How to solve Absolute Value abs() objective with Python Gekko?

Question

An optimization problem with a squared objective solves successfully with IPOPT in Python Gekko.

from gekko import GEKKO
import numpy as np
m = GEKKO()
x = m.Var(); y = m.Param(3.2)
m.Obj((x-y)**2)
m.solve()
print(x.value[0],y.value[0])

However, when I switch to an absolute value objective np.abs(x-y) (the numpy version of abs) or m.abs(x-y) (the Gekko version of abs), the IPOPT solver reports a failed solution. An absolute value approximation m.sqrt((x-y)**2) also fails.

Failed Solution

from gekko import GEKKO
import numpy as np
m = GEKKO()
x = m.Var(); y = m.Param(3.2)
m.Obj(m.abs(x-y))
m.solve()
print(x.value[0],y.value[0])

I understand that gradient-based solvers don't like functions without continuous first and second derivatives so I suspect that this is happening with abs() where 0 is a point that does not have continuous derivatives. Is there any alternative to abs() to reliably solve an absolute value with gradient-based solvers in Python Gekko?

Answer 1

You can use m.abs2 instead, It takes into account the issue with the derivative and should solve the issue.

Answer 2

Here is one possible solution using gekko's binary switch variable:

from gekko import GEKKO
import numpy as np
m = GEKKO()
y = m.Param(3.2)
x = m.Var()
#intermediate
difference = m.Intermediate(x - y)

f = m.if3(difference, -difference, difference)

m.Obj(f)
m.solve()
print(x.value[0],y.value[0])

Returns: 3.2 3.2

m.if3(condition, x1, x2) takes value as a condition, and returns x1 if condition >= 0 or x1 if condition < 0.

There are various functions to get around this problem in the logical functions section of the documentation, including m.abs2, m.abs3, and m.if2.

The type 2 functions use MPCC to solve and will continue using IPOPT. The type 3 functions will change to APOPT automatically.

https://github.com/BYU-PRISM/GEKKO/blob/master/docs/model_methods.rst https://gekko.readthedocs.io/en/latest/model_methods.html#logical-functions