【知识点】拉格朗日乘数法

简介：

在满足约束条件$\varphi(x_1 ,x_2 ,\cdots ,x_n )=0$时求$f(x_1 ,x_2 ,\cdots ,x_n )$的极值。

结论：

令$L(x_1 ,x_2 ,\cdots ,x_n )=f(x_1 ,x_2 ,\cdots ,x_n )+\lambda \varphi(x_1 ,x_2 ,\cdots ,x_n )$。

再令$\frac{\partial L}{\partial {x_i}}$为L关于$x_i$的偏导数。（即视其他$x_j$为常量，仅对$x_i$求导）

则方程组

$\begin{cases}\frac{\partial L}{\partial {x_1}}=0\\\frac{\partial L}{\partial {x_2}}=0\\ \cdots \\ \frac{\partial L}{\partial {x_n}}=0 \\ \varphi(x_1 ,x_2 ,\cdots ,x_n )=0\end{cases}$

的某个解$(\lambda_{k},x_{k,1} ,x_{k,2} ,\cdots ,x_{k,n} )$一定对应着f的极值点。

实现：

先通过枚举/二分/三分确定$\lambda$，然后根据题意算出一组合法解更新答案即可。

代码（NOI2012骑行川藏）：

#include<bits/stdc++.h>
#define maxn 200005
#define maxm 500005
#define inf 0x7fffffff
#define eps 1e-14
#define ll long long
#define rint register int
#define debug(x) cerr<<#x<<": "<<x<<endl
#define fgx cerr<<"--------------"<<endl
#define dgx cerr<<"=============="<<endl

using namespace std;
struct road{double s,k,v;}A[maxn];
int n; double E;

inline int read(){
    int x=0,f=1; char c=getchar();
    for(;!isdigit(c);c=getchar()) if(c=='-') f=-1;
    for(;isdigit(c);c=getchar()) x=x*10+c-'0';
    return x*f;
}

inline int dcmp(double x){return (abs(x)<=eps)?0:(x<0?-1:1);}
inline double phi(double lam){
    double res=0;
    for(int i=1;i<=n;i++){
        double l=max(A[i].v,0.0),r=1e5;
        while(dcmp(r-l)>0){
            double mid=(l+r)/2.0;
            if(dcmp(2.0*lam*A[i].k*(mid-A[i].v)*mid*mid-1.0)<=0) l=mid; else r=mid;
            //printf("%.8lf %.8lf %.8lf\n",2.0*lam*A[i].k*(mid-A[i].v)*mid*mid-1.0,l,r);
        }
        res+=A[i].k*(l-A[i].v)*(l-A[i].v)*A[i].s;
    }
    //printf("%lf\n",res-E);
    return res-E;
}
inline double calc(double lam){
    double res=0;
    for(int i=1;i<=n;i++){
        double l=max(A[i].v,0.0),r=1e5;
        while(dcmp(r-l)>0){
            double mid=(l+r)/2.0;
            if(dcmp(2*lam*A[i].k*(mid-A[i].v)*mid*mid-1)<=0) l=mid; else r=mid;
        }
        res+=A[i].s/l;
    }
    return res;
}


int main(){
    //freopen("bicycling17.in","r",stdin);
    n=read(),scanf("%lf",&E);
    //cout<<n<<":"<<E<<endl;
    for(int i=1;i<=n;i++)
        scanf("%lf%lf%lf",&A[i].s,&A[i].k,&A[i].v);
    double l=0,r=1e5;
    while(dcmp(r-l)>0){
        double mid=(l+r)/2.0;
        if(dcmp(phi(mid))>=0) l=mid; else r=mid;
    }
    //printf("%lf %lf\n",l,r);
    printf("%.6lf\n",calc(l));
    return 0;
}

骑行川藏

【知识点】拉格朗日乘数法

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