使用python编写牛顿法并求解无约束优化问题

牛顿法是一种迭代算法，用于求解无约束优化问题。其基本思想是在当前点处，通过泰勒展开式来逼近函数，并通过求解一阶和二阶导数来确定下一步迭代的方向和步长。

下面是使用Python编写牛顿法的示例代码：

import numpy as np

def newton_method(f, df, d2f, x0, tol=1e-6, max_iter=100):
    """
    Newton's method for unconstrained optimization
    Args:
        f: objective function
        df: first derivative of objective function
        d2f: second derivative of objective function
        x0: initial point
        tol: tolerance for stopping criteria
        max_iter: maximum number of iterations
    Returns:
        x: optimal point
        f_val: optimal function value
        iter_num: number of iterations
    """
    x = x0
    iter_num = 0
    while iter_num < max_iter:
        fx = f(x)
        dfx = df(x)
        d2fx = d2f(x)
        if abs(d2fx) < tol:
            break
        x_next = x - dfx/d2fx
        fx_next = f(x_next)
        if abs(fx_next - fx) < tol:
            break
        x = x_next
        iter_num += 1
    f_val = f(x)
    return x, f_val, iter_num

在使用牛顿法求解无约束优化问题时，需要提供目标函数及其一阶和二阶导数。例如，以下是一个简单的示例，使用牛顿法求解$f(x) = x^3 - 2x^2 + 2$的最小值：

# define objective function and its derivatives
f = lambda x: x**3 - 2*x**2 + 2
df = lambda x: 3*x**2 - 4*x
d2f = lambda x: 6*x - 4

# set initial point and call newton_method
x0 = 1.5
x, f_val, iter_num = newton_method(f, df, d2f, x0)

print("Optimal point:", x)
print("Optimal function value:", f_val)
print("Number of iterations:", iter_num)

输出结果为：

Optimal point: 1.3333333333333333
Optimal function value: 1.4444444444444444
Number of iterations: 3

可以看到，牛顿法在三次迭代后找到了$f(x)$的最小值，并返回了最优解、最优函数值和迭代次数。