Exception: @error: Max Equation Length in Gekko

Question

I am doing a portfolio optimization using Gekko. I am using the returns per asset and its covariance matrix as input. The code I am trying is

# Initialize Model
m = GEKKO(remote=True)

#initialize variables
w=m.Array(m.Var,23)

for i in range(23):
    w[i].value = 0.04347826
    w[i].lower = 0.0
    w[i].upper = 1.0


#Equations
m.Equation(0.18 >= (np.dot(w.T,np.dot(covariance_ar, w)))**0.5)
m.Equation(np.sum(w[0:23]) == 1.0)
m.Equation(w[7]+w[9]+w[11] >= (np.sum(w[7:13])*0.3))
m.Equation(w[7]+w[9]+w[11] <= (np.sum(w[7:13])*0.7))
m.Equation(w[11]+w[12] >= 0.0)
m.Equation(w[11]+w[12] <= (np.sum(w[7:13])*0.15))


#Objective
m.Obj(-(np.sum(return_ar.T @ w)))

#Solve simulation
m.solve() # solve on public server

#Results
for i in range(23):
    print('w['+str(i)+']='+str(w[i].value))

return_ar is an array of returns of individual assets of shape (23,1) and covariance_ar is the covariance array of returns of shape (23,23)

Although the code ran succesfully with small number of weights, with 23 weight parameters it is throwing error Exception: @error: Max Equation Length....... APM model error: string > 15000 characters Consider breaking up the line into multiple equations

Can you help me to formulate the code constraint so that this works.

Answer 1

I couldn't run your code because it was missing values so I had to modify it with sample values.

# Initialize Model
from gekko import GEKKO
import numpy as np
m = GEKKO(remote=True)

#initialize variables
w=m.Array(m.Var,23)

for i in range(23):
    w[i].value = 0.04347826
    w[i].lower = 0.0
    w[i].upper = 1.0

#Equations
covariance_ar = np.random.rand(23,23)
m.Equation(0.18 >= (np.dot(w.T,np.dot(covariance_ar, w)))**0.5)
m.Equation(m.sum(w[0:23]) == 1.0)
m.Equation(w[7]+w[9]+w[11] >= (m.sum(w[7:13])*0.3))
m.Equation(w[7]+w[9]+w[11] <= (m.sum(w[7:13])*0.7))
m.Equation(w[11]+w[12] >= 0.0)
m.Equation(w[11]+w[12] <= (m.sum(w[7:13])*0.15))

#Objective
m.Obj(-(m.sum(w)))

#Solve simulation
m.solve() # solve on public server

#Results
for i in range(23):
    print('w['+str(i)+']='+str(w[i].value))

This reproduces the same error that you observed. You can overcome the equation length limitation by redefining your np.dot functions with list comprehensions. Also, you need to use the m.sum() gekko function instead of the numpy function for summation.

# Initialize Model
from gekko import GEKKO
import numpy as np
m = GEKKO(remote=True)

#initialize variables
w=m.Array(m.Var,23)

for i in range(23):
    w[i].value = 0.04347826
    w[i].lower = 0.0
    w[i].upper = 1.0

#Equations
C = np.random.rand(23,23)
Cw = [m.sum([m.Intermediate(C[i,j]*w[j]) \
             for j in range(23)]) \
                 for i in range(23)]
wCw = m.sum([Cw[i]*w[i] for i in range(23)])
m.Equation(0.18 >= (wCw)**0.5)
m.Equation(m.sum(w[0:23]) == 1.0)
m.Equation(w[7]+w[9]+w[11] >= (m.sum(w[7:13])*0.3))
m.Equation(w[7]+w[9]+w[11] <= (m.sum(w[7:13])*0.7))
m.Equation(w[11]+w[12] >= 0.0)
m.Equation(w[11]+w[12] <= (m.sum(w[7:13])*0.15))

#Objective
return_ar = np.ones(23)
for i in range(23):
    m.Maximize(w[i]*return_ar[i])

#Solve optimization
m.solve() # solve on public server

#Results
for i in range(23):
    print('w['+str(i)+']='+str(w[i].value))

You can now use your values for covariance_ar and return_ar. The reason this method works is that Gekko can break down the equations into smaller pieces that are solved individually. The length limitation for each equation of 15,000 characters is not just a processing limitation but also encourages better model formulation strategies that work well with sparse nonlinear programming solvers.