求信赖域算法加局部二次近似求解 关于b级数i从1到nxi[yi-(e∧bxi/1+e∧bxi)])的最大值（b是P元向量）的Python代码

信赖域算法加局部二次近似求解的Python代码如下：

import numpy as np
from scipy.optimize import minimize

def trust_region(func, grad, hess, x0, radius=1.0, tol=1e-6):
    x = x0
    fval = func(x)
    gradval = grad(x)
    hessval = hess(x)

    while np.linalg.norm(gradval) > tol:
        # solve local quadratic model
        p = -np.linalg.solve(hessval, gradval)

        # calculate predicted and actual reduction
        pred = -0.5 * np.dot(gradval, p)
        actual = func(x + p) - fval

        # calculate rho
        rho = actual / pred

        if rho < 0.25:
            radius *= 0.25
        elif rho > 0.75 and np.linalg.norm(p) == radius:
            radius = min(2.0 * radius, 100.0)

        if rho > 0.1:
            x += p
            fval = func(x)
            gradval = grad(x)
            hessval = hess(x)

        if np.linalg.norm(p) < tol or np.linalg.norm(gradval) < tol:
            break

    return x

def b_series(b, x, y):
    n = len(x)
    s = 0.0
    for i in range(n):
        s += np.power(np.e, b.dot(x[i])) * (y[i] - np.power(np.e, np.dot(b, x[i]))) / (1 + np.power(np.e, b.dot(x[i])))
    return -s

def b_series_grad(b, x, y):
    n = len(x)
    s = np.zeros(len(b))
    for i in range(n):
        ebx = np.power(np.e, b.dot(x[i]))
        s += ebx * x[i] * (y[i] - ebx) / np.power(1 + ebx, 2)
    return -s

def b_series_hess(b, x, y):
    n = len(x)
    s = np.zeros((len(b), len(b)))
    for i in range(n):
        ebx = np.power(np.e, b.dot(x[i]))
        s += np.outer(x[i], x[i]) * ebx * (ebx - y[i]) / np.power(1 + ebx, 2)
    return -s

# example usage
x = np.array([[1, 2], [3, 4], [5, 6], [7, 8]])
y = np.array([2, 4, 6, 8])

b0 = np.zeros(2)
radius = 1.0
tol = 1e-6

b_opt = trust_region(lambda b: b_series(b, x, y), lambda b: b_series_grad(b, x, y), lambda b: b_series_hess(b, x, y), b0, radius, tol)

print("Optimal b:", b_opt)
print("Optimal value:", -b_series(b_opt, x, y))

其中，`trust_region` 函数是信赖域算法的实现，`b_series` 函数是目标函数，`b_series_grad` 函数是目标函数关于变量的梯度，`b_series_hess` 函数是目标函数关于变量的黑塞矩阵。