由于scipy SLSQP中不受约束的约束而导致的数学域错误最小化 [英] Math domain error due to disrespected constraint in scipy SLSQP minimize

问题描述

考虑一个简单的问题:

max log(x)
subject to x >= 1e-4

使用 scipy.optimize.minimize 解决问题:

import numpy as np
from scipy.optimize import minimize
from math import log

def func(x):
    return log(x[0])

def func_deriv(x):
    return np.array([1 / x[0]])

cons = ({'type': 'ineq',
         'fun' : lambda x: x[0] - 1e-4,
         'jac' : lambda x: np.array([1])})
minimize(func, [1.0], jac=func_deriv, constraints=cons, method='SLSQP')

该脚本遇到 ValueError ，因为 log(x)用负的 x 求值.似乎即使不满足约束条件也要对函数值进行评估.

The script encounters ValueError because log(x) is evaluated with negative x. It seems that the function value is evaluated even if the constraint is not satisfied.

我知道在 minimize()中使用 bounds 可以避免该问题，但这只是我原来问题的一种简化.在我最初的问题中，约束 x> = 1e-4 不能轻易地表示为 x 的边界，而是形式为 g(x)> = C ，所以 bounds 会无济于事.

I understand that using bounds in minimize() could avoid the problem, but this is just a simplification of my original problem. In my original problem, the constraint x >= 1e-4 cannot be represented easily as bounds of x, but rather of the form g(x) >= C, so bounds wouldn't help.

推荐答案

如果我们只关心 x>ε，可以定义扩展域的安全函数.

If we only care about the function value with x > ε, it is possible to define a safe function extending the domain.

以 log 函数为例.可以使用另一个三次方函数扩展 log ，同时使桥点ε平滑:

Take the log function as an example. It is possible to extend log with another cubic function, while making the bridge point ε smooth:

safe_log(x) = log(x) if x > ε else a * (x - b)**3

要计算 a 和 b ，我们必须满足:

To calculate a and b, we have to satisfy:

log(ε) = a * (ε - b)**3
1 / ε = 3 * a * (ε - b)**2

因此有safe_log函数:

Hence the safe_log function:

eps = 1e-3

def safe_log(x):
    if x > eps:
        return log(x)
    logeps = log(eps)
    a = 1 / (3 * eps * (3 * logeps * eps)**2)
    b = eps * (1 - 3 * logeps)
    return a * (x - b)**3

它看起来像这样:

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