工程数学二

一、实验内容

(1)求解无约束优化问题:f(#)=(x+10x)+5(x;-x)+(x-2x)+10-x);

(2)终止准则职|(x^) 10-“;

(3)完成最速下降法(负梯度法)的 MATLAB 编程、调试;

(4)要求选取多个不同的初始点，并给出选代次数，最优函数值等相关信息，有能力的同学尝试画出最优值随选代次数变化的曲线图;

(5)按照模板撰写实验报告，要求规范整洁。

 

2. 代码

f = @(x) x(1)-x(2)+2*x(1)^2 + 2*x(2)*x(1)+x(2)^2;

grad_f = @(x) [1+4*x(1)+2*x(2); -1+2*x(1)+2*x(2)];

initial_points = [0, 0;];

tol = 0.01;

results = cell(size(initial_points, 1), 1);

for i = 1:size(initial_points, 1)

x0 = initial_points(i, :)';

[x_opt, f_opt, iter, history] = steepest_descent_method(f, grad_f, x0, tol);

results{i} = struct('x0', x0, 'x_opt', x_opt, 'f_opt', f_opt, 'iter', iter, 'history', history);

fprintf('初始点 : (%.2f, %.2f), 极小点: (%.6f, %.6f), 极小值: %.6f, 迭代次数: %d\n', ...

x0(1), x0(2), x_opt(1), x_opt(2), f_opt, iter);

end

function [x_opt, f_opt, iter, history] = steepest_descent_method(f, grad_f, x0, tol)

max_iter = 10000;

iter = 0;

x = x0;

history = [];

while norm(grad_f(x)) > tol && iter < max_iter

d = -grad_f(x);

alpha = line_search(f, grad_f, x, d);

x = x + alpha * d;

iter = iter + 1;

history = [history; f(x)];

end

x_opt = x;

f_opt = f(x);

end

function alpha = line_search(f, grad_f, x, d)

alpha = 1;

rho = 0.5;

c = 1e-4;

while f(x + alpha * d) > f(x) + c * alpha * (grad_f(x)' * d)

alpha = rho * alpha;

end

fprintf("步长因子%.6f\n",alpha);

end

figure;

hold on;

for i = 1:length(results)

plot(results{i}.history, 'DisplayName', sprintf('Initial point: (%.2f, %.2f)', results{i}.x0(1), results{i}.x0(2)));

end

hold off;

xlabel('Iteration');

ylabel('Objective Function Value');

title('Objective Function Value vs. Iteration');

legend show;