基本粒子群算法

1、基本粒子群算法算法原理基本粒子群算法采用常数因子 c和c2及常惯性权重可，粒子根据如下的公式来更新自 己的速度和新的位置。匕(t 1) = wVj,j(t) cVm(t)%-沧(t)x 饵 1) = X.(t) V.(t 1),j =1,2,.d算法步骤基本粒子群算法的基本步骤如下：随机初始化种群中各微粒的位置和速度；评价每个微粒的适应度，将当前各微子的位置和适应值存储在各微子的pbest中,将所有的pbest中适应最优个体的位置和适应值存储在gbest中；用下式更新粒子的速度和位移：V (t 1) = wV (t) CrP X (t) gr P X (t)i,jj,j i i i,j- j,

2、j g 2L g,j- i,jX.,j(t 1)=Xi,j(t) V,j(t 1),j-1,2,d对每个微粒，将其适应值与其经历的最好位置作比较，如果较好，将其作为当前的最好位 置；比较当前所有pbest和gbest的值，更新gbest ；若满足停止条件(通常为预设的运算精度或迭代次数)，搜索停止，输出结果，否知返回(3)继续搜索。算法MATLAB实现在MATLAB中编程实现的基本粒子群算法优化函数为：PSO。功能：用基本粒子群算法求解无约束优化问题。调用格式：xm, fv = PSO( fitness N,c1,c2,w, M , D)其中，fitness :待优化的目标函数；N :粒子数目

3、；G:学习因子1 ；C2:学习因子2；w :惯性权重；M :最大迭代次数；D :自变量的个数；xm ：目标函数取最小值时的自变量值；fv ：目标函数的最小值。基本粒子群算法的MATLAB 代码如下：function xm,fv=PSO(fitness,N,c1,c2,w,M,D)% fitn ess :待优化的目标函数；% N：粒子数目；% cl：学习因子1;% c2：学习因子2;% w：惯性权重；% M:最大迭代次数；%。:自变量的个数；% xm：目标函数取最小值时的自变量值；% fv ：目标函数的最小值。format long；for i=1:Nforj=1:Dx(i,j)=randn；%

4、随机初始化位置v(i,j)=randn ；%随机初始化速度endendfor i=1:Np(i)=fitness(x(i,:)；y(i,:)=x(i,:)；endpg=x(N,:)； %pg为全局最优for1:(N-1)if fitness(x(i,:)fitness(pg)pg=x(i,:)；endendfor t=1:Mfor i=1:N%速度、位移更新v(i,:)=w*v(i,:)+c1*rand*(y(i,:)-x(i,:)+c2*rand*(pg-x(i,:)； x(i,:)=x(i,:)+v(i,:)； if fitness(x(i,:)p(i)p(i)=fitness(x(i,:

5、)；y(i,:)=x(i,:)；endif p(i)fitness(pg)pg=y(i,:)；end endendxm=pg;fv=fit ness(pg);例30采用基本粒子群算法求取 Sphere Model函数f(x)-7 x2的最小值。i 3解：本例题中函数的最小点为(为X2心)=(0 00),最小值为0,现在用PSO算法求最小值，首先看看不同迭代数对结果的影响，粒子群规模为40,学习因子都是2,惯性权重取0.5，迭代数分别取1000,5000,10000.首先建立目标函数文件fitness.m,输入下列内容：function F=fit ness(x)F=0;for i=1:30F=

6、F+x(i)A2;end在MATLAB命令窗口中输入：xm,fv=PSO(fit ness,40,2,2,0.5,1000,30)xm,fv=PSO(fit ness,40,2,2,0.5,5000,30)xm,fv=PSO(fit ness,40,2,2,0.5,10000,30)将上面求得的结果列表比较如下：迭代步数1000500010000 x1-0.1248159610.076616990.012519205x20.002098544-0.0344413010.023225914x30.3342652920.079611320.006409382x40.020930663-0.0205

