一维搜索

第四章 无约束最优化问题

一维搜索

前提：满足凸规划条件
搜索区间上有唯一的极小点

思想：序贯试验方法，即通过比较区间端点值，利用函数的单调性，缩小搜索区间长度直至小于规定的精度。
具体方法：fibonacci方法、黄金分割法

//
// Created by xylee on 2020/4/30.
//

#include <cstdio>
#include <iostream>
using namespace std;
double f[100] = {0};
double function(double x)
{
    double result = x*x-6*x-2;
    return result;
}
double fibonacci(double a,double b,double e)
{
    double x1,x2,f1,f2;
    int i = 0;
    while(f[i]<((b-a)/e))
    {
        i++;
    }
    x2 = a + f[i-1]/f[i]*(b-a);
    x1 = a+b - x2;
    f2 = function(x2);
    f1 = function(x1);
    while((x2 - x1)>e)
    {
        if(f2 > f1){
            b = x2;
            x2 = x1;
            x1 =a+b-x2;
            f2 = f1;
            f1 = function(x1);
        }
        else{
            a =x1;
            x1 = x2;
            x2 = a+b-x1;
            f1 = f2;
            f2 = function(x2);
        }
        cout <<"x1="<<x1<<" x2="<<x2<<" l="<<(x2-x1)<<endl;
    }
    return (x1+x2)/2;

}
double golden_section(double a,double b,double e)
{
    double x1,x2,f1,f2;
    int i = 0;
    while(f[i]<((b-a)/e))
    {
        i++;
    }
    x2 = a + 0.618*(b-a);
    x1 = a + 0.382*(b-a);
    f2 = function(x2);
    f1 = function(x1);
    while((x2 - x1)>e)
    {
        if(f2 > f1){
            b = x2;
            x2 = x1;
            x1 =a+b-x2;
            f2 = f1;
            f1 = function(x1);
        }
        else{
            a =x1;
            x1 = x2;
            x2 = a+b-x1;
            f1 = f2;
            f2 = function(x2);
        }
        cout <<"x1="<<x1<<" x2="<<x2<<" l="<<(x2-x1)<<endl;
    }
    return (x1+x2)/2;
}
int main() {
    f[0] = f[1] =1;
    for (int i = 2; i < 100; ++i) {
        f[i] = f[i-1]+f[i-2];
    }
    double a,b,e;
    cin >>a>>b>>e;
//    double result = fibonacci(a,b,e);
    double result = golden_section(a,b,e);
    printf("x= %.4lf",result);
    return 0;
}