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pythonnonlinear-optimizationgekko

Gekko optimisation error with a nonlinear function and 2 inputs

I want to maximise a concave function with two inputs

max 2 * x1 ** .8 + 1.4 * x2 ** .9

st x1 + x2 == C

using Gekko, but I get a error code -2.

 
from gekko import GEKKO

m = GEKKO()

C = m.Param(value=10)

x1, x2 = [m.Var(lb=0, ub=10) for i in range(2)]

x1.value = 5
x2.value = 5

m.Equation(x1 + x2 == C)

m.Obj(2 * x1 ** .8 + 1.4 * x2 ** .9)

m.options.IMODE = 3

m.solve()

print(x1.value)
print(x2.value)
Solution

  • There is a successful solution by switching to the APOPT solver m.options.SOLVER = 1. In this case the default solver, IPOPT, fails to find a solution but APOPT succeeds.

    from gekko import GEKKO
m = GEKKO()
x1, x2 = m.Array(m.Var,2,value=5,lb=0,ub=10)
m.Equation(x1+x2 == 10)
m.Minimize(2 * x1 ** .8 + 1.4 * x2 ** .9)

m.options.IMODE = 3
m.options.SOLVER = 1
m.solve()

print(x1.value)
print(x2.value)

    contour plot

    A contour plot with the solution shows that it did reach the optimal value along the black line (x1+x2=10 constraint).

    # Generate a contour plot
import numpy as np
import matplotlib.pyplot as plt

# Design variables at mesh points
xg = np.arange(0.0, 10.0, 0.1)
yg = np.arange(0.0, 10.0, 0.1)
x1g,x2g = np.meshgrid(xg, yg)

# Equation / Constraint
eq1 = x1g+x2g

# Objective
obj = 2*x1g**0.8 + 1.4*x2g**0.9

# Create a contour plot
plt.figure()
# Objective
CS = plt.contour(x1g,x2g,obj)
plt.clabel(CS, inline=1, fontsize=10)
# Equation
CS = plt.contour(x1g,x2g,eq1,[10.0],colors='k',linewidths=[4.0])
plt.clabel(CS, inline=1, fontsize=10)
# Plot optimal point
plt.plot(x1.value[0],x2.value[0],'o',color='orange',markersize=10)
plt.xlabel('x1'); plt.ylabel('x2')
plt.savefig('contour.png')
plt.show()

    The orange dot is the optimal solution at x1=0 and x2=10.

    Edit: Maximize instead of Minimize

    The problem statement is Maximize instead of Minimize. Thanks for the correction.

    Maximize Objective

    from gekko import GEKKO
m = GEKKO()
x1, x2 = m.Array(m.Var,2,value=5,lb=0,ub=10)
m.Equation(x1+x2 == 10)
m.Maximize(2 * x1 ** .8 + 1.4 * x2 ** .9)

m.options.IMODE = 3
m.options.SOLVER = 1
m.solve()

print(x1.value)
print(x2.value)

# Generate a contour plot
import numpy as np
import matplotlib.pyplot as plt

# Design variables at mesh points
xg = np.arange(0.0, 10.0, 0.1)
yg = np.arange(0.0, 10.0, 0.1)
x1g,x2g = np.meshgrid(xg, yg)

# Equation / Constraint
eq1 = x1g+x2g

# Objective
obj = 2*x1g**0.8 + 1.4*x2g**0.9

# Create a contour plot
plt.figure()
# Objective
CS = plt.contour(x1g,x2g,obj)
plt.clabel(CS, inline=1, fontsize=10)
# Equation
CS = plt.contour(x1g,x2g,eq1,[10.0],colors='k',linewidths=[4.0])
plt.clabel(CS, inline=1, fontsize=10)
# Plot optimal point
plt.plot(x1.value[0],x2.value[0],'o',color='orange',markersize=10)
plt.xlabel('x1'); plt.ylabel('x2')
plt.savefig('contour.png')
plt.show()