Constrained multi-variable non-linear regression using gekko

Question

I am trying to constrain multi-variable non-linear regression using gekko. I constrained parameters c[i], but can't constrain part of function having both parameters and variables.

Basically, 0 < c[i] (xd[10]**c[10]) (xd[11]**(c[11]+c[12]*xd[12])) < 1 is what I need

import numpy
import pandas as pd

#Dataframe from excel (imported x0-x12 from excel)

#Gekko:

from gekko import GEKKO

m = GEKKO(remote=False); m.options.IMODE=2#parameter regression

c  = m.Array(m.FV,13)#array of 13 fixed parameter

for ci in c:
    ci.STATUS=1#calculate fixed parameter

    ci.LOWER =0#constraint: lower limit

    ci.UPPER =1#constraint: lower limit

#Define variables

xd = m.Array(m.Param,13); xd[0].value=x0; xd[1].value=x1; xd[2].value=x2; xd[3].value=x3; 
xd[4].value=x4; xd[5].value=x5; xd[6].value=x6; xd[7].value=x7; xd[8].value=x8; xd[9].value=x9;
xd[10].value=x10; xd[11].value=x11; xd[12].value=x12


yd = m.Param(y); yp = m.Var()

#Equation of custom function

m.Equation(yp==(c[0]*xd[0]+c[1]*xd[1]+c[2]*xd[2]+c[3]*xd[3]+c[4]*xd[4]+c[5]*xd[5]+c[6]*xd[6]+c[7]*xd[7]+
                c[8]*xd[8]+c[9]*xd[9])*(xd[10]**c[10])*(xd[11]**(c[11]+c[12]*xd[12])))

#Minimize difference between actual and predicted y

m.Minimize((yd-yp)**2)

#Solve

m.solve(disp=False)

#Retrain parameter values

a = [c[i].value[0] for i in range(13)]

print(a) 

Answer 1

You can define new inequality constraints.

m.Equation(xd[10]**c[10])*(xd[11]**(c[11]+c[12]*xd[12])>=0)
m.Equation(xd[10]**c[10])*(xd[11]**(c[11]+c[12]*xd[12])<=1)

or define a new variable with lower and upper bounds.

#Inequality Constraints
z = m.Var(lb=0,ub=1)
m.Equation(z==(xd[10]**c[10])*(xd[11]**(c[11]+c[12]*xd[12])))

Here is a full script with the second approach.

import numpy as np
import pandas as pd

#Dataframe from excel (imported x0-x12 from excel)
x = [np.random.rand(10) for i in range(13)] 
y = np.random.rand(10)

#Gekko:
from gekko import GEKKO

m = GEKKO(remote=False); m.options.IMODE=2

c  = m.Array(m.FV,13)#array of 13 fixed parameter

for ci in c:
    ci.STATUS = 1 #calculate fixed parameter
    ci.LOWER  = 0 #constraint: lower limit
    ci.UPPER  = 1 #constraint: lower limit

#Define variables
xd = m.Array(m.Param,13)
for i in range(13):
    xd[i].value = x[i]
yd = m.Param(y); yp = m.Var()

#Equation of custom function
m.Equation(yp==(c[0]*xd[0]+c[1]*xd[1]+c[2]*xd[2]+\
                c[3]*xd[3]+c[4]*xd[4]+c[5]*xd[5]+\
                c[6]*xd[6]+c[7]*xd[7]+c[8]*xd[8]+\
                c[9]*xd[9])*(xd[10]**c[10])*\
                   (xd[11]**(c[11]+c[12]*xd[12])))

#Inequality Constraints
z = m.Var(lb=0,ub=1)
m.Equation(z==(xd[10]**c[10])*(xd[11]**(c[11]+c[12]*xd[12])))

#Minimize difference between actual and predicted y
m.Minimize((yd-yp)**2)

#Solve
m.solve(disp=True)

#Retrieve parameter values
a = [c[i].value[0] for i in range(13)]

print(a) 

This uses random numbers instead of the data from the Excel import that wasn't available from your question.