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1、function x,minf=minRosen(f,A,b,x0,var,eps)%目标函数：f;%约束矩阵：A;%约束右端力量：b;%初始可行点：x0;%自变量向量：var;%精度：eps;%目标函数取最小值时的自变量值：x;%目标函数的最小值：minf;format long;if nargin = 5 eps=1.0e-6;endsyms l;x00=transpose(x0);n=length(var);sz=size(A);m=sz(1);gf=jacobian(f,var);bConti=1;while bConti k=0; s=0; A1=A; A2=A; b1=b; b2=
2、b;for i=1:m dfun=A(i,:)*x00-b(i); if abs(dfun)0 A1=A1(1:k,:); b1=b1(1:k,:);endif s0 A2=A2(1:s,:); b2=b2(1:s,:);endwhile 1 P=eye(n,n); if k0 tM=transpose(A1); P=P-tM*inv(A1*tM)*A1; end gv=Funval(gf,var,x0); gv=transpose(gv); d=-P*gv ; if d=0 if k=0 x=x0; bConti=0; break; else w=inv(A1*tM)*A1*gv; if w
3、=0 x=x0; bConti=0; break; else u,index=min(w); sA1=size(A1); if sA1(1)=1 k=0; else k=sA1(2); A1=A1(1:(index-1),:);A1(index+1):sA1(2),:); %去掉A1对应的行 end end end else break; endendd1=transpose(d); y1=x0+l*d1; tmpf=Funval(f,var,y1); bb=b2-A2*x00; dd=A2*d; if dd=0 tmpI,lm=minJT(tmpf,0,0.1); else lm=inf;
4、for i=1:length(dd) if dd(i)0 if bb(i)/dd(i)lm lm=bb(i)/dd(i); end end end end xm,minf=minHJ(tmpf,0,lm,1.0e-14); tol=norm(xm*d); if toleps & k fu a = l; l = u; u = a + 0.618*(b - a); else b = u; u = l; l = a + 0.382*(b-a); end k = k+1; tol = abs(b - a);endif k = 100000 disp(找不到最小值！); x = NaN; minf =
5、NaN; return;endx = (a+b)/2;minf = subs(f, findsym(f),x);format short;function minx,maxx = minJT(f,x0,h0,eps)format long;if nargin = 3 eps = 1.0e-6;end x1 = x0;k = 0;h = h0;while 1 x4 = x1 + h; k = k+1; f4 = subs(f, findsym(f),x4); f1 = subs(f, findsym(f),x1); if f4 f1 x2 = x1; x1 = x4; f2 = f1; f1 =
6、 f4; h = 2*h; else if k=1 h = -h; x2 = x4; f2 = f4; else x3 = x2; x2 = x1; x1 = x4; break; end endendminx = min(x1,x3);maxx = x1+x3 - minx;format short;% syms x1 x2 x3 ;% f=x12+x1*x2+2*x22+2*x32+2*x2*x3+4*x1+6*x2+12*x3;% x,mf=minRosen(f,1 1 1 ;1 1 -2,6;-2,1 1 3,x1 x2 x3)% syms x1 x2;%f=x13+x22-2*x1-
7、4*x2+6; % x,mf=minRosen(f,2,-1;1,1;-1,0;0,-1,1;2;0;0,1 2,x1 x2)% syms x1 x2 x3;% f=-x1*x2*x3; % x,mf=minRosen(f,-1,-2,-2;1,2,2,0;72,10 10 10,x1 x2 x3)% syms x1 x2;%f=2*x12+2*x22-2*x1*x23-4*x17-6*x2; % x,mf=minRosen(f,1 1;1 5;-1 0;0 -1,2;5;0;0,-1 -1,x1 x2)-Main.msyms x1 x2 x3; % f=2*x12+2*x22-2*x1*x2
8、3-4*x17-6*x2; % var=x1,x2; % valst=-1,-1; % A=1 1;1 5;-1 0;0 -1; % b=2 5 0 0; % f=x13+x22-2*x1-4*x2+6; % var=x1,x2; % valst=0 0; % A=2,-1;1,1;-1,0;0,-1; % b=1 2 0 0; var=x1,x2,x3; valst=10,10,10; f=-x1*x2*x3; A=-1,-2,-2;1,2,2; b=0 72; x,mimfval=MinRosenGradientProjectionMethod(f,A,b,valst,var) x2,fv
9、al=fmincon(confun,valst,A,b) MinRosenGradientProjectionMethod.mfunction x,minf=MinRosenGradientProjectionMethod(f,A,b,x0,var,eps) %f is the objection function; %A is the constraint matrix; 约束矩阵 %b is the right-hand-side vector of the constraints; %x0 is the initial feasible point; 初始可行解%var is the v
10、ector of independent variable; 自变量向量 %eps is the precision; 精度 %x is the value of the independent variable when the objective function is minimum; 自变量的值是当目标函数最小%minf is the minimum value of the objective function; 目标函数的最小值 format long; if nargin = 5 eps=1.0e-6; end syms l; x00=transpose(x0); n=lengt
