(Python)RuntimeWarning：在 double_scalars 中遇到无效值

Question

我想用scipy.optimize.root求解静态平衡模型。具体来说，我需要找到三维函数系统的根。代码如下：

# Import package
import numpy as np
from scipy.optimize import root,fsolve

# Parameters
Kbar = 10 # Total capital
Lbar = 20 # Total labour
α = 0.3
β = np.array([0.3,0.6])
p = np.array([1.0,0.1]) 
# The first element of the price array is p1=1 (constant), the second is p2 which needs to be optimized.

# Defining the objective function system
def markets(x):
    # prices (excluded p1=1), x is a 3-dimentional array
    p[1] = x[0] # p2 is the first element of input x
    w    = x[1] # wage w is the second element of input x
    r    = x[2] # interest rate r is the third elemendt of input x
    # total income
    Ybar = w * Lbar + r * Kbar
    # get market equation
    sol[0] = 1/p[0]-(β[0]/w)**β[0]*((1-β[0])/r)**(1-β[0])
    sol[1] = 1/p[1]-(β[1]/w)**β[1]*((1-β[1])/r)**(1-β[1])
    sol[2] = β[0]*α*Ybar/w + β[1]*(1-α)*Ybar/w - Lbar
    return sol

# Initial guess x0 is a 3*1 array
x0 = np.zeros(3) + 5 

# Find market equilibrium
res = root(markets, x0)
print(res)

但是结果报错：

fjac: array([[1., 0., 0.],
       [0., 1., 0.],
       [0., 0., 1.]])
     fun: array([nan, nan, nan])
 message: 'The iteration is not making good progress, as measured by the \n  improvement from the last ten iterations.'
    nfev: 17
     qtf: array([ 0.89142371,  0.09796604, -4.69999992])
       r: array([nan, nan, nan, nan, nan, nan])
  status: 5
 success: False
       x: array([5., 5., 5.])
<ipython-input-37-15479e9f467a>:24: RuntimeWarning: invalid value encountered in double_scalars
  sol[0] = 1/p[0]-(β[0]/w)**β[0]*((1-β[0])/r)**(1-β[0])
<ipython-input-37-15479e9f467a>:25: RuntimeWarning: invalid value encountered in double_scalars
  sol[1] = 1/p[1]-(β[1]/w)**β[1]*((1-β[1])/r)**(1-β[1])

我搜索了一些关于“RuntimeWarning: invalid value encountered in double_scalars”的信息，知道除以零时会出现这个错误。但是，我没有在我的函数中发现任何可能的错误。

谁能帮我解决这个问题？非常感谢。

Answer 1

首先，这不是一个最小工作示例 (MWE)，因为您没有在 markets 函数中定义 sol。所以让我们假设

# Defining the objective function system
def markets(x):
    # prices (excluded p1=1), x is a 3-dimentional array
    p[1] = x[0] # p2 is the first element of input x
    w    = x[1] # wage w is the second element of input x
    r    = x[2] # interest rate r is the third elemendt of input x
    # total income
    Ybar = w * Lbar + r * Kbar
    # get market equation
    sol = np.zeros(3)
    sol[0] = 1/p[0]-(β[0]/w)**β[0]*((1-β[0])/r)**(1-β[0])
    sol[1] = 1/p[1]-(β[1]/w)**β[1]*((1-β[1])/r)**(1-β[1])
    sol[2] = β[0]*α*Ybar/w + β[1]*(1-α)*Ybar/w - Lbar
    return sol

那么，正如您已经提到的，错误是由于 1/p[1] 项中被零除造成的。因此，你需要找到一个根 x > 0，即我们寻找一个点 x > 0 使得 markets(x) = 0。但是，scipy.optimize.root 方法不支持简单的变量边界。

通过将根问题写成等效约束最小化问题：

min ||markets(x)|| s.t. x >= eps

其中eps > 0，可以借助scipy.optimize.minimize解决问题：

from scipy.optimize import minimize

# Variable bounds
eps = 1.0e-4
bnds = [(eps, None) for _ in range(3)]

# initial point
x0 = np.zeros(3) + 5

# Solve the minimization problem
res = minimize(lambda x: np.linalg.norm(markets(x)), x0=x0, bounds=bnds)

(Python)RuntimeWarning：在 double_scalars 中遇到无效值

(Python) RuntimeWarning: invalid value encountered in double_scalars

python

numpy

scipy-optimize