28209      18.4067     -22.6786     -15.0283      37.9383     0.432471      6.61393     -8.05875     -20.0565      19.8547      4.37133     -42.5954   -0.0334405     -3.40251      6.88389      62.2091     0.172499      11.2198     -48.6923      -67.423    0.0211212     -12.6448      12.6342      29.2959
最优值: 79525.8485
WOA：
最优位置: -12.6378      19.9852     -4.62757      3.59142     -6.46627      -31.574      4.70859      1.53809     -13.6483     0.232594       18.055      -25.147     -2.65681      21.1192     -38.5041      28.7567    0.0453173       18.099     -19.2165     0.282751      3.06679       6.9108     -33.1866     0.761728      2.79618      6.13306      55.1233     -8.32065    -0.670031      -14.593     0.214735      8.92709     0.272594      -50.363      15.3282     -18.0643     -10.9522     -36.5814     -22.6927     -35.7718     -6.65932      2.67766      45.7359     -19.3691       1.5423      21.7817      36.1599      18.6607      12.4744      19.6498
最优值: 65693.5784
GWO：
最优位置: -27.7834     -5.24425      -35.293      64.4224     -32.5214        1.077      48.2816      19.4635    -0.557019     0.257393      1.52473     0.192639     0.851719       67.459     -57.3794     -0.13128    0.0110039     -1.30238     -2.40503    -0.398193      2.11672     0.191146      1.07182     -14.1447      7.01075     -77.2355     0.197914    -0.807279   0.00823467     -33.9185     -3.95848    -0.773544     -6.48287       2.7642    -0.112889      -61.311       3.1723     -76.9479    -0.325617      -35.783    0.0573579     0.399328   -0.0480134      2.29034      17.7053  -0.00594777      44.9558    0.0412872      55.2416      -1.7177
最优值: 48198.7315
函数：F14
GEO-LDS：
最优位置: -43.7351      27.0136     -56.7333      30.6321      -3.4445       9.9693     -43.3637      13.7246       24.607      75.1036      81.3301     -19.3758     -2.10495       31.526     -9.73582      36.4996      7.55315      45.8211      26.0758      8.91723     -49.3405      21.7165     -52.6656     -26.0443       41.684      14.6784      34.9787      43.6951     -11.5749     -9.29551      14.1007     -20.8437      20.2867     -13.0295      27.6306      11.9496     -32.7651      98.4918      1.33309      13.0363      80.4435      15.0453      33.3567      19.6106     -7.84032      23.1032     -13.2641     0.265775     -21.3092      -11.554
最优值: 5032.4492
GEO：
最优位置: -3.76906      30.2953     -24.1555     0.625027      21.8598     -30.0569      14.9426      9.92074     -49.3269    -0.346503    0.0501711      8.71941     -1.04017     -22.4421      27.1301      21.9484      20.2562      71.6476      26.7301      40.8427     -8.02334        23.72     -9.84784      12.0766     -10.7027      4.04017      -6.4356      17.7153     -1.81598      37.2728     -25.2327      58.5315     -10.0652     -12.0243     -8.26211      21.5521     -2.22174     -10.8748      46.2582      10.8785     -5.66359       22.628     -7.66075     -17.9555     -28.2952     -47.4989     -23.1912      30.5568      9.94708     -95.4552
最优值: 11965.894
SCA：
最优位置: -69.24462          -100      74.50064          -100      10.60458           100      -6.85089      32.18563      94.28918     -33.98307      11.63641           100      74.91779      -13.2907       37.5248      27.99677      -3.82076      10.74292      42.61168      38.74201     -12.20832     -45.54497      30.36606      14.69586     -8.485569           100      26.23182     -17.79896    -0.7361026      57.33561     -17.86026          -100     -93.81027     0.9810884     -32.58549      -22.6844      -58.9952      70.93067      45.39334      91.92309       -73.977     -1.763877     -17.46874     -98.95774     -2.038037     -98.10532      24.31561           100      46.75155      16.64608
最优值: 14141.5566
WOA：
