DFP算法

python实现：

import sympy
import numpy as np
from numpy import matlib as mb

#func为要求极值的函数，x0为初始位置，max_iter为最大迭代次数，epsilon为相邻两次迭代的x改变量
def DFP_x(func, x0, max_iter, epsilon):
    i = 0 # 记录迭代次数的变量
    x0 = float(x0) # 浮点数计算更快
    d_func = sympy.diff(func, x) # 定义一阶导数
    beta = 0.5 #beta 0~1
    delta = 0.25 #delta 0~0.5
    Hk = 1 #初始对称正定矩阵H0通常取为1或单位矩阵
    while i < max_iter:
        gk = d_func.subs(x, x0)
        dk = -Hk*gk

        mk = 0
        while mk < 10:
            if func.subs(x, x0+beta**mk*dk) < func.subs(x,x0) + delta*beta**mk*gk*dk:
                break
            mk += 1
        xnew = x0 + beta**mk*dk

        sk = xnew - x0
        yk = d_func.subs(x, xnew) - gk

        if sk*yk > 0:
            Hk = Hk - (Hk*yk*yk*Hk)/(yk*Hk*yk) + (sk*sk)/(sk*yk)
            
        i += 1
        print('迭代第%d次：%.5f' %(i, xnew))      
        if abs(func.subs(x, xnew)-func.subs(x, x0)) < epsilon:
            break
        x0 = xnew
    return xnew

#func为要求极值的函数，X0为初始位置，max_iter为最大迭代次数，epsilon为相邻两次迭代的x改变量
def DFP_x0x1(func, X0, max_iter, epsilon):
    i = 0 # 记录迭代次数的变量
    X0[0], X0[1] = float(X0[0]), float(X0[1]) #浮点数计算更快
    dx0_func = sympy.diff(func, x0) #定义一阶导数
    dx1_func = sympy.diff(func, x1)
    beta = 0.5 #beta 0~1  
    delta = 0.25 #delta 0~0.5
    Hk = mb.identity(len(X0)) #初始对称正定矩阵H0通常取为1或单位矩阵
    while i < max_iter:
        gk = np.mat([float(dx0_func.subs([(x0, X0[0]), (x1, X0[1])])), float(dx1_func.subs([(x0, X0[0]), (x1, X0[1])]))]).T #梯度矩阵
        dk = -Hk*gk

        mk = 0
        while mk < 10:
            if func.subs([(x0, X0[0]+beta**mk*dk[0,0]), (x1, X0[1]+beta**mk*dk[1,0])]) < func.subs([(x0, X0[0]), (x1, X0[1])]) + delta*beta**mk*gk.T*dk:
                break
            mk += 1
        Xnew = [X0[0] + beta**mk*dk[0,0], X0[1] + beta**mk*dk[1,0]]

        sk = np.mat([beta**mk*dk[0,0], beta**mk*dk[1,0]]).T
        yk = np.mat([float(dx0_func.subs([(x0, Xnew[0]), (x1, Xnew[1])])), float(dx1_func.subs([(x0, Xnew[0]), (x1, Xnew[1])]))]).T - gk

        if sk.T*yk > 0:
            Hk = Hk - (Hk*yk*yk.T*Hk)/(yk.T*Hk*yk) + (sk*sk.T)/(sk.T*yk)

        i += 1
        print('迭代第%d次：[%.5f, %.5f]' %(i, Xnew[0], Xnew[1]))      
        if abs(func.subs([(x0, Xnew[0]), (x1, Xnew[1])])-func.subs([(x0, X0[0]), (x1, X0[1])])) < epsilon:
            break
        X0 = Xnew
    return Xnew


if __name__ == '__main__':      
    x = sympy.symbols("x") 
    x0 = sympy.symbols("x0")
    x1 = sympy.symbols("x1")
    result = DFP_x(x**4-4*x, 10, 50, 1e-5)
    print('最佳迭代的位置：%.5f' %result)
    result = DFP_x0x1((x0-1)**2+(x1-1)**4, [10,10], 50, 1e-5)
    print('最佳迭代位置：[%.5f, %.5f]' %(result[0], result[1]))

