目标 function 不使用 Gekko 收敛

Question

I tried running this code but the objective function diverges even when I raise the maximum of iterations (raised it to 2000 and still didn't find a solution).

Any help is welcome.

The code:

from __future__ import division
import numpy as np
import matplotlib.pyplot as plt
from gekko import GEKKO

m = GEKKO()

plt.rcParams['font.size'] = 15

# Parameters
nu_float = 0.5
beta_E_float = 10.0
tau_E_float=0.05
delta_E_float=0.03
beta_F_float=0.01
s_h_float = 0.9951
delta_F_float = 0.04
eta_float = 0.95
delta_float = 1.25
F_star = 5106.0
b_float = nu_float*beta_F_float*beta_E_float/(tau_E_float+delta_E_float)#3.125
print('b='+str(b_float))
K_float = F_star/(nu_float*beta_F_float/delta_F_float-nu_float*beta_F_float/b_float)
print('K='+str(K_float))
nu=m.Param(value=nu_float)
beta_E=m.Param(value=beta_E_float)
tau_E=m.Param(value=tau_E_float)
delta_E=m.Param(value=delta_E_float)
beta_F=m.Param(value=beta_F_float)
b=m.Param(value=b_float)
s_h=m.Param(value=s_h_float)
delta_F=m.Param(value=delta_F_float)
eta=m.Param(value=eta_float)
delta=m.Param(value=delta_float)
K=m.Param(value=K_float)
T = 80
C = 1000
U_bar = 150


# Time interval
nt = 500
m.time = np.linspace(0,T,nt)
delta_t = T/nt


# Control
u = m.MV(lb=0,ub=U_bar)
u.STATUS = 1
u.DCOST = 0


# Variables
E_u = m.Var(value=K*(1-delta_F/b))
F_u = m.Var(value=F_star)
E_i = m.Var(value=0.0)
F_i = m.Var(value=0.0)
U = m.Var(value=0.0)


# Vector to extract the final value
final_array = np.zeros(nt)
final_array[-1] = 1.0
final = m.Param(value=final_array)


# Equations
m.Equation(E_u.dt()==beta_E*F_u*(1-s_h*F_i/(F_u+F_i))*(1-(E_u+E_i)/K)-(tau_E+delta_E)*E_u)
m.Equation(F_u.dt()==nu*beta_F*E_u-delta_F*F_u)
m.Equation(E_i.dt()==eta*beta_E*F_i*(1-(E_u+E_i)/K)-(tau_E+delta_E)*E_i)
m.Equation(F_i.dt()==nu*beta_F*E_i-delta*delta_F*F_i+u)
m.Equation(U.dt()==u)
m.Equation(final*U<=C)


# Objective Function
m.Obj(0.5*(final*E_u)**2+0.5*(0.5*(abs(K*(1-delta*delta_F/(b*eta))-final*E_i)+K*(1-delta*delta_F/(b*eta))-final*E_i))**2+0.5*(final*F_u)**2+0.5*(0.5*(abs(K*(nu*beta_F/(delta*delta_F)-nu*beta_F/(b*eta))-final*F_i)+K*(nu*beta_F/(delta*delta_F)-nu*beta_F/(b*eta))-final*F_i))**2)


# Resolution
m.options.IMODE = 6
m.options.NODES = 4
m.options.MV_TYPE = 1
m.options.SOLVER = 3
print('beginning of computations')
m.solve()
print('end of computations')


# Print results
E_u_array=np.transpose(np.matrix(E_u))
F_u_array=np.transpose(np.matrix(F_u))
E_i_array=np.transpose(np.matrix(E_i))
F_i_array=np.transpose(np.matrix(F_i))
u_array=np.transpose(np.matrix(u))
U_array=np.transpose(np.matrix(U))
t_array = np.arange(nt)*T/nt
E_u_T = float(final_array*E_u_array)
F_u_T = float(final_array*F_u_array)
E_i_T = float(final_array*E_i_array)
F_i_T = float(final_array*F_i_array)
print(r'$E_u(T):$ ' + str(E_u_T))
print(r'$F_u(T):$ ' + str(F_u_T))
print(r'$E_i(T):$ ' + str(E_i_T))
print(r'$F_i(T):$ ' + str(F_i_T))

The error

 
 An error occured.
 The error code is           -1
 
 
 ---------------------------------------------------
 Solver         :  IPOPT (v3.12)
 Solution time  :    124.735700000005      sec
 Objective      :    11931117306673.5
 Unsuccessful with error code            0
 ---------------------------------------------------
 
 Creating file: infeasibilities.txt
 Use command apm_get(server,app,'infeasibilities.txt') to retrieve file
 @error: Solution Not Found
Traceback (most recent call last):
  File "C:\Users\dave\tipe 2k22.py", line 86, in <module>
    m.solve()
  File "c:\users\dave\appdata\local\programs\python\python37\lib\site-packages\gekko\gekko.py", line 2185, in solve
    raise Exception(response)
Exception:  @error: Solution Not Found

Answer 1

The abs() functions in the objective function are not differentiable at x=0 . Try switching to m.abs3() for a binary variable switch that is continuously differentiable.

