Powell方法的代码实现

Powell's method，也称为拟牛顿法，是一种迭代优化算法，用于寻找函数的局部最小值，尤其是在目标函数不可微的情况下。它结合了梯度信息和二分搜索策略。这里提供一个基本的Python版本的Powell方法实现示例：

import numpy as np

def armijo_condition(f, x_k, g_k, alpha):
    return f(x_k + alpha * g_k) <= f(x_k) + c * alpha * g_k.dot(g_k)

def line_search(f, x_k, d_k, tolerance=1e-4):
    alpha = 0.5
    while not armijo_condition(f, x_k, d_k, alpha):
        alpha *= 0.5
    if abs(alpha * d_k.dot(g_k)) < tolerance:
        return alpha
    else:
        return min(1, alpha)

def powell_method(target_function, initial_guess, max_iter=100, tolerance=1e-6):
    x = initial_guess
    for _ in range(max_iter):
        gradient = approximate_gradient(target_function, x)
        Hessian_approximation = approximate_hessian(target_function, x)
        # 使用Broyden-Fletcher-Goldfarb-Shanno(BFGS)公式更新方向向量d_k
        beta = 1 / (d_k.T @ Hessian_approximation @ d_k)
        direction = -Hessian_approximation @ d_k + beta * (gradient - previous_direction)
        
        alpha = line_search(target_function, x, direction)
        x += alpha * direction
        
        if np.linalg.norm(approximate_gradient(target_function, x)) < tolerance:
            break
        previous_direction = direction
    return x

# 这里假设target_function接受一个位置参数并返回目标函数值，以及approximate_gradient和approximate_hessian是你自定义的函数，分别计算目标函数的一阶和二阶近似梯度