数学建模作业-第五章 非线性最小二乘问题 （1）最小二乘方法无约束问题为：minz=编写lingo程序：sets:quantity/1.15/:x,y;endsetsdata: x=2,5,7,10,14,19,26,31,34,38,45,52,53,60,65; y=54,50,45,37,35,25,20,16,18,13,8,11,8,4,6;enddatamin=sum(quantity:(a+b*exp(x*c)-y)2);free(a);free(b);free(c);结果：Local optimal solution found. Objective value: 44.78049 Extended solver steps: 5 Total solver iterations: 68 Variable Value Reduced Cost A 2.430177 0.000000B 57.33209 0.000000 C -0.4460383E-01 0.000000（2）最小一乘法无约束问题为：min(a,b) z=编写lingo程序写出相应的LINGO程序如下：sets:quantity/1.15/:x,y;endsetsdata: x=2,5,7,10,14,19,26,31,34,38,45,52,53,60,65; y=54,50,45,37,35,25,20,16,18,13,8,11,8,4,6;enddatamin=sum(quantity:abs(a+b*exp(x*c)-y);free(a);free(b);free(c);运行结果：Local optimal solution found. Objective value: 20.80640 Extended solver steps: 2 Total solver iterations: 643 Variable Value Reduced CostA 3.398267 0.000000B 57.11461 0.000000 C -0.4752126E-01 0.000000（3）最大偏差最小的方法编写程序：sets:quantity/1.15/: x,y;endsetsdata: x = 2,5,7,10,14,19,26,31,34,38,45,52,53,60,65; y = 54,50,45,37,35,25,20,16,18,13,8,11,8,4,6;enddatamin=max(quantity: abs(a+b*exp(c*x)-y);free(a); free(b); free(c);结果：Linearization components added: Constraints: 91 Variables: 76 Integers: 30 Local optimal solution found. Objective value: 2.774408 Extended solver steps: 38 Total solver iterations: 7654 Variable Value Reduced CostA 2.885594 0.000000B 55.86246 0.000000C -0.4441314E-01 0.000000（4）画出散点图和曲线图编写matlab程序如下： x=2 5 7 10 14 19 26 31 34 38 45 52 53 60 65;y=54 50 45 37 35 25 20 16 18 13 8 11 8 4 6;scatter(x,y,k*);hold onx=0:0.1:100;y=2.43+exp(-0.0446*x).*57.33;plot(x,y, g);xlabel(x);ylabel(y);hold onx=0:0.1:100;y=3.40+exp(-0.048*x).*57.11;plot(x,y, r);xlabel(x);ylabel(y);hold onx=0:0.1:100;y=2.89+exp(-0.044*x).*55.86;plot(x,y, b);xlabel(x);ylabel(y);得到图形：分析：三条曲线的结果大致相同，但是略有差异。最小二乘法当回归残差满足正态分布时，它有很好的统计性质，但是他的回归性质不稳定。最小一乘法和最小二乘法近似，更靠近主流数据。最大偏差最小回归法因为受到最大平偏差的影响，曲线始终偏移。 非线性优化问题：（1）设汽油由桶A类原油和桶B类原油化合而成，民用燃料油由桶A类原油和桶B类原油化合而成，汽油广告费为元，民用燃料的广告费为元。由题意知道，汽油产量为桶，销量为桶；民用燃料油产量为桶，销量为桶。约束条件：销售约束：，原料约束：，指标约束：，目标函数：利润：编写lingo程序如下：max=0.5*z1*250+z2*200-z1-z2;x1+x2=5000;y1+y2=8*(x1+y1);10*x2+5*y2=6*(x2+y2);0.5*z1=x1+y1;z2=x2+y2;得到： Global optimal solution found. Objective value: 3230000. Total solver iterations: 2 Variable Value Reduced Cost Z1 10000.00 0.000000 Z2 10000.00 0.000000 X1 3000.000 0.000000 X2 2000.000 0.000000 Y1 2000.000 0.000000 Y2 8000.000 0.000000 Row Slack or Surplus Dual Price1 3230000. 1.000000 2 0.000000 297.0000 3 0.000000 174.5000 4 0.000000 -24.50000 5 0.000000 -24.50000 6 0.000000 248.0000 7 0.000000 199.0000结果分析：汽油由3000桶A类原油和2000桶B类原油化合而成；民用燃料油产品由2000桶A类原油和8000桶B类原油化合而成；广告费分别为1000元；石油公司获得最大利润3230000元。（2）根据问题的需要，设汽油中增加SQ量,民用燃料油增加SQ量.根据题意，新的约束条件为：销售约束：，SQ约束：，原料约束：，指标约束：，新的目标函数：利润：编写lingo程序如下：在LINGO中输入：max=0.5*z1*250+z2*200-z1-z2-200*(k1+k2);x1+x2=5000;y1+y2=(8-k10.5)*(x1+y1);10*x2+5*y2=(6-0.6*k20.6)*(x2+y2);0.5*z1=x1+y1+k1;z2=x2+y2+k2;k1=(x1+y1)*0.05;k2=(x2+y2)*0.05;得到： Local optimal solution found.Objective value: 3756000. Total solver iterations: 26 Variable Value Reduced Cost