优化方法Word格式.docx

1、slope = dot（gk,dk）;while f1 f0 + 0.1*ak*slopeak = ak/2;xk = x0 + ak*dk;f1 = fun（xk）;endk = k+1;x0 = xk;gk = grad（xk）;fprintf（-The %d-th iter, the residual is %fn,k,res）;x_star = xk;function f = fun（x）f = （1-x（1）2 + 100*（x（2）-x（1）2）2;function g = grad（x）g = zeros（2,1）;g（1）=2*（x（1）-1）+400*x（1）*（x（1）2-

2、x（2）;g（2） = 200*（x（2）-x（1）2）;运行结果：x0=0,0;eps=1e-4 eps =1.0000e-004xk=steepest（x0,eps）-The 1-th iter, the residual is 3.271712-The 2-th iter, the residual is 2.504194-The 3-th iter, the residual is 2.073282-The 998-th iter, the residual is 0.449447-The 999-th iter, the residual is 0.449447-The 1000-t

3、h iter, the residual is 0.449447-The 1001-th iter, the residual is 0.449447xk =0.63690.40382.牛顿法function x_star = newton（x0,eps）bk = grad2（x0）（-1）;dk=-bk*gk;xk=x0+dk;bk = grad2（xk）（-1）;function g = grad2（x）g = zeros（2,2）;g（1,1）=2+400*（3*x（1）2-x（2）;g（1,2）=-400*x（1）;g（2,1）=-400*x（1）;g（2,2）=200;eps=1e-

4、4;xk=newton（x0,eps）-The 1-th iter, the residual is 447.213595-The 2-th iter, the residual is 0.0000001.00003. BFGS方法function x_star = bfgs（x0,eps）g0 = grad（x0）;gk=g0;Hk=eye（2）;dk = -Hk*gk;fa0=xk-x0;g0=gk;y0=gk-g0;Hk=（eye（2）-fa0*（y0）/（fa0）*（y0）*（eye（2）-（y0）*（fa0）*（y0）+（fa0*（fa0）*（y0）;function f=fun（x

5、）f=（1-x（1）2 + 100*（x（2）-x（1）2）2; xk=bfgs（x0,eps）-The 2-th iter, the residual is 2.381565-The 3-th iter, the residual is 3.448742-The 998-th iter, the residual is 0.004690-The 999-th iter, the residual is 0.008407-The 1000-th iter, the residual is 0.005314-The 1001-th iter, the residual is 0.0108800.

6、99550.99114.共轭梯度法f unction x_star =conj（x0,eps）d0=dk;k=k+1;x0=xk;gk=grad（xk）;f=（norm（gk）/norm（g0）2;res=norm（gk）;dk=-gk+f*d0;f=（1-x（1）2+100*（x（2）-x（1）2）2;function g=grad（x）g=zeros（2,1）;g（1）=400*x（1）3-400*x（1）*x（2）+2*x（1）-2;g（2）=-200*x（1）2+200*x（2）;xk=Conj（x0,eps）-The 2-th iter, the residual is 1.3805

7、42-The 3-th iter, the residual is 4.527780-The 4-th iter, the residual is 0.850596-The 73-th iter, the residual is 0.001532-The 74-th iter, the residual is 0.000402-The 75-th iter, the residual is 0.000134-The 76-th iter, the residual is 0.0000570.9999上机大作业：编写程序利用增广拉格朗日方法求解下述问题min 4x 1- x2 - 122 2s.

8、t. 25 - x1 - x2 = 010x1x1 + 10x 2- x2 - 34 0x ,x 0初始点取 x0 = 0, 精度取 = 1e - 4.主程序：function x,mu,lamda,output=main（fun,hf,gf,dfun,dhf,dgf,x0）maxk=2000;theta=0.8; eta=2.0;k=0; ink=0;ep=1e-4;sigma=0.4;x=x0; he=feval（hf,x）; gi=feval（gf,x）;n=length（x）; l=length（he）; m=length（gi）;mu=0.1*ones（l,1）; lamda=0.1

9、*ones（m,1）;betak=10; betaold=10;while （betakep &maxk）ik,x,val=bfgs（ lagrang ,dlagrang ,x0,fun,hf,gf,dfun,dhf,dgf,mu,lamda,sigma）;ink=ink+ik;he=feval（hf,x）;betak=sqrt（norm（he,2）2+norm（min（gi,lamda/sigma）,2）2）;if betakepmu=mu-sigma*he;lamda=max（0.0,lamda-sigma*gi）;if （k=2 & betaktheta*betaold）sigma=et

