算法导论29（线性规划）

29.1 标准型和松弛型

标准型：

maximize:∑j=1ncjxjsubjectto:∑j=1naijxj≤bi,i=1,2,⋯,mxj≥0,j=1,2,⋯,n

松弛型：

z=v+∑j∈Ncjxjxi=bi−∑j∈Naijxj,i∈B

29.3 单纯形算法

转动：
(1) Compute the coefficients of equation for new basic variable xe .

xl=bl−∑j∈N−{e}aljxj−alexe⇒xe=blale−∑j∈N−{e}aljalexj−1alexl⇒xe=b^e−∑j∈N−{e}∪{l}a^ejxj

(2) Compute the coefficients of the remaining constraints.

xi=bi−∑j∈N−{e}aijxj−aiexe(i∈B−{l})⇒xi=bi−∑j∈N−{e}aijxj−aie(b^e−∑j∈N−{e}∪{l}a^ejxj)⇒xi=bi−aieb^e−∑j∈N−{e}(aij−aiea^ej)xj−(−aiea^el)xl

(3) Compute the objective function.

z=v+∑j∈N−{e}cjxj+cexe⇒z=v+∑j∈N−{e}cjxj+ce(b^e−∑j∈N−{e}∪{l}a^ejxj)⇒z=v+ceb^e+∑j∈N−{e}(cj−cea^ej)xj+(−cea^el)xl

(4) Compute new sets of basic and nonbasic variables.

N^=N−{e}∪{l}B^=B−{l}∪{e}

struct slackForm
{
    unordered_set<int> N,B;
    vector<vector<double> > A;
    vector<double> b,c;
    double v;
};

slackForm pivot(slackForm s,int l,int e)
{
    slackForm s0;
    //Compute the coefficients of equation for new basic variable xe
    s0.b[e]=s.b[l]/s.A[l][e];
    for(unordered_set<int>::iterator j=s.N.begin();j!=s.N.end();++j)
    {
        if(*j!=e)s0.A[e][*j]=s.A[l][*j]/s.A[l][e];
    }
    s0.A[e][l]=1/s.A[l][e];
    // Compute the coefficients of the remaining constraints
    for(unordered_set<int>::iterator i=s.B.begin();i!=s.B.end();++i)
    {
        if(*i!=l)
        {
            s0.b[*i]=s.b[*i]-s.A[*i][e]*s0.b[e];
            for(unordered_set<int>::iterator j=s.N.begin();j!=s.N.end();++j)
            {
                if(*j!=e)s0.A[*i][*j]=s.A[*i][*j]-s.A[*i][e]*s0.A[e][*j];
            }
            s0.A[*i][l]=-s.A[*i][e]*s0.A[e][l];
        }
    }
    s0.A[e][l]=1/s.A[e][l];
    //Compute the objective function
    s0.v=s.v+s.c[e]*s0.b[e];
    for(unordered_set<int>::iterator j=s.N.begin();j!=s.N.end();++j)
    {
        if(*j!=e)s0.c[*j]=s.c[*j]-s.c[e]*s0.A[e][*j];
    }
    s0.c[l]=-s.c[e]*s0.A[e][l];
    //Compute new sets of basic and nonbasic variables
    s.N.erase(e);
    s.N.insert(l);
    s0.N=s.N;
    s.B.erase(l);
    s.B.insert(e);
    s0.B=s.B;
    return s0;
}

正式的单纯形算法：

z=v+∑j∈Ncjxj⇒e=maxj∈N{cj|cj>0}xi=bi−∑j∈Naijxj(i∈B)⇒xi=bi−∑j∈N−{e}aijxj−aiexe⇒l=mini∈B{biaie|aie>0}

vector<double> simplex(slackForm s)
{
    while(true)
    {
        double delta=0;
        int e=0;
        for(unordered_set<int>::iterator j=s.N.begin();j!=s.N.end();++j)
        {
            if(s.c[*j]>delta)
            {
                delta=s.c[*j];
                e=*j;
            }
        }
        if(e==0)break;
        delta=INT_MAX;
        int l=0;
        for(unordered_set<int>::iterator i=s.B.begin();i!=s.B.end();++i)
        {
            if(s.A[*i][e]>0)
            {
                double t=s.b[*i]/s.A[*i][e];
                if(t<delta)delta=t;
                l=*i;
            }
        }
        s=pivot(s,l,e);
    }
    vector<double> x(s.c.size()+1,0);
    for(int i=1;i<=s.c.size();++i)
    {
        if(s.B.find(i)!=s.B.end())x[i]=s.b[i];
    }
    return x;
}