```python #!/usr/bin/env python3 # -*- coding: utf-8 -*- import sys import numpy as np import matplotlib.pyplot as plt def rosenbrock(x): x0, x1 = x[0], x[1] return 100 * (x0**2 - x1)**2 + (x0 - 1)**2 def rosenbrock_grad(x): x0, x1 = x[0], x[1] df_dx0 = 400 * x0 * (x0**2 - x1) + 2 * (x0 - 1) df_dx1 = -200 * (x0**2 - x1) return np.array([df_dx0, df_dx1]) def armijo_line_search(xk, dk, rho=0.5, c=1e-4, alpha0=1.0, max_iter=100): fk = rosenbrock(xk) gk = rosenbrock_grad(xk) alpha = alpha0 for _ in range(max_iter): x_new = xk + alpha * dk f_new = rosenbrock(x_new) if f_new <= fk + c * alpha * (gk @ dk): g_new = rosenbrock_grad(x_new) return alpha, x_new, f_new, np.linalg.norm(g_new) alpha *= rho if alpha < 1e-12: break x_new = xk + alpha * dk f_new = rosenbrock(x_new) g_new = rosenbrock_grad(x_new) return alpha, x_new, f_new, np.linalg.norm(g_new) def gradient_descent_armijo(x0, max_iter=50, tol=1e-5): x = np.array(x0, dtype=float) history = {'x': [x.copy()], 'f': [], 'g_norm': [], 'alpha': []} for k in range(max_iter): g = rosenbrock_grad(x) g_norm = np.linalg.norm(g) history['g_norm'].append(g_norm) if g_norm < tol: break d = -g alpha, x_new, f_new, g_new_norm = armijo_line_search(x, d) history['x'].append(x_new.copy()) history['f'].append(f_new) history['alpha'].append(alpha) x = x_new history['f'].insert(0, rosenbrock(history['x'][0])) return history def plot_convergence(history, save_path=None): xs = np.array(history['x']) fs = np.array(history['f']) gs = np.array(history['g_norm']) fig, axes = plt.subplots(1, 3, figsize=(18, 5)) x0_grid = np.linspace(-1.5, 2.0, 400) x1_grid = np.linspace(-1.0, 3.0, 400) X0, X1 = np.meshgrid(x0_grid, x1_grid) Z = 100 * (X0**2 - X1)**2 + (X0 - 1)**2 contour = axes[0].contour(X0, X1, Z, levels=np.logspace(-1, 3, 15), alpha=0.6, cmap='viridis') axes[0].clabel(contour, inline=True, fontsize=8) axes[0].plot(xs[:, 0], xs[:, 1], 'ro-', markersize=4, linewidth=1.2, label='迭代轨迹') axes[0].quiver(xs[:-1, 0], xs[:-1, 1], xs[1:, 0]-xs[:-1, 0], xs[1:, 1]-xs[:-1, 1], angles='xy', scale_units='xy', scale=1, color='r', width=0.003) axes[0].set_title('Rosenbrock等高线与Armijo迭代路径') axes[0].set_xlabel('$x_0$') axes[0].set_ylabel('$x_1$') axes[0].grid(True, alpha=0.3) axes[0].legend() axes[1].semilogy(range(len(fs)), fs, 'b-o', markersize=3) axes[1].set_title('目标函数值 $f(x_k)$ 下降（对数坐标）') axes[1].set_xlabel('迭代次数 $k$') axes[1].set_ylabel('$f(x_k)$') axes[1].grid(True, alpha=0.3) axes[2].semilogy(range(len(gs)), gs, 'g-s', markersize=3) axes[2].set_title('梯度模长 $\\|\\nabla f(x_k)\\|$ 衰减') axes[2].set_xlabel('迭代次数 $k$') axes[2].set_ylabel('$\\|\\nabla f(x_k)\\|$') axes[2].grid(True, alpha=0.3) plt.tight_layout() if save_path: plt.savefig(save_path, dpi=300, bbox_inches='tight') else: plt.show() if __name__ == "__main__": save_flag = "--save" in sys.argv save_path = "armijo_rosenbrock_convergence.png" if save_flag else None hist = gradient_descent_armijo(x0=[-1.0, 1.0], max_iter=60, tol=1e-6) plot_convergence(hist, save_path) ```