最优位置: -80.5277     -12.2657     -16.5304     -79.4314      24.1014      53.1003      67.9514     -36.4269      85.7727     -5.01952     -84.0117     -15.0635     -52.8853     -81.3078      13.3535     -53.3007      53.8215      14.4238      93.0922      18.1309     -9.39963      51.9132      36.5194      40.3329     -55.2527     -1.64794      90.3879     -46.5522     -35.5632     -99.1939       46.286     -29.1767     -39.7799      2.94062     -18.4251      20.9258     -53.8467     -68.5768     -68.2482      13.6861      60.6555      96.9366      50.5592     -1.88594      54.8516      -80.782     -75.0975      1.24588      63.6357      55.4788
最优值: 10563.9272
GWO：
最优位置: -22.0808      7.35851     -20.5511      18.3915       21.284    0.0700074      6.36936     -47.5503     -23.0613     -4.58269     -59.0224     -19.6178   -0.0951267     -9.67598      98.2822      23.6001      52.6457      70.9322      91.6015     -51.1964       85.391      27.7096      16.4036     -3.51308     -38.0555     -13.2801      3.29112     -76.9961      22.4085      -23.733      14.0629     -20.9061      12.7199      17.3659      5.51834      99.7334      6.12606      23.6832     -14.9005     -12.6903      2.09288      29.3521     0.157768      9.07919       -1.499     -19.1535     -13.0936      99.7174      23.8007      2.78367
最优值: 5962.238
函数：F15
GEO-LDS：
最优位置: 21.5339      99.9889    -0.847176      17.6204      -18.757      9.65648      36.1828      14.2402     -21.9435      6.59367     -6.31285     -29.8018      46.4143      50.2329     -31.9685      16.2408     -41.2883      54.7278      -21.947     -21.0795      35.2275      12.0997      73.3549      50.1766     -52.9359     -11.3798     -45.6433      22.4486      45.9601     -35.3785     -3.47906      8.22706      16.2492      6.55272     -52.3316      40.2246      28.7817     -4.08364      21.6207      21.1035     -88.5771     -50.5235      21.0818     -59.9047     -33.7641     -58.5505      33.4972      4.18717      60.9326      90.0145
最优值: 7516.1439
GEO：
最优位置: -3.41323      27.5561      -10.029      48.4933      24.2686    -0.825723     -14.9393      5.34725      12.2427      34.6006      31.3388    -0.936982      22.7579      44.1764      -27.862     -21.2499     -35.1294       20.632       12.216      7.62839       -9.084      12.9661     -19.4618      6.99102     -16.1714      3.84809      10.7528      51.0482      11.7863     -28.6826      -25.691     -2.86375     -17.9675     -44.8348     -22.5039     -22.0656       11.023      37.3985       23.371      -2.5542     -25.3349     -21.6416     -21.0466      3.04622      22.6017     -9.31363      3.77909     -8.21748      9.39909     -6.62638
最优值: 8229.6264
SCA：
最优位置: 93.939     -5.41163      33.9303     -67.5785    0.0253671      4.95402      88.8756      44.5899     -86.0073      37.6093      64.7361        58.98      1.63393    0.0640917     -76.0167      49.7673     -28.8294     -9.61742     -23.0069      58.6043      26.2929      22.3383      94.2308       68.503      73.5475     -43.0836      63.4714      5.29156     -29.9537     -27.9008     -40.2191      18.1633      99.8265      80.5964     -66.7173     -65.4555      13.1626     -18.4144     0.491043     -99.3127     -48.6242      50.5542     -76.9295    -0.726453     -21.0906     -38.9715     -0.52737     -57.8117     -62.4769      34.5954
最优值: 15583.6223
WOA：
最优位置: -1.73528     -45.5971     -69.3559     -56.4238      47.0161    -0.643685     -10.2287       19.303      67.5535     -5.77306     -76.4076     -82.0591      52.6012      89.8877      -74.092       17.341      91.5299      24.1324     -63.6004     -53.9604     -75.1564      88.5743     -52.9166      97.9548       6.6836     -11.2572      63.1698      75.0188     -89.8159      -22.839      99.7016     -53.6446     -41.3976     -55.1454      40.2558      98.5282       44.359     -30.4721     -85.1322       51.807     -16.8975      98.7756      30.4766      50.1836     -90.4629     -75.7297      62.6728      8.46596      9.53325     -16.8181
最优值: 12530.1724
