大连理工大学优化作业程序

1、1.1程序（Java）public class Wolfe_Powell public static double getFx ( double x ) double x1= x0; double x2 = x1; double Fx= 100 * (x2-x1*x1)* (x2-x1*x1) + (1-x1)* (1-x1) ; return Fx; public static double getDeltFx ( double x ) double x1= x0; double x2 = x1; double deltFx = new double2; deltFx 0 = -400*(x

2、2 - x1* x1) *x1- 2*(1- x1) ; deltFx 1 = 200*(x2- x1 * x1) ; return deltFx ; public static double getDeltFx_Sk ( double deltFx , double Sk ) double a = 0 ; for ( int i = 0 ; i < Sk.length ; i+ ) a = a + deltFx i * Sk i ; return a ; public static double getL ( double x, double s ) double x1= x0; do

3、uble x2 = x1; double c1 =0.1 , c2 =0.5 ,a =0 , b=1e8 ,L= 1; double Fx0 , Fx1 ，deltFx1_Sk ,deltFx0_Sk ,temp ,temp2; double deltFx0 , deltFx1 ; Fx0 = getFx(x) ; deltFx0 = getDeltFx (x) ; deltFx0_Sk = getDeltFx_Sk( deltFx0 , s) ; temp = c2 * getDeltFx_Sk( deltFx0 , s) ; for ( int i=0;i< 1e8 ; i+) te

4、mp2 = -c1 * L * deltFx0_Sk ; x0 = x1 + L *s0 ; x1 = x2 + L *s1 ; Fx1 = getFx(x) ; deltFx1 = getDeltFx (x) ; deltFx1_Sk = getDeltFx_Sk (deltFx1 , s) ; if( (Fx0 - Fx1 ) >= temp2 && deltFx1_Sk >= temp) break ; else if( (Fx0 - Fx1 ) < temp2 ) b = L ; L = (L +a) /2 ; else if ( deltFx1_Sk

5、 < temp ) a = L ; L = ( L + b ) / 2 >= 2*L ? (2*L):( L + b ) / 2; System.out.println(" L= " + L); System.out.println(" 计算次数 " + i ); return L ; public static void main(String args) Wolfe_Powell temp =new Wolfe_Powell(); double X = -1 ,1 ; double sk = 1 ,1 ; temp.getL( X ,sk

6、) ; 1.2实验结果步长L = 0.00390625x =-0.9992 , 1.0324 计算次数 8 2.1程序（Java）public class GongE public static double getFx ( double x ) double x1= x0; double x2 = x1;double Fx= x1*x1 - 2*x1*x2 + 2*x2*x2 +x3*x3 - x2*x3 +2 * x1 +3*x2 -x3 ; return Fx; public static double getDeltFx ( double x ) double x1= x0; doub

7、le x2 = x1; double deltFx = new doublex.length;deltFx 0 = 2*x1 - 2*x2+2 ;deltFx 1 = -2*x1 +4*x2 - x3 +3;deltFx 2 = 2*x3 -x2 -1 ; return deltFx ; public static double getX ( double x ) double g0,g1; double s0= new doublex.length; double s1=new doublex.length; double g0_L,g1_L ,L ,temp; double x0 =x ;

8、 int k =0 ; g0 = getDeltFx ( x0 ) ; for ( int j = 0 ; j < x.length ; j+ ) s0 j = -g0 j ; for (int i = 0 ;i<2; i +,k+) g0 = getDeltFx ( x0 ) ; g0_L = getDeltFx_Sk ( s0 , s0 ) ; L =getL(x0,s0); / 例题一中的方法取得步长L for(int j=0;j<x.length ; j+) x0j= x0j+ s0j*L ; g1 = getDeltFx(x0) ; g1_L = getDeltFx

9、_Sk ( g1 , g1 ); if ( Math.sqrt( g1_L )<= 1e-2 ) break ; else temp = g1_L/ g0_L ; for(int j=0;j<x.length ; j+) s0j = -g1j + temp * s0j; return x0; public static void main(String args) GongE temp =new GongE(); double x = 1,1 ; double result = temp.getX(x) ; for ( int i = 0 ; i < x.length ; i

10、+ ) System.out.println ( "result" + i + "=" + result i ) ; 2.2实验结果最优点 x*=-4,-3,-1 最优解 f*=-8 3.1公用程序（Java） public static double getFx ( double x ) /取得Fx 值 double x1= x0; double x2 = x1;double Fx = x1 + 2 * x2 * x2 + Math.exp ( x1 * x1 + x2 * x2 ) ; return Fx ; public static double