C++实现：

#include <iostream>
#include <vector>
#include <Eigen/Dense>

const double dx = 1e-3;

double f(double x)
{
    
    
	return pow(x, 4) - 4 * x;
}

double df(double x)
{
    
    
	//return 4 * pow(x, 3) - 4;
	return (f(x + dx) - f(x)) / dx;
}

double f(std::vector<double> X)
{
    
    
	return pow(X[0] - 1, 2) + pow(X[1] - 1, 4);
}

double df0(std::vector<double> X)
{
    
    
	//return 2 * (X[0] - 1);
	return (f({
    
     X[0] + dx, X[1] }) - f(X)) / dx;
}

double df1(std::vector<double> X)
{
    
    
	//return 4 * pow(X[1] - 1, 3);
	return (f({
    
     X[0], X[1] + dx }) - f(X)) / dx;
}

double DFP_x(double x0, int max_iter, double epsilon)
{
    
    
	int i = 0;
	double beta = 0.5;
	double delta = 0.25;
	double Hk = 1;
	double xnew;
	while (i < max_iter)
	{
    
    
		double gk = df(x0);
		double dk = -Hk*gk;
		int mk = 0;
		while (mk < 10)
		{
    
    
			if (f(x0 + pow(beta, mk)*dk) < f(x0) + delta*pow(beta, mk)*gk*dk)
				break;
			++mk;
		}
		xnew = x0 + pow(beta, mk)*dk;

		float sk = xnew - x0;
		float yk = df(xnew) - gk;

		if (sk*yk > 0)
			Hk = Hk - (Hk*yk*yk*Hk) / (yk*Hk*yk) + (sk*sk) / (sk*yk);

		++i;
		std::cout << "迭代次数：" << i << " " << x0 << std::endl;
		if (abs(f(xnew) - f(x0)) < epsilon)
			break;
		x0 = xnew;
	}
	return xnew;
}

std::vector<double> DFP_x0x1(std::vector<double> X0, int max_iter, double epsilon)
{
    
    
	int i = 0;
	double beta = 0.5;
	double delta = 0.25;
	Eigen::Matrix2d Hk = Eigen::Matrix2d::Identity();
	std::vector<double> Xnew;
	while (i < max_iter)
	{
    
    
		Eigen::Vector2d gk;
		gk << df0(X0), df1(X0);
		Eigen::Vector2d dk = -Hk*gk;

		int mk = 0;
		while (mk < 10)
		{
    
    
			Xnew = {
    
     X0[0] + pow(beta, mk)*dk(0), X0[1] + pow(beta, mk)*dk(1) };
			if (f(Xnew) < f(X0) + delta*pow(beta, mk)*gk.transpose()*dk)
				break;
			++mk;
		}
		Xnew = {
    
     X0[0] + pow(beta, mk)*dk(0), X0[1] + pow(beta, mk)*dk(1) };

		Eigen::Vector2d sk;
		sk << pow(beta, mk)*dk(0), pow(beta, mk)*dk(1);
		Eigen::Vector2d yk;
		yk << df0(Xnew), df1(Xnew);
		yk -= gk;

		if (sk.transpose()*yk> 0)
			Hk = Hk - (Hk*yk*yk.transpose()*Hk) / (yk.transpose()*Hk*yk) + (sk*sk.transpose()) / (sk.transpose()*yk);

		++i;
		std::cout << "迭代次数：" << i << " " << X0[0] << " " << X0[1] << std::endl;
		if (abs(f(Xnew) - f(X0)) < epsilon)
			break;
		X0 = Xnew;
	}
	return X0;
}


int main(int argc, char* argv[])
{
    
    
	double result = DFP_x(10, 50000, 1e-5);
	std::cout << "最佳迭代位置：" << result << std::endl;

	std::vector<double> results = DFP_x0x1({
    
     10,10 }, 50000, 1e-5);
	std::cout << "最佳迭代位置：" << results[0] << " " << results[1] << std::endl;

	system("pause");
	return EXIT_SUCCESS;
}