用python实现精确线搜索+最速下降方法 求解以下二次规划问题 min f(x) =1/2x1^2+x2^2-x1x2-x1

为了用Python实现针对给定二次规划问题`min f(x) = 1/2*x1^2 + x2^2 - x1*x2 - x1`的精确线搜索和最速下降法，你可以按照以下步骤进行：

首先，定义函数`f`和它的导数`df`（梯度）：

import numpy as np

def objective_function(x):
    x1, x2 = x
    return 0.5 * x1**2 + x2**2 - x1*x2 - x1

def gradient_function(x):
    x1, x2 = x
    return np.array([x1 - x2 - 1, x2 - x1])

接着，编写精确线搜索的牛顿线性化部分和步长搜索：

def line_search(f, gradient, current_x, initial_stepsize=1e-4, max_iterations=50, tolerance=1e-6):
    df_dx = gradient(current_x)
    alpha = initial_stepsize
    for i in range(max_iterations):
        candidate_x = current_x - alpha * df_dx
        if f(candidate_x) <= f(current_x) and np.linalg.norm(candidate_x - current_x) < tolerance:
            break
        curvature = np.dot(df_dx.T, df_dx)
        alpha *= -1 / curvature
    else:
        print("Failed to find a suitable step size after", max_iterations, "iterations.")
    return alpha, candidate_x

# 更新点函数
def update_x(gradient_func, current_x, alpha):
    return current_x - alpha * gradient_func(current_x)

def quasi_newton_descent(objective_function, initial_guess, max_iterations=100):
    x = initial_guess
    for _ in range(max_iterations):
        gradient = gradient_function(x)
        alpha, next_x = line_search(objective_function, gradient, x)
        x = update_x(gradient_function, x, alpha)
    return x

现在你可以调用`quasi_newton_descent`函数，传入初始猜测值，得到优化结果：

initial_guess = np.array([0.0, 0.0])  # 或者根据需要设置其他初始值
optimized_solution = quasi_newton_descent(objective_function, initial_guess)
print(f"Optimized solution: {optimized_solution}")