5.23

Newton法程序设计

function [x_opt, f_opt, iterations] = newtons_method(f, gradient, hessian, x0, epsilon)

    x = x0;

    iterations = 0;

    

    while norm(gradient(x)) > epsilon

        H = hessian(x);

        grad = gradient(x);

        p = -inv(H)*grad;

        alpha = backtracking_line_search(f, gradient, x, p);

        x = x + alpha*p;

        iterations = iterations + 1;

    end

    

    x_opt = x;

    f_opt = f(x);

end

 

% 定义函数f(x)

f = @(x) (x(1) + 10*x(2))^2 + 5*(x(3) - x(4))^2 + (x(2) - 2*x(3))^4 + 10*(x(1) - x(4))^4;

 

% 定义梯度向量gradient(x)

gradient = @(x) [2*(x(1) + 10*x(2)) + 40*(x(1) - x(4))^3;

    20*(x(1) + 10*x(2)) + 4*(x(2) - 2*x(3))^3;

    10*(x(3) - x(4)) - 8*(x(2) - 2*x(3))^3;

    -10*(x(3) - x(4)) - 40*(x(1) - x(4))^3];

 

% 定义Hessian矩阵hessian(x)

hessian = @(x) [2 + 120*(x(1) - x(4))^2, 20, 0, -120*(x(1) - x(4))^2;

    20, 200 + 12*(x(2) - 2*x(3))^2, -24*(x(2) - 2*x(3))^2, 0;

    0, -24*(x(2) - 2*x(3))^2, 10 + 48*(x(2) - 2*x(3))^2, -10;

    -120*(x(1) - x(4))^2, 0, -10, 10 + 120*(x(1) - x(4))^2];

% 设置初始点和终止准则

x0 = [1; 1; 1; 1];

epsilon = 1e-6;

 

% 调用Newton法求解优化问题

[x_opt, f_opt, iterations] = newtons_method(f, gradient, hessian, x0, epsilon);

disp('最优解:');

disp(x_opt);

disp('最优值:');

disp(f_opt);

disp('迭代次数:');

disp(iterations);

 

function alpha = backtracking_line_search(f, gradient, x, p, rho, c)

    % 参数说明:

    % f: 目标函数，输入参数为x

    % gradient: 目标函数的梯度函数，输入参数为x

    % x: 当前点的位置向量

    % p: 搜索方向向量

    % rho: 回溯因子，通常取值在0到1之间，如0.5

    % c: 充分下降系数，一个正数，通常较小，如0.1

     rho=0.5

     c=0.1

    % 初始化步长alpha为1

    alpha = 1;

    

    % Armijo-Goldstein条件检查

    while f(x + alpha*p) > f(x) + c*alpha*gradient(x)'*p

        % 若不满足充分下降条件，则按rho的比例减小步长

        alpha = alpha * rho;

    end

end