# Objective Function
m.Minimize(0.5*(final*E_u)**2\
      +0.5*(0.5*(m.abs3(K*(1-delta*delta_F/(b*eta))-final*E_i)\
      +K*(1-delta*delta_F/(b*eta))-final*E_i))**2+0.5*(final*F_u)**2\
      +0.5*(0.5*(m.abs3(K*(nu*beta_F/(delta*delta_F)\
      -nu*beta_F/(b*eta))-final*F_i)+K*(nu*beta_F/(delta*delta_F)\
      -nu*beta_F/(b*eta))-final*F_i))**2)

You'll also need to switch to the solver APOPT to calculate the Mixed Integer Nonlinear Programming (MINLP) solution with m.options.SOLVER=1 . If you leave out the m.options.SOLVER line then Gekko automatically selects it when the m.abs3() function is used. You can override it with another solver selection, if desired, but it won't necessarily return an integer solution. More information on the binary form of the abs() function is in the Optimization online course (Logical Conditions in Optimization) .

from __future__ import division
import numpy as np
import matplotlib.pyplot as plt
from gekko import GEKKO

m = GEKKO()

plt.rcParams['font.size'] = 15

# Parameters
nu_float = 0.5
beta_E_float = 10.0
tau_E_float=0.05
delta_E_float=0.03
beta_F_float=0.01
s_h_float = 0.9951
delta_F_float = 0.04
eta_float = 0.95
delta_float = 1.25
F_star = 5106.0
b_float = nu_float*beta_F_float*beta_E_float/(tau_E_float+delta_E_float)#3.125
print('b='+str(b_float))
K_float = F_star/(nu_float*beta_F_float/delta_F_float-nu_float*beta_F_float/b_float)
print('K='+str(K_float))
nu=m.Param(value=nu_float)
beta_E=m.Param(value=beta_E_float)
tau_E=m.Param(value=tau_E_float)
delta_E=m.Param(value=delta_E_float)
beta_F=m.Param(value=beta_F_float)
b=m.Param(value=b_float)
s_h=m.Param(value=s_h_float)
delta_F=m.Param(value=delta_F_float)
eta=m.Param(value=eta_float)
delta=m.Param(value=delta_float)
K=m.Param(value=K_float)
T = 80
C = 1000
U_bar = 150

# Time interval
nt = 100
m.time = np.linspace(0,T,nt)
delta_t = T/nt

# Control
u = m.MV(lb=0,ub=U_bar)
u.STATUS = 1
u.DCOST = 0

# Variables
E_u = m.Var(value=K*(1-delta_F/b))
F_u = m.Var(value=F_star)
E_i = m.Var(value=0.0)
F_i = m.Var(value=0.0)
U = m.Var(value=0.0)

# Vector to extract the final value
final_array = np.zeros(nt)
final_array[-1] = 1.0
final = m.Param(value=final_array)

# Equations
m.Equation(E_u.dt()==beta_E*F_u*(1-s_h*F_i/(F_u+F_i))\
                     *(1-(E_u+E_i)/K)-(tau_E+delta_E)*E_u)
m.Equation(F_u.dt()==nu*beta_F*E_u-delta_F*F_u)
m.Equation(E_i.dt()==eta*beta_E*F_i*(1-(E_u+E_i)/K)\
                     -(tau_E+delta_E)*E_i)
m.Equation(F_i.dt()==nu*beta_F*E_i-delta*delta_F*F_i+u)
m.Equation(U.dt()==u)

m.Equation(final*U<=C)

# Objective Function
m.Minimize(0.5*(final*E_u)**2\
      +0.5*(0.5*(m.abs3(K*(1-delta*delta_F/(b*eta))-final*E_i)\
      +K*(1-delta*delta_F/(b*eta))-final*E_i))**2+0.5*(final*F_u)**2\
      +0.5*(0.5*(m.abs3(K*(nu*beta_F/(delta*delta_F)\
      -nu*beta_F/(b*eta))-final*F_i)+K*(nu*beta_F/(delta*delta_F)\
      -nu*beta_F/(b*eta))-final*F_i))**2)

# Resolution
m.options.IMODE = 6
m.options.NODES = 3
m.options.MV_TYPE = 1
m.options.SOLVER = 1
print('beginning of computations')
m.solve()
print('end of computations')

# Print results
E_u_array=np.transpose(np.matrix(E_u))
F_u_array=np.transpose(np.matrix(F_u))
E_i_array=np.transpose(np.matrix(E_i))
F_i_array=np.transpose(np.matrix(F_i))
u_array=np.transpose(np.matrix(u))
U_array=np.transpose(np.matrix(U))
t_array = np.arange(nt)*T/nt
E_u_T = float(final_array*E_u_array)
F_u_T = float(final_array*F_u_array)
E_i_T = float(final_array*E_i_array)
F_i_T = float(final_array*F_i_array)
print(r'$E_u(T):$ ' + str(E_u_T))
print(r'$F_u(T):$ ' + str(F_u_T))
print(r'$E_i(T):$ ' + str(E_i_T))
print(r'$F_i(T):$ ' + str(F_i_T))

This produces a successful solution:

 --------- APM Model Size ------------
 Each time step contains
   Objects      :            0
   Constants    :            0
   Variables    :           27
   Intermediates:            0
   Connections  :            0
   Equations    :           13
   Residuals    :           13
 
 Number of state variables:           4356
 Number of total equations: -         3861
 Number of slack variables: -          990
 ---------------------------------------
 Degrees of freedom       :           -495
 
 * Warning: DOF <= 0
 ----------------------------------------------
 Dynamic Control with APOPT Solver
 ----------------------------------------------
Iter:     1 I:  0 Tm:      5.72 NLPi:    7 Dpth:    0 Lvs:    0 Obj:  1.61E+11 Gap:  0.00E+00
 Successful solution
 
 ---------------------------------------------------
 Solver         :  APOPT (v1.0)
 Solution time  :    5.74739999999292      sec
 Objective      :    160555572371.779     
 Successful solution
 ---------------------------------------------------
 
end of computations
$E_u(T):$ 40801.053817
$F_u(T):$ 5092.8119666
$E_i(T):$ 39.66478887
$F_i(T):$ 63.713604115