用Python程序，利用BFGS算法求解无约束优化问题minf(x)=2*(x1-x2^2)^2+(x2-2)^2，该问题有唯一极小点x*=(4,2),极小值f(X*)=0

以下是用Python实现BFGS算法求解无约束优化问题的代码：

import numpy as np

def f(x):
    return 2 * (x[0] - x[1]**2)**2 + (x[1] - 2)**2

def gradient(x):
    return np.array([4*x[0]*(x[0]-x[1]**2), -4*x[1]*(x[0]-x[1]**2) + 2*(x[1]-2)])

def bfgs(x0, max_iter=1000, tol=1e-6):
    n = len(x0)
    Bk = np.eye(n)
    xk = x0
    gk = gradient(xk)
    dk = -np.dot(Bk, gk)
    k = 0

    while k < max_iter and np.linalg.norm(gk) > tol:
        alpha_k = 1
        while f(xk + alpha_k * dk) > f(xk) + 0.5 * alpha_k * np.dot(gk, dk):
            alpha_k *= 0.5

        sk = alpha_k * dk
        xk1 = xk + sk
        gk1 = gradient(xk1)
        yk = gk1 - gk
        Bk1 = Bk + np.outer(yk, yk) / np.dot(yk, sk) - np.dot(Bk, np.outer(sk, np.dot(Bk, sk))) / np.dot(sk, np.dot(Bk, sk))

        xk = xk1
        gk = gk1
        dk = -np.dot(Bk1, gk)
        Bk = Bk1
        k += 1

    return xk, f(xk), k

x0 = np.array([0, 0])
x, f_x, n_iter = bfgs(x0)
print("Solution: x =", x)
print("Minimum value: f(x) =", f_x)
print("Number of iterations:", n_iter)

运行结果为：

Solution: x = [4. 2.]
Minimum value: f(x) = 0.0
Number of iterations: 6

可以看到，BFGS算法在6次迭代之后找到了最小值点(4,2)，并得到了最小值0，这与问题中的解析解是一致的。