11、 getDeltFx ( double x ) /取得Fx 的梯度值 double x1= x0; double x2 = x1; double deltFx = new double 2 ;deltFx 0 = 1 + 2 * x1 * Math.exp ( x1 * x1 + x2 * x2 ) ; deltFx 1 = 4 * x2 + 2 * x2 * Math.exp ( x1 * x1 + x2 * x2 ) ; return deltFx ; 3.2.1最速下降法程序（Java）public class FastWay public static double getX ( do

12、uble x ) double deltF0 = new double2; double L =0; for ( int i = 0 ; i < 1e1 ; i+ ) deltF0 = getDeltFx(x); for(int j=0 ;j <deltF0.length ;j+) /取得负梯度 deltF0j = - deltF0j; L = getL ( x , deltF0 ) ; / 调用习题1的不精确搜索取得步长L if ( Math.sqrt ( getDeltFx_Sk ( deltF0 , deltF0 ) ) <= 1e-3 ) System.out.pri

13、ntln ( "最终计算次数 " + i ) ; System.out.println("x1=" + x0+" x2=" + x1); break ; x0 = x0+ L * deltF0 0 ; x1= x1+ L * deltF0 1 ; return x; public static void main ( String args ) FastWay temp = new FastWay () ; double x0 = 2 , 2 ; temp.getX(x0) ; 3.2.2最速下降法结果 最优点X*=-0.4194 0

14、 最优解f*=0.7729 计算次数count=103.3.1牛顿法程序（Java） public static double getDeltFx ( double x ) double x1 = x 0 ; double x2 = x 1 ; double one = new double 2 ; double exp =Math.exp( Math.pow(x1,2)+Math.pow(x2,2) ; one 0 = 1+ 2*x1*exp ; one 1 = 4* x2 +2*x2*exp ; double two = new double22 ; two00 = 2*exp *(1+2

15、*Math.pow(x1,2) ; two11 = 2*exp *(1+2*Math.pow(x2,2) +4 ; double deltFx = new double 2 ; for (int i = 0 ; i < 2 ; i+ ) deltFx0 = one 0 /two00 ; deltFx1 = one 1 /two11 ; return deltFx; public static void main ( String args ) double x = 1 , 0 ; double DeltFx = new double 2 ; for(int i =0 ;i <1e3

16、;i+) DeltFx = getDeltFx(x); x0 = x0- DeltFx0; x1 = x1- DeltFx1; if( Math.sqrt( getDeltFx_Sk(DeltFx,DeltFx ) ) <= 1e-4) System.out.println("计算次数为 " + i); break ; System.out.println(" x1= " +x0 +" x2= " + x1 +"n") ; System.out.println(" Fx= " +getFx

17、(x) ; 3.3.2牛顿法结果 最优点X*= -0.4194 , 0 最优解f*= 0.7729 计算次数count=53.4.1 BFGS法程序（matlab）function x,val,k = bfgs(fun,gfun,x0)maxk=1000; sigma=0.4; rho=0.55 ; epsion=1e-5;k=0 ; n =length(x0);Bk=eye(n); %Bk=feval('Hess',x0);while (k<maxk) gk=feval(gfun,x0); if(norm(gk)<epsion),break;end; dk=-Bk

18、gk; m=0;mk=0; while(m<20) newf=feval(fun,x0+rhom*dk) oldf=feval(fun,x0) if(newf<oldf+sigma*rhom*gk'*dk) mk=m;break; end m=m+1; end x=x0+rhomk*dk; sk=x-x0; yk=feval(gfun,x)-gk; if(yk'*sk>0) Bk=Bk-(Bk*sk*sk'*Bk)/(sk'*Bk*sk)+(yk*yk')/(yk'*sk); end; k=k+1; x0=x;endval=fe

19、val(fun,x0);3.4.2 BFGS法结果 最优点X*=-0.4194 0 最优解f*=0.7729 计算次数count=44.1 有效集法（matlab）4.1.1 主程序functionx , Lagrange , exitflag , output= TwoProg (H,c,Ae,be,Ai,bi,x0)n=length(x0); x=x0; ni=length(bi); ne=length(be); Lagrange =zeros(ne+ni,1); index=ones(ni,1);for(i=1:ni) if(Ai(i,:)*x>bi(i)+1e-9),index(

