一维最优化：二次插值法（Quadratic interpolation method）

理论见参考《Practical Optimization》中4.5节，代码实现如下：（

可以将黄金分割法和二次插值法结合起来，想用黄金分割法快速搜索得到最优解，然后用该最优解作为二次插值的初值，可以提高计算精度。Matlab中就是采用此组合策略。

）

OneDimensionalOptimization.h

#ifndef _OneDimensionalOptimization_
#define _OneDimensionalOptimization_
#include <algorithm>
using namespace std;

class OneDimensionalOptimization
{
public:
	//参考《Practical Optimization》中4.5节
	double quadraticInterpolationSearch(double(*p)(double x),double& l, double& u, double& x, double tol = 1e-6);
};
#endif

OneDimensionalOptimization.cpp

double OneDimensionalOptimization::quadraticInterpolationSearch(double(*p)(double x),double& l, double& u, double& x, double tol)
{
	int mm = 0;
	//1)
	double x1 = l;
	double x3 = u;
	double x0_aver = 1e99;

	//2)
	double x2 = 0.5 * (x1 + x3);
	double f1 = p(x1);
	double f2 = p(x2);
	double f3 = p(x3);

	//3)
	while(true)
	{
		mm++;
		double x_aver = ((x2*x2 - x3*x3)*f1 + (x3*x3 - x1*x1)*f2 + (x1*x1 - x2*x2)*f3 )/(2*((x2 - x3)*f1 + (x3 - x1)*f2 + (x1 - x2)*f3));
		double f_aver = p(x_aver);
		if(abs(x_aver - x0_aver) < tol)
		{
			x = x_aver;
			return f_aver;
		}
		//4)
		if(x1 < x_aver && x_aver < x2)
		{
			if(f_aver <= f2)
			{
				x3 = x2;
				f3 = f2;
				x2 = x_aver;
				f2 = f_aver;
			}
			else
			{
				x1 = x_aver;
				f1 = f_aver;
			}
		}
		else if(x2 < x_aver && x_aver < x3)
		{
			if(f_aver <= f2)
			{
				x1 = x2;
				f1 = f2;
				x2 = x_aver;
				f2 = f_aver;
			}
			else
			{
				x3 = x_aver;
				f3 = f_aver;
			}
		}
		x0_aver = x_aver;
	}
}


测试代码如下：
main.cpp
#include "src\Optimizer\OneDimensionalOptimization.h"

#include <fstream>
using namespace std;

double p(double x)
{
	return -sin(x);
}
int main(int argv, char** argc)
{
	OneDimensionalOptimization* one = new OneDimensionalOptimization();
	double l = 0;
	double u = 2*3.1415926;
	double x;
	double res1 = one->quadraticInterpolationSearch(p,l,u,x,1e-10);
	return 1;
}