GWO：
最优位置: 9.93431     -87.6679     -71.3577     -23.4231      6.00532     -52.2604      99.7194      20.0658      38.6318      71.2373     -13.9751      1.02715      26.3028      10.6711     -99.9162     -95.6555      44.1449      69.5486     -38.8808      56.3182      60.0796     -11.2899      12.4873      48.4252      92.3832      11.4973     -10.9312      4.63922     -31.9895      70.3222      46.8481     -16.8736     -34.9169     -4.71674      10.9703      61.5062      20.9681     -88.8163      66.4723      50.5689     -61.9262     -45.5958      20.8048     -12.2781     -41.4722      68.9329     -87.9526      68.1865     -30.9816      28.7123
最优值: 6384.1745
函数：F21
GEO-LDS：
最优位置: 43.6057     -48.0131      -52.493      20.9374     -13.8264      57.9169      44.5924      57.4724     -39.3128     -50.7102     -64.8045     -21.3885       8.7491     -11.5378     -14.9631     -2.68755      62.0057      61.0645      43.9932      63.5178     -49.3715       53.688     -24.1174      46.1882      12.7909      22.3134         29.1     -34.6654     -3.38658     -21.9525     -22.1739     -1.69416     -59.5238     -23.5768     -44.7196      30.2115      2.54926      31.6001      17.6825      3.63489      47.1061      63.3675     -60.8181     -72.3612       62.476      19.0393      43.8664      -19.297      66.6345      38.2773
最优值: 1822.1858
GEO：
最优位置: 43.6458     -47.8909     -52.5481      20.8878     -13.8172      57.8967      44.5939         57.6     -39.3012     -50.6601     -64.7526     -21.4299      8.72982     -11.5937     -14.8841     -2.76873      61.9259       60.981      43.8654      63.5283     -49.3411      53.6819     -24.1384      46.0746      12.7018      22.3067       29.162     -34.6664     -3.51927     -21.8788     -22.1142     -1.57956     -59.4292     -23.4688      -44.655      30.2137      2.40605      31.5368      17.6717      3.66099      47.0637      63.1996     -60.7518     -72.2683      62.4217      19.0822      43.8082     -19.2947      66.6666      38.1621
最优值: 1822.2729
SCA：
最优位置: 13.8781     -0.17216     -24.1798     -22.0382    -0.850722     0.056863     -7.97552     -0.56194     -8.96246     -21.7527     -21.0047 -2.78121e-05    0.0540387     -25.4635      8.77992      6.54084      26.3454    -0.238422      2.65756      10.2358     -6.55224      -3.5936       5.3155      38.4363     -19.3846      4.70705    -0.384079      5.84609     0.139506     -6.69332     -19.9177      9.42012      -40.665    -0.310301    -0.543802     0.520411     -6.02138      32.0765      10.2908      -16.046      63.4329      60.9088     0.967684     -39.7719      47.9571     -7.51832      11.1701     -1.41002      12.2865      19.3073
最优值: 4863.9844
WOA：
最优位置: 38.7403     -50.0555     -49.9969      5.49994     -7.48045       61.763      40.8886      53.7572     -39.7599     -51.6194     -64.1895     -12.0763      15.3103     -1.72339     0.575032      8.40034     0.824714      47.1408      41.1214      57.6658     -47.2859      54.0322     -27.0857      55.7058      8.44923      27.6781      22.9386     -27.8682      6.01298     -19.9348     -28.8917     -3.65185      -56.089     -14.5869     -39.7287      25.5658     -17.5431      26.0572      20.2151     -4.09432      37.7633      70.3212     -52.2899      -61.194      66.0455    -0.101958      44.6036    -0.633496      62.1184      42.0982
最优值: 2875.6355
GWO：
最优位置: 52.9487       35.162      56.4676     -55.6298     0.460488     -47.8437      0.43581      31.9911      5.64976      35.6211       63.646    -0.714823   -0.0977563     -44.9748      1.64882      29.9157     -45.9224     -67.3347     -71.1511       39.812      77.9031     0.471097      51.4396     0.152626     -74.5342       44.901      45.2911    0.0278234     -58.6451     -5.53916     -70.1457      66.1085      27.3827     -57.7642      38.6609      56.3439     -39.3532    -0.304049     0.944901    -0.580236     0.411809      44.7773      74.4617      55.2389      55.5212      63.4795      -57.871      24.9811      55.4366      31.2059
