刚刚定义了一个新变量，现在程序陷入了无限循环

Question

我是编程的初学者，我写了一个实现优化算法的脚本。起初效果不错。但是后来我试图通过定义一个新变量使其更快，但是由于某种原因，它似乎陷入了无限循环。这是第一个版本（我已在评论中指出更改将发生的地方）：

# This program uses the Steepest Descent Method to 
# minimize the Rosenbrock function
import numpy as np
import time

# Define the Rosenbrock Function
def f(x_k):
    x, y = x_k[0, 0], x_k[0, 1] 
    return 100 * (y - x**2)**2 + (1 - x)**2

# Gradient of f 
def gradient(x_k):
    x, y = x_k[0, 0], x_k[0, 1] 
    return  np.array([[-400*x*(y-x**2)-2*(1-x), 200*(y-x**2)]])




def main():
    start = time.time()
    # Define the starting guess
    x_k = np.array([[2, 2]])
    # Define counter for number of steps
    numSteps = 0

    # Keep iterating until both components of the gradient are less than 0.1 in absolute value
    while abs((gradient(x_k)[0, 0])) > 0.1 or abs((gradient(x_k))[0, 1]) > 0.1:
        numSteps = numSteps + 1

        # Step direction
        p_k = - gradient(x_k)
        gradTrans = - p_k.T

        # Now we use a backtracking algorithm to find a step length
        alpha = 1.0
        ratio = 0.8
        c = 0.01 # This is just a constant that is used in the algorithm

        # This loop selects an alpha which satisfies the Armijo condition  

        #####################################
        ###### CHANGE WILL HAPPEN HERE ######
        #####################################

        while f(x_k + alpha * p_k) > f(x_k) + (alpha * c * (gradTrans  @ p_k))[0, 0]:
            alpha = ratio * alpha

        x_k = x_k + alpha * p_k
    end =  time.time()
    print("The number of steps is: ", numSteps)
    print("The final step is:", x_k)
    print("The gradient is: ", gradient(x_k))
    print("The elapsed time is:", round(end - start, 2), "seconds.")



main()

现在，该程序的效率非常低，因为在第二个while循环中，即使每次迭代都计算量f(x_k) + (alpha * c * (gradTrans @ p_k))[0, 0]:，尽管它是恒定的。因此，我决定将该数量命名为RHS = f(x_k) + (alpha * c * (gradTrans @ p_k))[0, 0]:，并将其放入while循环中。新代码如下。我所做的只是将这个数量定义为变量，现在程序陷入了无限循环。非常感谢您的帮助。

# This program uses the Steepest Descent Method to 
# minimize the Rosenbrock function
import numpy as np
import time

# Define the Rosenbrock Function
def f(x_k):
    x, y = x_k[0, 0], x_k[0, 1] 
    return 100 * (y - x**2)**2 + (1 - x)**2

# Gradient of f 
def gradient(x_k):
    x, y = x_k[0, 0], x_k[0, 1] 
    return  np.array([[-400*x*(y-x**2)-2*(1-x), 200*(y-x**2)]])




def main():
    start = time.time()
    # Define the starting guess
    x_k = np.array([[2, 2]])
    # Define counter for number of steps
    numSteps = 0

    # Keep iterating until both components of the gradient are less than 0.1 in absolute value
    while abs((gradient(x_k)[0, 0])) > 0.1 or abs((gradient(x_k))[0, 1]) > 0.1:
        numSteps = numSteps + 1

        # Step direction
        p_k = - gradient(x_k)
        gradTrans = - p_k.T

        # Now we use a backtracking algorithm to find a step length
        alpha = 1.0
        ratio = 0.8
        c = 0.01 # This is just a constant that is used in the algorithm

        # This loop selects an alpha which satisfies the Armijo condition  
        RHS = f(x_k) + (alpha * c * (gradTrans  @ p_k))[0, 0]

        #####################################
        ###### CHANGE HAS OCCURED ###########
        #####################################

        while f(x_k + alpha * p_k) > RHS:
            alpha = ratio * alpha

        x_k = x_k + alpha * p_k
    end =  time.time()
    print("The number of steps is: ", numSteps)
    print("The final step is:", x_k)
    print("The gradient is: ", gradient(x_k))
    print("The elapsed time is:", round(end - start), "seconds.")



main()

Answer 1

RHS需要使用新的alpha值在循环内重新计算。 （不确定如何加快速度。）