11、h(var); sz=size(A); m=sz(1);% m is the number of rows of A 行数gf=jacobian(f,var);%calculate the jacobian matrix of the objective function 计算目标函数的雅可比矩阵bConti=1; while bConti k=0; s=0; A1=A; A2=A; b1=b; b2=b; for i=1:m dfun=A(i,:)*x00-b(i); %separate matrix A and b 分离矩阵A和b if abs(dfun)0.0000001 %find m
12、atrixs that satisfy A1 x_k=b1 找到满足的矩阵 k=k+1; A1(k,:)=A(i,:); b1(k,1)=b(i); else%find matrixs that satisfy A2 x_k0 A1=A1(1:k,:); b1=b1(1:k,:); end if s0 A2=A2(1:s,:); b2=b2(1:s,:); end while 1 P=eye(n,n); if k0 tM=transpose(A1); P=P-tM*inv(A1*tM)*A1; %calculate P; end gv=Funval(gf,var,x0); gv=transpo
13、se(gv); d=-P*gv; %calculate the searching direction 计算搜索方向% flg=1; % if(P=zeros(n) % flg =0; % end % if flg=1 % d=d/norm(d); %normorlize the searching direction 搜索方向% end % 加入这部分会无止境的循环 if d=0 if k=0 x=x0; bConti=0; break; else w=-inv(A1*tM)*A1*gv; if w=0 x=x0; bConti=0; break; else u,index=min(w);%
14、find the negative component in w sA1=size(A1); if sA1(1)=1 k=0; else k=sA1(2); A1=A1(1:(index-1),:);A1(index+1):sA1(1),:); %delete corresponding row in A1 删除对应的行A1 end end end else break; end end d1=transpose(d); y1=x0+l*d1; %new iteration variable 新的迭代变量 tmpf=Funval(f,var,y1); bb=b2-A2*x00; dd=A2*d
15、; if dd=0 tmpI,lm= ForwardBackMethod(tmpf,0,0.001); %find the searching interval 找到搜索区间 else lm=inf; %find lambda_max for i=1:length(dd) % if(dd(i)0) % % if dd(i)0% % if bb(i)/dd(i)0 保证 0 %find the minimizer by one dimension searching method 找到由一维搜索方法得到目标 tol=norm(xm*d); if toleps&kfu a=l; l=u; u=a+
16、0.618*(b-a); else b=u; u=l; l=a+0.382*(b-a); end k=k+1; tol=abs(b-a); end if k=100000 disp(找不到最小值！); x=NaN; minf=NaN; return; end x=(a+b)/2; %return the minimizer 返回最小值minf=subs(f, findsym(f),x); format short; ForwardBackMethod.m 进退法确定搜索区间function left,right=ForwardBackMethod(f,x0,step) if nargin=2
17、step = 0.01 end if nargin=1 x0=0; step = 0.01 end f0 =subs(f,findsym(f),x0); x1=x0+step; f1=subs(f,findsym(f),x1); if(f1f2) x0=x1; x1=x2; f0=f1; f1=f2; step=2*step; x2=x1+step; f2=subs(f,findsym(f),x2); end left=min(x0, x2); right=max(x0, x2); else step=2*step; x2=x1-step; f2=subs(f,findsym(f),x2);
18、while(f0f2) x1=x0; x0=x2; f1=f0; f0=f2; step=2*step; x2=x1-step; f2=subs(f,findsym(f),x2); end left=min(x1,x2);%left end point right=max(x1,x2);%right end point end Funval.mfunction fv=Funval(f,varvec,varval) var=findsym(f); %找出表达式包含的变量t,s f=t2+s+1 varc=findsym(varvec); %找出传递参数的变量t s中的 t s s1=length
19、(var); %函数的个数 s2=length(varc); %变量的个数 m=floor(s1-1)/3+1); %靠近左边的整数 varv=zeros(1,m); if s1 = s2%if the number of variable is different, deal with it specially 如果变量的数量是不同的,专门处理它 for i=0: (s1-1)/3) k=findstr(varc,var(3*i+1); index=(k-1)/3; varv(i+1) = varval(index+1); end fv=subs(f,var,varv); else fv=s
20、ubs(f,varvec,varval); %return the value of the function 如果原来函数变量个数与传递参数中变量个数一致，调用subs函数，计算给定点处函数值end % disp(here) % f % varvec % varval %fv = subs(f,varvec,varval); 书是我们时代的生命别林斯基书籍是巨大的力量列宁书是人类进步的阶梯高尔基书籍是人类知识的总统莎士比亚书籍是人类思想的宝库乌申斯基书籍举世之宝梭罗好的书籍是最贵重的珍宝别林斯基书是唯一不死的东西丘特书籍使人们成为宇宙的主人巴甫连柯书中横卧着整个过去的灵魂卡莱尔人的影响短暂而微弱，书的影响则广泛而深远普希金人离开了书，如同离开空气一样不能生活科洛廖夫书不仅是生活，而且是现在、过去和未来文化生活的源泉 库法耶夫书籍把我们引入最美好的社会，使我们认识各个时代的伟大智者史美尔斯书籍便是这种改造灵魂的工具。人类所需要的，是富有启发性的养料。而阅读，则正是这种养料雨果