最陡峭的下降轨迹行为

Question

我已经编写了在公式1/2 * (x1^2 +γ* x2^2)给出的二次形式上执行最陡峭下降的代码。在数学上，我将Boyd's Convex Optimization一书中给出的方程作为指导，并希望重现给定的例子。我的问题是，我的最陡峭下降路径的轨迹看起来有点奇怪，与书中显示的轨迹不匹配，尽管它似乎是收敛的。

例如，我的代码输出了什么：

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它应该是什么样子：

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另一个：

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Vs它应该是什么样子：

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下面是一个可重现的例子，尽管有点长：

import numpy as np
import matplotlib.pyplot as plt

# Function value at a given point
def f(x1,x2):
    return np.power(np.e,x1 + 3*x2 - 0.1) + np.power(np.e,x1 - 3*x2 - 0.1) + np.power(np.e,- x1 - 0.1)

# Partial Derivatives
def g(x1, x2):
    g1 = np.power(np.e, x1 + 3 * x2 - 0.1) + np.power(np.e, x1 - 3 * x2 - 0.1) - np.power(np.e, - x1 - 0.1)
    g2 = np.power(3*np.e, x1 + 3 * x2 - 0.1) - np.power(3 * np.e, x1 - 3 * x2 - 0.1)
    return np.asarray([g1[0], g2[0]])

# Descent Parameters
alpha = 0.1
beta = 0.7

# Starting point
x0 = np.array([-1, 1], ndmin = 2, dtype = np.float64 ).T
f0 = f(x0[0], x0[1]) # Initial function value

# Calculate the minimum
xmin = np.array([0,0], ndmin = 2).T # Analytic minimum to the problem
fmin = f(0,0)

nangles = 1000
nopts = 1000
thetas = np.linspace(0, 2 * np.pi, nangles)
cont = np.zeros((5,2,nangles))
Delta = (f0 - fmin)/4

# function that plots the level curves or contour lines
for i in range(nangles):
    ray = np.vstack([np.linspace(0,4,nopts) * np.cos(thetas[i]),
                        np.linspace(0,4,nopts) * np.sin(thetas[i])])

    fray = f(ray[0], ray[1])

    def get_ind(expression):
        ind = np.nonzero(expression)[0].max(0)
        return(ray[:,ind - 1] + ray[:,ind]) / 2

    cont[0,:,i] = get_ind(fray < f0)
    cont[1,:,i] = get_ind(fray < f0 - 3 * Delta)
    cont[2,:,i] = get_ind(fray < f0 - 2 * Delta)
    cont[3,:,i] = get_ind(fray < f0 - Delta)
    cont[4,:,i] = get_ind(fray < f0 + Delta)

def plt_contour():
    fig, ax = plt.subplots()
    ax.plot(cont[0,0,:], cont[0,1,:], '--',
            cont[1,0,:], cont[1,1,:], '--',
            cont[2,0,:], cont[2,1,:], '--',
            cont[3,0,:], cont[3,1,:], '--',
            cont[4,0,:], cont[4,1,:], '--')
    ax.axis('equal')
    ax.axis('off')
    return fig, ax

maxiters = 100
xs = np.zeros((2, maxiters))
fs = np.zeros((1, maxiters))
x = x0.copy()
Pnorm = np.diag([8,2])

for i in range(maxiters):
    print(i)
    fval = f(x[0],x[1])
    xs[:,i:i + 1] = x
    fs[:,i] = fval
    if fval - fmin < 1e-10: #stopping criterion
        break
    # Backtracking Line Search
    gval = g(x[0],x[1])
    v = np.dot(-np.linalg.inv(Pnorm), gval) # I think this is the step that I am not doing right
    s = 1
    for k in range(10):
        xnew = x + (s * v).reshape(2,1)
        fxnew = f(xnew[0], xnew[1])
        if fxnew < fval + s * alpha * gval.T @ v:
            break
        else:
            s = s * beta
    x = x + (s * v).reshape(2,1)

# Plot path
fig, ax = plt_contour()
fig.set_size_inches(5,5)
ax.plot( xs[0,0:i + 1], xs[1, 0:i + 1], 'o')
ax.plot( xs[0,0:i + 1], xs[1, 0:i + 1], '-')
plt.show()

我有一种预感，我做错的地方是最陡峭的下降步骤，也就是变量v，也许我实际上没有在做我认为我正在做的事情。为任何帮助干杯。

Answer 1

g2计算是否正确？3不应该在幂函数之外吗？

g2 = 3*np.power(np.e, x1 + 3 * x2 - 0.1) - 3*np.power(np.e, x1 - 3 * x2 - 0.1)