最优值: 1869.1088
函数：F22
GEO-LDS：
最优位置: 15.5189      11.6827     -20.9062      84.3389       22.447      9.85011       6.3882      45.5208      40.3509     -18.7546      12.1415     -19.5814     -39.7479        83.47     -9.80026      24.2371      18.3873      45.8629      15.8638       31.626      22.8335     -30.0932     -13.5704     -68.2549      41.7757     -40.9191      64.5707      17.1918      22.5789      16.0398      22.0077     -20.8643     -8.25849      10.6129     -27.1063     -10.1405      74.3277       65.597      37.4304     -7.02302      41.3536      1.47917      8.81101      -16.799     -7.87821     -1.83881      29.1547      24.0285      11.1146     -20.3124
最优值: 6540.0547
GEO：
最优位置: -21.88263      27.51143     -20.42188      31.31235     -3.886591     -49.15006      35.18544      45.64519           100     -18.76029      12.12445      19.21227     -1.913582      55.77675       -45.515     -4.053829       18.4935      60.43215      15.57595      17.62849     -63.21298      64.88478     -73.94595     -16.91573      54.10505      -41.1038      52.99973     -9.053922     -11.53739      15.95159     -12.81138      21.30118      12.89003      27.34889     -27.09933     -22.39351      13.20826     -6.395092      37.40773     -6.984376      2.270901      14.97481      41.40158     -22.59984         59.15     -11.62591       53.0807      16.83777      52.28236      23.37623
最优值: 5862.6682
SCA：
最优位置: 0.0001407738      82.04954      85.85223      82.71172     -60.21952          -100     -46.40856     -30.97981      11.84077     -13.21566     -28.57523     -24.44247     -1.450442       4.70007        30.639     -25.41826       87.7662      14.75946      62.70316     -29.25474     -22.26571     -19.35276      28.67689      15.14577       89.0869     -77.08639      67.16187      1.927299     -11.86775     -1.088279      6.972082     -64.53798     -94.22216      17.29005     -3.829649     -35.95513      21.17044     -18.59632      42.19403     -83.77436     -49.56271     -45.63128     -11.92292     -99.90383      23.57444     -17.72157      48.36624     -6.363874     -5.276323      87.75783
最优值: 15848.7106
WOA：
最优位置: 43.434      99.4349       24.831      68.9171     -26.4633      99.3151      71.6107      85.0768      95.9871      32.1473      43.5886      12.8917     -11.7677     -61.2826     -43.4118       99.724      57.8102      68.2949      57.9921     -4.70471     -38.1371      99.4546     -53.2659      34.3079      -83.702      -57.181      24.9874     -8.58608      99.4895      67.0233      47.2894     -3.59536      44.6916      86.9058      18.1591      99.7374     -18.8576      -38.452      48.6913      32.8675      99.3887      -54.019      21.4319     -31.3941      66.1093      64.7936      70.6321     -67.8373      11.1734      22.2466
最优值: 12784.2909
GWO：
最优位置: -18.7247      33.6702      55.4135     -40.6084      88.4392       55.798      51.3369      14.2401      3.91541      4.25104      62.5726   -0.0247625     -11.3325     -43.8366     -11.1569      30.3589     -35.7087      44.3069     -45.5275     -64.7129      76.5476      4.25657      60.5278      13.3075     -7.31614      81.0803      53.0039      42.0402     -48.9057      8.27586     -36.2967      11.3174      34.4106      19.9685      77.2307       37.277      3.17119     -94.8852      33.2876      -1.2158      3.24372     0.591608     -15.8459        60.37      60.2439       62.292     -9.51614      25.1501      2.19109     -5.41538
最优值: 7125.9226

实验结果表明：对于大多数基准函数，GEO-DLS算法在收敛速度和精度上都优于其他算法。

三、参考文献

[1] Jeng-Shyang Pan, Ji-Xiang Lv, Li-Jun Yan, et al. Golden eagle optimizer with double learning strategies for 3D path planning of UAV in power inspection[J]. Mathematics and Computers in Simulation, 2022, 193, 509-532.