function [x,lamk,exitflag,output]=qpact(H,c,Ae,be,Ai,bi,x0)
% 功能: 用有效集方法解一般约束二次规划问题:
% min f(x)=0.5*x'*H*x+c'*x,
% s.t. a'_i*x-b_i=0,(i=1,...,l),
% a'_i*x-b_i>=0,(i=l+1,...,m)
%输入: x0是初始点, H, c分别是目标函数二次型矩阵和向量；
% Ae=(a_1,...,a_l)', be=(b_1,...,b_l)';
% Ai=(a_{l+1},...,a_m), bi=(b_{l+1},...,b_m)'.
%输出: x是最优解， lambda是对应的乘子向量；output是结构
% 变量, 输出极小值f(x), 迭代次数k等信息, exitflag是算法终止类型
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 初始化
epsilon=1.0e-9; err=1.0e-6;
k=0; x=x0; n=length(x); kmax=1.0e3;
ne=length(be); ni=length(bi); lamk=zeros(ne+ni,1);
index=ones(ni,1);
for (i=1:ni)
if(Ai(i,:)*x>bi(i)+epsilon), index(i)=0; end
end
%算法主程序
while (k<=kmax)
%求解子问题
Aee=[ ];
if(ne>0), Aee=Ae; end
for(j=1:ni)
if(index(j)>0), Aee=[Aee; Ai(j,:)]; end
end
gk=H*x+c;
[m1,n1] = size(Aee);
[dk,lamk]=qsubp(H,gk,Aee,zeros(m1,1));
if(norm(dk)<=err)
y=0.0;
if(length(lamk)>ne)
[y,jk]=min(lamk(ne+1:length(lamk)));
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
end
end
alpha=min(alpha,tm);
x = x+alpha*dk;
%修正有效集
if(tm<1), index(ti)=1; end
end
if(exitflag==0), break; end
k=k+1;
end
output.fval=0.5*x'*H*x+c'*x;
output.iter=k;
%%%%%%%% 求解子问题 %%%%%%%%%%%%%%%
function [x,lambda]=qsubp(H,c,Ae,be)
ginvH=pinv(H);
[m,n]=size(Ae);
if (m>0)
rb = Ae*ginvH*c + be;
lambda = pinv(Ae*ginvH*Ae')*rb;
x = ginvH*(Ae'*lambda-c);
else
x = -ginvH*c;
lambda = zeros(m,1);
end
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