GEKKO gives @error: Solution Not Found when constraint has equality

when I solve the following problem I get: Exception: @error: Solution Not Found

from gekko import GEKKO
import pandas as pd
import numpy as np

m = GEKKO(remote=True)



alpha=[6.97, 4.97, 6.55]
beta = [-0.04, -0.11, -0.1]
MP = [2707.2, 2213.3, 1904.1]
Plower = [1700, 1400, 1100]
Pupper = [5000, 5000, 5000]

h= 250 ; b= 150 ; c= 100
T= len(alpha)

D = m.Array(m.Var, T ,lb=-0, integer=False)
P = m.Array(m.Var, T ,lb=-0,  integer=False)


m.Maximize( sum(D[t]*(P[t]-MP[t]) for t in range (T)))


m.Equations(D[t]==alpha[t]+beta[t]*P[t] for t in range (T))


#bound
m.Equations(P[t]>=  Plower[t]for t in range (T))
m.Equations(P[t]<= Pupper[t] for t in range (T))
m.options.SOLVER=1
m.solve(disp=True) # solve on public server

then, I reformulate the objective function as m.Maximize( sum((alpha[t]+beta[t]*P[t])(P[t]-MP[t]) for t in range (T))) and eliminate the constraint and I get the solution. In other words, I replace D[t] using the constraint.

Can you please explain the logic behind the error. Is there any problem with equality constraint?

Solvers can often perform better with simplification of the model. This is an excellent example of a simple reformulation to eliminate variables D[t]. Another way to do this in Gekko is to use an Intermediate variable.

D = [m.Intermediate(alpha[t]+beta[t]*P[t]) for t in range (T)]
m.Maximize(sum(D[t]*(P[t]-MP[t]) for t in range (T)))

There is also the m.options.REDUCE=3 option that automates the elimination of variables from the model. It analyzes simple equations that can be removed and variable values fixed with a pre-solve before the problem is given to the solver. This option wouldn't help with this model. Here are the three approaches:

1. Original (Infeasible):

from gekko import GEKKO
import pandas as pd
import numpy as np

m = GEKKO(remote=True)
alpha=[6.97, 4.97, 6.55]
beta = [-0.04, -0.11, -0.1]
MP = [2707.2, 2213.3, 1904.1]
Plower = [1700, 1400, 1100]
Pupper = [5000, 5000, 5000]
h= 250 ; b= 150 ; c= 100
T= len(alpha)
D = m.Array(m.Var, T ,lb=-0, integer=False)
P = m.Array(m.Var, T ,lb=-0,  integer=False)
m.Maximize( sum(D[t]*(P[t]-MP[t]) for t in range (T)))
m.Equations(D[t]==alpha[t]+beta[t]*P[t] for t in range (T))
m.Equations(P[t]>=  Plower[t]for t in range (T))
m.Equations(P[t]<= Pupper[t] for t in range (T))
m.options.SOLVER=1
m.solve(disp=True) # solve on public server

2. Objective Function Substitution (Successful):

from gekko import GEKKO
import pandas as pd
import numpy as np

m = GEKKO(remote=True)
alpha=[6.97, 4.97, 6.55]
beta = [-0.04, -0.11, -0.1]
MP = [2707.2, 2213.3, 1904.1]
Plower = [1700, 1400, 1100]
Pupper = [5000, 5000, 5000]
h= 250 ; b= 150 ; c= 100
T= len(alpha)
D = m.Array(m.Var, T ,lb=-0, integer=False)
P = m.Array(m.Var, T ,lb=-0,  integer=False)
m.Maximize(sum((alpha[t]+beta[t]*P[t])*(P[t]-MP[t]) for t in range (T)))
m.Equations(P[t]>=Plower[t] for t in range (T))
m.Equations(P[t]<=Pupper[t] for t in range (T))
m.options.SOLVER=1
m.solve(disp=True) # solve on public server

3. Intermediate Variable (Successful):

from gekko import GEKKO
import pandas as pd
import numpy as np

m = GEKKO(remote=True)
alpha=[6.97, 4.97, 6.55]
beta = [-0.04, -0.11, -0.1]
MP = [2707.2, 2213.3, 1904.1]
Plower = [1700, 1400, 1100]
Pupper = [5000, 5000, 5000]
h= 250 ; b= 150 ; c= 100
T= len(alpha)
D = m.Array(m.Var, T ,lb=-0, integer=False)
P = m.Array(m.Var, T ,lb=-0,  integer=False)
D = [m.Intermediate(alpha[t]+beta[t]*P[t]) for t in range (T)]
m.Maximize(sum(D[t]*(P[t]-MP[t]) for t in range (T)))
m.Equations(P[t]>=Plower[t] for t in range (T))
m.Equations(P[t]<=Pupper[t] for t in range (T))
m.options.SOLVER=1
m.solve(disp=True) # solve on public server

I prefer Intermediate variables because they keep the model easier to read and diagnose than using very long equations or objective function definitions.