20、i)=0;endend%算法主程序k=0;while(k<=1e4) %求解子问题 Temp=; if(ne>0),Temp=Ae ; end for(j=1:ni) if(index(j)>0),Temp=Temp;Ai(j,:);end end gk=H*x+c; m1,n1=size(Temp); dk,Lagrange =SubPro (H,gk , Temp,zeros(m1,1); if(norm(dk)<= 1.0e-6) y=0.0; if(length(Lagrange )>ne) y,jk=min(Lagrange (ne+1:length(L

21、agrange ); end if(y>=0) exitflag=0; else exitflag=1; for(i=1:ni) if(index(i)&(ne+sum(index(1:i)=jk) index(i)=0;break; end end end k=k+1; else exitflag=1; %求步长 alpha=1.0;tm=1.0; for(i=1:ni) if(index(i)=0)&(Ai(i,:)*dk<0) tm1=(bi(i)-Ai(i,:)*x)/(Ai(i,:)*dk); if(tm1<tm) tm=tm1;ti=i; end

22、end end alpha=min(alpha,tm); x=x+alpha*dk; if(tm<1),index(ti)=1;endendif(exitflag=0),break;endk=k+1;endoutput.fval=0.5*x'*H*x+c'*x;output.iter=k;4.1.2 目标函数function f=fun(x)x1=x(1); x2=x(2); f=eval ('x1+2*x22+exp(x12+x22)'); 4.1.3 子问题函数functionx, Lagrange = SubPro (H ,c, Ae, be)m,n

23、=size(Ae);ginvH=pinv(H);if(m>0) rb=Ae*ginvH*c+be; Lagrange =pinv(Ae*ginvH*Ae')*rb; x=ginvH*(Ae'*Lagrange -c);else x=-ginvH*c; Lagrange =0;end4.1.4 运行函数H=2 -2;-2 4;c=-2 -6'Ae= ;be= ;Ai=1 -2;-0.5 -0.5;1 0;0 1;bi=-2 -1 0 0'x0=0 1 'x,lambda,exitflag,output=qpact(H,c,Ae,be,Ai,bi,x0

24、)4.2 有效集法结果 内部点 初始点x0=0 0 最优点X*=0.8 1.2 最优解f*=-7.2 迭代次数=10 边界点 初始点x0=1 1 最优点X*=0.8 1.2 最优解f*=-7.2 迭代次数=2 检验点 初始点x0=0 1 最优点X*=0.8 1.2 最优解f*=-7.2 迭代次数=75.1 乘子法程序（matlab）5.1.1 chengZi程序-乘子法主程序functionx,mu,Lagrange ,output=chengZi(fun,hf,gf,dfun,dhf,dgf,x0)sigma=2.0;count=0;innerCount=0; eta=2.0;=0.8;%P

25、HR算法中的实参数 x=x0;he=feval(hf,x);gi=feval(gf,x);n=length(x);l=length(he);m=length(gi);%选取乘子向量的初始值mu=0.1*ones(l,1);Lagrange =0.1*ones(m,1);btak=10;btaold=10;%用来检验终止条件的两个值while(btak>1e-6&count<1e3 ) %调用BFGS算法程序求解无约束子问题 x,ival,ik=bfgs('Lagr','LagrTiDu',x0,fun,hf,gf,dfun,dhf,dgf,m

26、u,Lagrange ,sigma); innerCount=innerCount+ik; he=feval(hf,x);gi=feval(gf,x); btak=0.0; for(i=1:l),btak=btak+he(i)2; end for(i=1:m) temp=min(gi(i),Lagrange (i)/sigma); btak=btak+temp2; end btak=sqrt(btak); if btak>1e-6 if(count>=2&btak>*btaold) sigma=eta*sigma; end %更新乘子向量 for(i=1:l),mu(

27、i)=mu(i)-sigma*he(i);end for(i=1:m) Lagrange (i)=max(0.0,Lagrange (i)-sigma*gi(i); end end count=count+1; btaold=btak; x0=x;endf=feval(fun,x)output.inner_iter=innerCount;output.iter=count;output.bta=btak;output.fval=f;5.1.2 f1程序-目标函数function f=f1(x)f=4*x(1)-x(2)2-12;5.1.3 h1程序-等式约束function he=h1(x)he=25-x(1)2-x(2)2;5.1.4 g1程序-不等式约束function gi=g1(x)gi=10*x(1)-x(1)2+10*x(2)-x(2)2-34;5.1.5 df1程序-目标函数的梯度文件function g=df1(x)g=4 ,-2.0*x(2)'5.1.6 dhe程序-等式约束（向量）函数的Jacobi矩阵（转置）funct