7、89443-0.01245984x5-0.0223425970.0519288970.01045483x60.0710595280.0146866-0.002044968x7-0.0102492010.071374031-0.021047682x8-0.173703820.0539362690.074049957x90.044030849-0.0227154410.034692645x100.1152608770.014670858-0.014452009x11-0.0338816130.142809736-0.039527693x120.246496620.0062825960.008809

8、395x130.106152816-0.030329827-0.003126436x14-0.041429312-0.0750854530.007888331x15-0.0923389240.001383922-0.01880419x160.165928197-0.016592079-0.074531144x17-0.300160719-0.0934554250.055319274x18-0.2536121490.00385701-0.008023665x190.1897073210.027221239-0.019769054x20-0.126004501-0.0701443590.00844

9、6675x21-0.253708939-0.014838222-0.008821635x22-0.0491421820.034546978-0.033772348x230.1242394570.0188413610.010630827x24-0.068640510.012666381-0.016788305x250.193477783-0.047085223-0.019466059x26-0.082514364-0.0439247480.049907028x270.030547537-0.063624144-0.007000314x280.034872683-0.0150186970.0112

10、33546x290.017222532-0.002414611-0.023183183x300.062528682-0.0725939270.04341795函数极值0.6335670050.0827690110.026632793从上面的求解结果可以看出，在其他参数不变的情况下，一般迭代步数越大，求得的解得精度越高，但 这并不是绝对的，因为PSO算法本质上也是一种随机算法，即使用同样的参数，每一次求解也可能得出不 同的结果，同时如果对于多峰函数，PSO算法还可能陷入局部最优点。下面看看粒子群规模对结果的影响，学习因子都是2,惯性权重都是0.5，迭代步数都是10000，粒子种群规模分别取50,

11、60，80和100.在MATLAB命令窗口中输入：xm,fv=PSO(fitness,50,2,2,0.5,10000,30)xm,fv=PSO(fitness,60,2,2,0.5,10000,30)xm,fv=PSO(fitness,80,2,2,0.5,10000,30) xm,fv=PSO(fitness,100,2,2,0.5,10000,30)将上面求得点的结果列表比较如下:粒子规模506080100 x1-0.0038516160.009602536-0.021677781-0.01417639x20.028916006-0.00399180.0020802390.0066393

12、52x3-0.020816339-0.029155107-0.0146794980.006570414x4-0.013996327-0.032994703-0.023606005-0.009206691x5-0.0611956080.0153159820.0106670870.004333063x60.0594251370.0045642580.0052266920.000746219x70.025469967-0.00507466-0.0073936420.024755412x8-0.0581338050.0312299710.0111372730.001984843x9-0.0454399

13、76-0.0187316480.009383178-0.056376597x10-0.0220096990.036518558-0.0140111470.015999913x110.0565898640.005337156-0.001308054-0.026670753x120.041709956-0.0098912050.028213036-0.031919309x13-0.0107801540.0025090070.010189469-0.028577535x14-0.0061500350.0296137270.0100253470.004485906x15x16x17x18x19x20

14、x21x22x23x24x25x26x27x28x29x30函数极值-0.048458880.0063039080.009540240.0477722780.0173931080.0049338590.010141597-0.0197414150.0007111520.025022711-0.026329590.05270640.065753630.0205180430.0385902750.001238250.036343855-0.026871907 0.001120755 0.00642773 -0.022967437 -0.004758758-0.028100017 -0.000726

15、7440.0135315030.0190519490.0021108210.028880996-0.0185979560.0049487770.0063944370.006699072-0.0170227330.0102328770.0064641450.0072406060.0046881560.0040298950.009560694-0.0028791570.0077468230.005322445-0.012783715-0.008991788-0.012629268-0.017530155-0.039854583-0.026901063-0.019755395-0.0524164540.0093345340.003708717 -0.002