最优化共轭梯度法matlab代码

以下是两种MATLAB编写的最优化共轭梯度法的代码：

1. 使用黄金分割法精确一维搜索的最优化共轭梯度法代码：

function [x, fval] = conjugate_gradient_golden(x0, epsilon)
    % 初始化
    x = x0;
    fval = objective_function(x);
    grad = gradient(x);
    d = -grad;
    
    % 迭代
    while norm(grad) > epsilon
        alpha = golden_section_search(x, d);
        x = x + alpha * d;
        grad_new = gradient(x);
        beta = norm(grad_new)^2 / norm(grad)^2;
        d = -grad_new + beta * d;
        grad = grad_new;
        fval = objective_function(x);
    end
end

function fval = objective_function(x)
    % 目标函数
    fval = x(1)^2 + 2 * x(2)^2;
end

function grad = gradient(x)
    % 数值微分法计算梯度
    h = 1e-6;
    grad = zeros(size(x));
    for i = 1:length(x)
        x_plus_h = x;
        x_plus_h(i) = x_plus_h(i) + h;
        grad(i) = (objective_function(x_plus_h) - objective_function(x)) / h;
    end
end

function alpha = golden_section_search(x, d)
    % 黄金分割法精确一维搜索
    a = 0;
    b = 1;
    rho = 0.618;
    epsilon = 1e-6;
    
    while abs(b - a) > epsilon
        x1 = a + (1 - rho) * (b - a);
        x2 = a + rho * (b - a);
        f1 = objective_function(x + x1 * d);
        f2 = objective_function(x + x2 * d);
        
        if f1 < f2
            b = x2;
        else
            a = x1;
        end
    end
    
    alpha = (a + b) / 2;
end

2. 使用Wolfe-Powell非精确一维搜索的最优化共轭梯度法代码：

function [x, fval] = conjugate_gradient_wolfe_powell(x0, epsilon)
    % 初始化
    x = x0;
    fval = objective_function(x);
    grad = gradient(x);
    d = -grad;
    
    % 迭代
    while norm(grad) > epsilon
        alpha = wolfe_powell_search(x, d);
        x = x + alpha * d;
        grad_new = gradient(x);
        beta = norm(grad_new)^2 / norm(grad)^2;
        d = -grad_new + beta * d;
        grad = grad_new;
        fval = objective_function(x);
    end
end

function fval = objective_function(x)
    % 目标函数
    fval = x(1)^2 + 2 * x(2)^2;
end

function grad = gradient(x)
    % 数值微分法计算梯度
    h = 1e-6;
    grad = zeros(size(x));
    for i = 1:length(x)
        x_plus_h = x;
        x_plus_h(i) = x_plus_h(i) + h;
        grad(i) = (objective_function(x_plus_h) - objective_function(x)) / h;
    end
end

function alpha = wolfe_powell_search(x, d)
    % Wolfe-Powell非精确一维搜索
    alpha = 1;
    c1 = 0.1;
    c2 = 0.9;
    rho = 0.618;
    epsilon = 1e-6;
    
    while true
        f0 = objective_function(x);
        g0 = gradient(x);
        f1 = objective_function(x + alpha * d);
        
        if f1 > f0 + c1 * alpha * g0' * d || (f1 >= objective_function(x + alpha * d) && alpha < epsilon)
            alpha = zoom(x, d, alpha);
            break;
        end
        
        g1 = gradient(x + alpha * d);
        
        if abs(g1' * d) <= -c2 * g0' * d
            break;
        end
        
        if g1' * d >= 0
            alpha = zoom(x, d, alpha);
            break;
        end
        
        alpha = rho * alpha;
    end
end

function alpha = zoom(x, d, alpha_lo)
    % Wolfe-Powell非精确一维搜索中的zoom函数
    alpha_hi = alpha_lo * 2;
    c1 = 0.1;
    c2 = 0.9;
    epsilon = 1e-6;
    
    while true
        alpha = (alpha_lo + alpha_hi) / 2;
        f0 = objective_function(x);
        g0 = gradient(x);
        f1 = objective_function(x + alpha * d);
        
        if f1 > f0 + c1 * alpha * g0' * d || f1 >= objective_function(x + alpha * d)
            alpha_hi = alpha;
        else
            g1 = gradient(x + alpha * d);
            
            if abs(g1' * d) <= -c2 * g0' * d
                break;
            end
            
            if g1' * d * (alpha_hi - alpha_lo) >= 0
                alpha_hi = alpha_lo;
            end
            
            alpha_lo = alpha;
        end
        
        if abs(alpha_hi - alpha_lo) < epsilon
            break;
        end
    end
end