bfgs算法python代码实现

以下是BFGS算法的Python代码实现：

import numpy as np

def bfgs(f, grad_f, x0, max_iter=1000, tol=1e-6):
    n = len(x0)
    Bk = np.eye(n)
    xk = x0
    gk = grad_f(xk)
    dk = -np.dot(Bk, gk)
    k = 0
    while k < max_iter and np.linalg.norm(gk) > tol:
        alpha = backtracking_line_search(f, grad_f, xk, dk)
        xk_new = xk + alpha * dk
        sk = xk_new - xk
        yk = grad_f(xk_new) - gk
        rho = 1 / np.dot(yk, sk)
        if np.isnan(rho):
            break
        Bk = (np.eye(n) - rho * np.outer(sk, yk)) @ Bk @ (np.eye(n) - rho * np.outer(yk, sk)) + rho * np.outer(sk, sk)
        xk = xk_new
        gk = grad_f(xk)
        dk = -np.dot(Bk, gk)
        k += 1
    return xk

def backtracking_line_search(f, grad_f, x, d, alpha=1, rho=0.5, c=1e-4):
    while f(x + alpha * d) > f(x) + c * alpha * np.dot(grad_f(x), d):
        alpha *= rho
    return alpha

其中，`f`是目标函数，`grad_f`是目标函数的梯度函数，`x0`是初始点，`max_iter`是最大迭代次数，`tol`是收敛精度。`backtracking_line_search`是回溯线性搜索函数，用于确定每次迭代的步长。