FR共轭梯度法matlab

FR共轭梯度法是一种无约束优化算法，用于求解目标函数的最小值。在MATLAB中，可以编写相应的函数来实现FR共轭梯度法。下面是两个函数的介绍：

1. 编写FR共轭梯度法求解无约束优化问题的函数，采用黄金分割法精确一维搜索，用数值微分法计算梯度，函数式M文件，精度设为epson可调。

function [x, fval, k] = FR_conjugate_gradient(f, x0, epson)
% f: 目标函数
% x0: 初始点
% epson: 精度
% x: 最优解
% fval: 最优解对应的函数值
% k: 迭代次数

% 初始化
k = 0;
x = x0;
g = gradient(f, x);
d = -g;
alpha = golden_section(f, x, d);
x = x + alpha * d;
g_new = gradient(f, x);
beta = (g_new' * g_new) / (g' * g);
d = -g_new + beta * d;

% 迭代
while norm(g_new) > epson
    k = k + 1;
    g = g_new;
    alpha = golden_section(f, x, d);
    x = x + alpha * d;
    g_new = gradient(f, x);
    beta = (g_new' * g_new) / (g' * g);
    d = -g_new + beta * d;
end

fval = f(x);

end

% 数值微分法计算梯度
function g = gradient(f, x)
h = 1e-6;
n = length(x);
g = zeros(n, 1);
for i = 1:n
    x1 = x;
    x1(i) = x1(i) + h;
    g(i) = (f(x1) - f(x)) / h;
end
end

% 黄金分割法精确一维搜索
function alpha = golden_section(f, x, d)
a = 0;
b = 1;
epsilon = 1e-6;
phi = (1 + sqrt(5)) / 2;
resphi = 2 - phi;
x1 = a + resphi * (b - a);
x2 = b - resphi * (b - a);
f1 = f(x + x1 * d);
f2 = f(x + x2 * d);
while abs(b - a) > epsilon
    if f1 < f2
        b = x2;
        x2 = x1;
        x1 = a + resphi * (b - a);
        f2 = f1;
        f1 = f(x + x1 * d);
    else
        a = x1;
        x1 = x2;
        x2 = b - resphi * (b - a);
        f1 = f2;
        f2 = f(x + x2 * d);
    end
end
alpha = (a + b) / 2;
end

2. 编写n步重新开始的FR共轭梯度法求解无约束优化问题的函数，采用黄金分割法精确一维搜索，用数值微分法计算梯度，函数式M文件，精度设为epson可调。

function [x, fval, k] = n_restart_FR_conjugate_gradient(f, x0, epson, n)
% f: 目标函数
% x0: 初始点
% epson: 精度
% n: 重新开始的步数
% x: 最优解
% fval: 最优解对应的函数值
% k: 迭代次数

% 初始化
k = 0;
x = x0;
g = gradient(f, x);
d = -g;
alpha = golden_section(f, x, d);
x = x + alpha * d;
g_new = gradient(f, x);
beta = (g_new' * g_new) / (g' * g);
d = -g_new + beta * d;

% 迭代
while norm(g_new) > epson
    k = k + 1;
    if mod(k, n) == 0
        d = -g_new;
    end
    g = g_new;
    alpha = golden_section(f, x, d);
    x = x + alpha * d;
    g_new = gradient(f, x);
    beta = (g_new' * g_new) / (g' * g);
    d = -g_new + beta * d;
end

fval = f(x);

end

% 数值微分法计算梯度
function g = gradient(f, x)
h = 1e-6;
n = length(x);
g = zeros(n, 1);
for i = 1:n
    x1 = x;
    x1(i) = x1(i) + h;
    g(i) = (f(x1) - f(x)) / h;
end
end

% 黄金分割法精确一维搜索
function alpha = golden_section(f, x, d)
a = 0;
b = 1;
epsilon = 1e-6;
phi = (1 + sqrt(5)) / 2;
resphi = 2 - phi;
x1 = a + resphi * (b - a);
x2 = b - resphi * (b - a);
f1 = f(x + x1 * d);
f2 = f(x + x2 * d);
while abs(b - a) > epsilon
    if f1 < f2
        b = x2;
        x2 = x1;
        x1 = a + resphi * (b - a);
        f2 = f1;
        f1 = f(x + x1 * d);
    else
        a = x1;
        x1 = x2;
        x2 = b - resphi * (b - a);
        f1 = f2;
        f2 = f(x + x2 * d);
    end
end
alpha = (a + b) / 2;
end