优化问题无图表输出且gp_minimize报错的技术求助

我编写了一段集成**牛顿法（Newton's Method）和高斯过程优化（Gaussian Process）**的Python代码，运行后无法生成图表，多次修改仍会触发两类错误。以下是完整代码及错误信息：

import numpy as np
from scipy.linalg import solve, cholesky
from scipy.optimize import minimize
from skopt import gp_minimize
import matplotlib.pyplot as plt

def gaussian_rbf(x, x_prime, beta):
    return np.exp(-beta * np.linalg.norm(x - x_prime)**2)

def construct_interpolation_matrix(nodes, beta):
    N = len(nodes)
    K = np.zeros((N, N))
    for i in range(N):
        for j in range(N):
            K[i, j] = gaussian_rbf(nodes[i], nodes[j], beta)
    return K

def conditioning_analysis(N, m, beta):
    nodes = np.linspace(0, 1, N)
    K = construct_interpolation_matrix(nodes, beta)

    selected_indices = np.random.choice(N, m, replace=False)
    selected_nodes = nodes[selected_indices]

    condition_full = np.linalg.cond(K)
    condition_partial = np.linalg.cond(K[selected_indices][:, selected_indices])

    return condition_full, condition_partial

def objective_function(x):
    x_scalar = np.atleast_1d(x)[0]
    return -(x_scalar**2 + np.sin(5 * x_scalar))

def gradient_hessian(x):
    df_dx = 2 * x * np.exp(-(1 - x)**2) - 4 * x * (1 - x) * np.exp(-(1 - x)**2)
    d2f_dx2 = -2 * np.exp(-(1 - x)**2) + 4 * x * (1 - x) * np.exp(-(1 - x)**2) - 4 * (1 - x) * np.exp(-(1 - x)**2)
    return df_dx, d2f_dx2

def optimize_with_newton(initial_guess, max_iter=10):
    x_opt = initial_guess
    for _ in range(max_iter):
        df_dx, d2f_dx2 = gradient_hessian(x_opt)
        x_opt = x_opt - df_dx / d2f_dx2
    return x_opt

def gaussian_process_optimization(initial_points, objective_function, bounds, n_iter=10):
    def objective_function_gp(X):
        return np.array([objective_function(x) for x in X])

    result = gp_minimize(objective_function_gp, bounds, acq_func="LCB", n_calls=n_iter + 1, random_state=42, x0=initial_points)
    return result.x[0]

# Task 1: Analyze conditioning
N = 10
m = 5
beta = 1.0
condition_full, condition_partial = conditioning_analysis(N, m, beta)
print(f"Conditioning for full matrix: {condition_full}")
print(f"Conditioning for partial matrix: {condition_partial}")

# Optimize with Newton's method
initial_guess_newton = 0.5
x_opt_newton = optimize_with_newton(initial_guess_newton)
print(f"Optimal solution with Newton's method: {x_opt_newton}")

# Gaussian process optimization
initial_points_gp = np.random.rand(5)
bounds_gp = [(0.0, 1.0)]

x_opt_gp = gaussian_process_optimization(initial_points_gp, objective_function, bounds_gp, n_iter=10)
print(f"Optimal solution with Gaussian process optimization: {x_opt_gp}")

# Compare methods
x_values = np.linspace(0, 1, 1000)
y_true = objective_function(x_values)

plt.plot(x_values, y_true, label="True Function")
plt.scatter(x_opt_newton, objective_function(x_opt_newton), color="red", label="Newton's Method")
plt.scatter(x_opt_gp, objective_function(x_opt_gp), color="green", label="Gaussian Process")
plt.legend()
plt.xlabel("x")
plt.ylabel("f(x)")
plt.title("Comparison of Optimization Methods")
plt.show()

错误类型1（频繁出现） #

result = gp_minimize(objective_function_gp, bounds, acq_func="LCB", n_calls=n_iter + 1, random_state=42, x0=initial_points)

  File ~\anaconda3\Lib\site-packages\skopt\optimizer\gp.py:259 in gp_minimize
    return base_minimize(

  File ~\anaconda3\Lib\site-packages\skopt\optimizer\base.py:266 in base_minimize
    raise RuntimeError("Optimization space (%s) and initial points in x0 "

RuntimeError: Optimization space (Space([Real(low=0.0, high=1.0, prior='uniform', transform='normalize')])) and initial points in x0 use inconsistent dimensions.

错误类型2（调整后偶发） #

File ~\anaconda3\Lib\site-packages\skopt\optimizer\gp.py:259 in gp_minimize
    return base_minimize(

  File ~\anaconda3\Lib\site-packages\skopt\optimizer\base.py:291 in base_minimize
    result = optimizer.tell(x0, y0)

  File ~\anaconda3\Lib\site-packages\skopt\optimizer\optimizer.py:482 in tell
    check_x_in_space(x, self.space)

  File ~\anaconda3\Lib\site-packages\skopt\utils.py:187 in check_x_in_space
    if not np.all([p in space for p in x]):

  File ~\anaconda3\Lib\site-packages\skopt\utils.py:187 in <listcomp>
    if not np.all([p in space for p in x]):

  File ~\anaconda3\Lib\site-packages\skopt\space\space.py:1035 in __contains__
    if component not in dim:

  File ~\anaconda3\Lib\site-packages\skopt\space\space.py:364 in __contains__
    return self.low <= point <= self.high

ValueError: The truth value of an array with more than one element is ambiguous. Use a.any() or a.all()

编辑补充 #

调整代码后成功生成图表，但仅能看到牛顿法的结果，高斯过程的结果未显示。调整后的图表代码如下：

x_values = np.linspace(0, 1, 1000)
y_true = [objective_function(x) for x in x_values]
plt.plot(x_values, y_true, label="True Function")
plt.scatter(x_opt_newton, objective_function(x_opt_newton), color="red", label="Newton's Method")
plt.scatter(x_opt_gp, objective_function(x_opt_gp), color="green", label="Gaussian Process")
plt.legend()
plt.xlabel("x")
plt.ylabel("f(x)")
plt.title("Comparison of Optimization Methods")
plt.show()

内容的提问来源于stack exchange，提问作者Kurapika