10、a*sigma;betaold=betak;x0=x;f=feval（fun,x）;output.fval=f;output.iter=k;output.inner_iter=ink;output.beta=betak;增广拉格朗日函数function lag=lagrang（x,fun,hf,gf,dfun,dhf,dgf,mu,lamda,sigma）l=length（he）;lag=f; s1=0.0;for （i=1:l）lag=lag-he（i）*mu（i）;s1=s1+he（i）2;lag=lag+0.5*sigma*s1;s2=0.0;m）s3=max（0.0,lamda（i）-

11、sigma*gi（i）;s2=s2+s32-lamda（i）2;lag=lag+s2/（2.0*sigma）;增广拉格朗日梯度函数function dlag=dlagrang（x,fun,hf,gf,dfun,dhf,dgf,mu,lamda,sigma）dlag=feval（dfun,x）;dhe=feval（dhf,x）; dgi=feval（dgf,x）;dlag=dlag+（sigma*he（i）-mu（i）*dhe（:,i）;dlag=dlag+（sigma*gi（i）-lamda（i）*dgi（:线搜索程序 基于 BFGS方法function k,x,val=bfgs（fun,gf

12、un,x0,varargin）Max=1000;ep=1.e-4;beta=0.55; sigma1=0.4;n=length（x0）; Bk=eye（n）;while （kMax）gk=feval（gfun,x0,varargin:）;if （norm（gk）ep）, break ; enddk=-Bkgk;m=0; mk=0;while （m20）newf=feval（fun,x0+betam*dk,varargin:oldf=feval（fun,x0,varargin:if （newf0）Bk=Bk-（Bk*sk*sk*Bk）/（sk*Bk*sk）+（yk*yk）/（yk*sk）;val

13、=feval（fun,x0,varargin:目标函数文件function f=f1（x）f=4*x（1）-x（2）2-12;等式约束文件function he=h1（x）he=25-x（1）2-x（2）2;不等式约束function gi=g1（x）gi=zeros（3,1）;gi（1）=10*x（1）-x（1）2+10*x（2）-x（2）2-34;gi（2）=x（1）;gi（3）=x（2）;目标函数梯度文件function g=df1（x）g=4;-2*x（1）;等式函数梯度文件function dhe=dh1（x）dhe=-2*x（1）;-2*x（2）;不等式函数梯度文件function

14、 dgi=dg1（x）dgi=10-2*x（1）,1,0； 10-2*x（2）， 0,1;输入数据X0=0;0x,mu,lamda,output=main（f1,h1g1df1dh1dg1,x0）输出数据x =1.00134.8987mu =0.0158lamda =0.5542output =fval: -31.9924iter: 5inner_iter: 33beta: 8.4937e-005上机大作业：1.解：将目标函数改写为向量形式：x*a*x-b*x程序代码：n=2;a=0.5,0;0,1;b=2 4;c=1 1;cvx_beginvariable x（n）minimize（ x*a

15、*x-b*x）subject toc * x =0cvx_end运算结果 :Calling SDPT3 4.0: 7 variables, 3 equality constraintsFor improved efficiency, SDPT3 is solving the dual problem.-num. of constraints = 3dim. of socp var = 4, num. of socp blk = 1dim. of linear var = 3*SDPT3: Infeasible path-following algorithmsversion predcorr

16、gam expon scale_dataNT1 0.000it pstep dstep pinfeas dinfeas gapprim-objdual-objcputime-0|0.000|0.000|8.0e-001|6.5e+000|3.1e+002| 1.000000e+001 0.000000e+000| 0:0:00| chol 1 11|1.000|0.987|4.3e-007|1.5e-001|1.6e+001| 9.043148e+000 -2.714056e-001| 0:01| chol 1 12|1.000|1.000|2.6e-007|7.6e-003|1.4e+000

17、| 1.234938e+000 -5.011630e-002| 0:3|1.000|1.000|2.4e-007|7.6e-004|3.0e-001| 4.166959e-001 1.181563e-001| 0:4|0.892|0.877|6.4e-008|1.6e-004|5.2e-002| 2.773022e-001 2.265122e-001| 0:5|1.000|1.000|1.0e-008|7.6e-006|1.5e-002| 2.579468e-001 2.427203e-001| 0:6|0.905|0.904|3.1e-009|1.4e-006|2.3e-003| 2.511

18、936e-001 2.488619e-001| 0:7|1.000|1.000|6.1e-009|7.7e-008|6.6e-004| 2.503336e-001 2.496718e-001| 0:8|0.903|0.903|1.8e-009|1.5e-008|1.0e-004| 2.500507e-001 2.499497e-001| 0:9|1.000|1.000|4.9e-010|3.5e-010|2.9e-005| 2.500143e-001 2.499857e-001| 0:10|0.904|0.904|5.7e-011|1.3e-010|4.4e-006| 2.500022e-001 2.499978e-001| 0:01| chol 2 211|1.000|1.000|5.2e-013|1.1e-0