Math domain error due to disrespected constraint in scipy SLSQP minimize

Question

Consider a simple problem:

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

To solve the problem with scipy.optimize.minimize:

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')

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.

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.

Answer 1

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

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

To calculate a and b, we have to satisfy:

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

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

And it looks like this: