機(jī)械優(yōu)化設(shè)計(jì)作業(yè)——梯度法求解

1、梯度法求最優(yōu)解問(wèn)題姓名：李剛學(xué)號(hào)：20162103091、所求問(wèn)題用梯度法求目標(biāo)函數(shù)F(x)二1.5x2+0.5x2-xx-2x的無(wú)約束最優(yōu)解，已知初始點(diǎn)12121X(0)=2,4t,8=0.02或=0.0022、Matlab編程實(shí)現(xiàn)建立了一個(gè)Stair(e).m的子函數(shù)來(lái)求解，e為函數(shù)的輸入變量，即問(wèn)題中的8。只需要在Matlab“命令行窗口”去調(diào)用此函數(shù)即可，用8二0.02或8二0.002去代替Stair(e).m函數(shù)中的輸入變量e即可。程序源碼：functionz=Stair(e)%輸入變量e來(lái)代替求解精度問(wèn)題，方便求解不同靜精度問(wèn)題symsx1x2af=1.5*x2+0.5*x2“2-

2、xl*x2-2*xl;%目標(biāo)函數(shù)dx1=diff(f,x1);dx2=diff(f,x2);g=dx1,dx2;%對(duì)x1求偏導(dǎo)%對(duì)x2求偏導(dǎo)%梯度X=x1,x2;X0=-2,4;%建立坐標(biāo)變量%初始點(diǎn)坐標(biāo)g0=subs(g,X,X0);G=norm(g0);k=0;F=subs(f,X,X0);F=double(F);%subs函數(shù)，用X0代替g函數(shù)中的X%梯度的模%迭代次數(shù)變量%subs函數(shù)，用X0代替f函數(shù)中的X%數(shù)據(jù)類(lèi)型轉(zhuǎn)換fprintf(迭代次數(shù)k=%d；,k);fprintf(坐標(biāo)值xl=%5.7f,XO(1);fprintf(x2=%5.7f；,X0(2);%fprintf(坐標(biāo)值

3、X=%d%dn,X0);fprintf(函數(shù)值F=%5.7fnn,F)fori=1:20if(Ge)%迭代求解%判斷是否滿(mǎn)足精度要求k=k+1;d=-g0/G;%確定搜索方向F=subs(f,X,X0);fm=subs(f,X,X0+a*d);fmd=diff(fm);a0=solve(fmd);X0=X0+a0*d;g0=subs(g,X,X0);G=norm(g0);g0=double(g0);%計(jì)算F(x)目標(biāo)函數(shù)的值%求最優(yōu)步長(zhǎng)，以計(jì)算新的迭代點(diǎn)%求偏導(dǎo)%求出偏導(dǎo)函數(shù)的零解，求得最優(yōu)步長(zhǎng)%求得新的迭代點(diǎn)%計(jì)算新的梯度%計(jì)算新的梯度的模%數(shù)據(jù)類(lèi)型轉(zhuǎn)化G=double(G);X0=doub

4、le(X0);F=double(subs(f,X,X0);a0=double(a0);%結(jié)果輸出fprintf(迭代次數(shù)k=%d；,k);fprintf(坐標(biāo)值xl=%5.7f,XO(1);fprintf(x2=%5.9f；,X0(2);fprintf(函數(shù)值F=%5.7fn,F);fprintf(梯度g(k)=%5.7f%5.7f；,g0);fprintf(梯度的模|g(k)|=%5.7f；,G);fprintf(步長(zhǎng)a(k)=%5.7fnn,a0);endendz=X0F;end運(yùn)行結(jié)果當(dāng)=0.02時(shí)，命令窗口輸入Stair(0.02),運(yùn)行結(jié)果見(jiàn)下圖:Stair(0.02)迭代次敎曰；坐

5、標(biāo)值sl=-2.0000000x2=4.OOOOOOO；函數(shù)值F=26.0000000迭代次數(shù)El；坐標(biāo)值sl=l.5294118z2=2.235294118；函數(shù)值F=-0.4705882梯度g(k)=0.35294120.7058824；梯度的|g(k)|=0.7892005；步a(k)=3.9460023迭代次k=2；坐標(biāo)值sl=0.9411765x2=l.058823529；函數(shù)值F=-0.9896194梯度g(k)=-0.23529410.1176471；梯度的|g(k)|1=0.2630668；步長(zhǎng)a(k)=l.3153341迭代次數(shù)坐標(biāo)值sl=l.0103806z2=1.0242

6、21453；函數(shù)值F=-0.9997965梯度g(k)=0.00692040.0138408；梯度的|g(k)|=0.0154745；步長(zhǎng)a(k)=0.0773726當(dāng)=0.002時(shí)，命令窗口輸入Stair(0.02),運(yùn)行結(jié)果見(jiàn)下圖:Stair(0.002)迭代次數(shù)k=O；坐標(biāo)值sl=-2.0000000x2=4.OOOOOOO；函數(shù)值F=26.0000000迭優(yōu)次數(shù)心1；坐標(biāo)值xl=l.,5254118z2=2-2352941IS；函數(shù)值F=-0.4705882梯度g(k)=0.35294120.7058824；梯度的|g(k)|=0.7892005；-a(k)=3.S46M23迭優(yōu)灰數(shù)心

7、2；坐標(biāo)值xl=OL5411765x2=l.058823529；函數(shù)值F=-0.9896194梯度E(k)=-O.23529410.1176471；梯度的|g(k)|1=0.2630668；步a(k)=l.3153341迭代次數(shù)1=3；坐標(biāo)值Kl=l.010380612=1.024221453；函數(shù)值F=-0.9997965梯度g(k)=0.00692040.0138408；梯度的|g(k)|1=0.0154745；步長(zhǎng)a(k)=0.0773726迭代次數(shù)k=4；坐標(biāo)值sl=0.9988466x2=1.001153403；函數(shù)值F=-0.9999960梯度g(k)=-0.00461360.00

8、23068；梯度的|g(k)|1=0.0051582；-a(k)=0.0257909迭代次數(shù)E5；坐標(biāo)值Kl=l.0002035x2=1.OOG474930；函數(shù)值F=-0.9999999梯度g(k)=0.00013570.0002714；梯度的|g(k)|1=0.0003034；步a(k)=0.0015171具體見(jiàn)表格：(=0.02)k0123X(k)-2，1.529411&0.9411765,1.0103806,42.235294111.058823531.02422145F(X(k)26-0.4705882-0.9896194-0.9997965g(k)0.3529412,0.70588

9、24-0.235294,0.11764710.0069204,0.0138408g(k)0.78920050.26306680.0154745a(k)3.94600231.31533410.0773726具體見(jiàn)表格：(=0.002)k012345X(k)-21.5294118,0.9411765,1.0103806,0.9988466,1.0002035,42.235294111.058823531.024221451.0011534031.000474930F(X(k)26-0.4705882-0.9896194-0.9997965-0.9999960-0.9999999g(k)0.3529412,-0.235294,0.0069204,-0.0046136,0.0001357,0