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APMontor solution of Volterra fishing problem
 
 
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== Results ==
== Results with APOPT (MINLP) ==


=== IPOPT ===
An MINLP solution is calculated with [https://apopt.com APOPT] with an objective function value of <math>x_2(t_f) = 1.36</math>. APOPT requires 50 NLP solutions to find an integer solution (111 seconds of processing time). Each NLP solution in the branch and bound method requires an average of 2.2 seconds to complete with a range between 12.99 and 0.57 seconds.
The relaxed solution calculated by IPOPT (Ipopt 3.12, Linux x86_64, default settings, 4 GHz quadcore, Linux 4.2.0-23-generic) has an objective function value of <math>x_2(t_f) = 1.34428</math>. IPOPT requires 22 iterations (6.062 seconds of processing time). The following is a Gnuplot compatible tabular version of the solution:
 
[[File:Volterra_fishing_APMonitor.png]]


<pre>
<pre>
#        t,        x_0,        x_1,          u
----------------------------------------------------------------
          0,        0.5,        0.7, 5.18436e-09
APMonitor, Version 0.7.9
      0.12,    0.519594,    0.659989, 5.42459e-09
APMonitor Optimization Suite
      0.24,    0.542429,    0.623852, 5.79496e-09
----------------------------------------------------------------
      0.36,    0.568601,    0.591429, 6.30563e-09
      0.48,    0.59823,    0.562571, 6.97415e-09
        0.6,    0.631456,    0.537143,  7.827e-09
--------- APM Model Size ------------
      0.72,    0.66843,    0.515028, 8.90236e-09
Each time step contains
      0.84,    0.709312,    0.496134, 1.02545e-08
   Objects      :            0
      0.96,    0.75426,    0.480401, 1.19611e-08
  Constants    :            2
      1.08,    0.803421,      0.4678, 1.41346e-08
  Variables    :            5
        1.2,    0.856919,    0.458344, 1.69417e-08
  Intermediates:            1
      1.32,    0.914842,    0.452091, 2.06377e-08
   Connections  :            0
      1.44,    0.97722,    0.449154, 2.56285e-08
   Equations    :            5
      1.56,      1.044,    0.449708,  3.2592e-08
  Residuals   :            4
      1.68,    1.11503,   0.454003, 4.27322e-08
        1.8,    1.18998,    0.462372, 5.83658e-08
Number of state variables:          2178
      1.92,    1.26837,    0.475251, 8.44574e-08
Number of total equations: -        2057
      2.04,    1.34942,    0.493189, 1.33425e-07
Number of slack variables: -            0
      2.16,    1.43209,    0.516862, 2.44977e-07
---------------------------------------
      2.28,    1.51492,    0.547087, 6.24464e-07
Degrees of freedom      :            121
        2.4,    1.59605,    0.584816,    0.715281
      2.52,    1.61776,    0.61833,    0.999999
----------------------------------------------
      2.64,    1.61116,      0.6499,    0.999999
Dynamic Control with APOPT Solver
      2.76,    1.59844,    0.68229,          1
----------------------------------------------
      2.88,    1.57963,   0.714939,          1
Iter:     1 I:  0 Tm:    12.99 NLPi:  93 Dpth:   0 Lvs:   2 Obj:  1.34E+00 Gap:      NaN
          3,    1.55497,   0.747196,          1
Iter:     2 I:  0 Tm:      2.82 NLPi:  18 Dpth:   1 Lvs:   3 Obj:  1.34E+00 Gap:      NaN
      3.12,    1.52487,   0.778343,   0.999999
Iter:     3 I:  0 Tm:      3.39 NLPi:  30 Dpth:   2 Lvs:   4 Obj:  1.34E+00 Gap:      NaN
      3.24,    1.48993,    0.807625,    0.999999
Iter:    4 I:  0 Tm:     10.37 NLPi:  131 Dpth:   3 Lvs:   5 Obj:  1.34E+00 Gap:      NaN
      3.36,      1.4509,    0.83429,    0.999999
Iter:     5 I:  0 Tm:      1.81 NLPi:  10 Dpth:    4 Lvs:   6 Obj:  1.34E+00 Gap:      NaN
      3.48,    1.40865,    0.857635,    0.999998
Iter:     6 I:  0 Tm:      7.93 NLPi:  102 Dpth:   5 Lvs:    7 Obj:  1.34E+00 Gap:      NaN
        3.6,    1.36412,    0.877047,    0.999997
Iter:     7 I:  0 Tm:      3.22 NLPi:  28 Dpth:   6 Lvs:   8 Obj:  1.34E+00 Gap:      NaN
      3.72,    1.31827,    0.892039,    0.999991
Iter:     8 I:  0 Tm:      1.85 NLPi:  12 Dpth:   7 Lvs:   9 Obj:  1.34E+00 Gap:      NaN
      3.84,    1.27203,    0.90228,    0.999646
Iter:    9 I:  0 Tm:    10.24 NLPi:  140 Dpth:    8 Lvs:  10 Obj:  1.34E+00 Gap:      NaN
      3.96,    1.22628,    0.907616,    0.536112
Iter:   10 I:  0 Tm:      2.04 NLPi:    9 Dpth:    9 Lvs:  11 Obj:  1.35E+00 Gap:      NaN
      4.08,    1.20755,    0.919653,    0.581667
Iter:    11 I:  0 Tm:      6.50 NLPi:  86 Dpth:  10 Lvs:  12 Obj:  1.35E+00 Gap:      NaN
        4.2,    1.18502,    0.928525,    0.503883
Iter:   12 I:  0 Tm:      1.94 NLPi:  17 Dpth:  11 Lvs:  13 Obj:  1.35E+00 Gap:      NaN
      4.32,     1.16606,   0.936899,   0.464153
Iter:   13 I:  0 Tm:      2.17 NLPi:  18 Dpth:  12 Lvs:  14 Obj:  1.35E+00 Gap:      NaN
      4.44,     1.14851,   0.944178,   0.418428
Iter:   14 I:  0 Tm:      3.56 NLPi:  44 Dpth:  13 Lvs:  15 Obj:  1.35E+00 Gap:      NaN
      4.56,     1.13278,   0.950658,   0.378617
Iter:    15 I:  0 Tm:      2.34 NLPi:  23 Dpth:  14 Lvs:  16 Obj:  1.35E+00 Gap:      NaN
      4.68,     1.11859,   0.956378,   0.341493
Iter:   16 I:  0 Tm:      2.46 NLPi:  26 Dpth:  15 Lvs:  17 Obj:  1.35E+00 Gap:      NaN
        4.8,     1.10582,   0.961433,      0.3073
Iter:   17 I:  0 Tm:     2.72 NLPi:  26 Dpth:  16 Lvs:  18 Obj:  1.35E+00 Gap:      NaN
      4.92,     1.09438,    0.965904,   0.276462
Iter:   18 I:  0 Tm:      3.99 NLPi:  60 Dpth:  17 Lvs:  19 Obj:  1.35E+00 Gap:      NaN
      5.04,     1.08411,   0.969849,   0.248219
Iter:   19 I:  0 Tm:      1.91 NLPi:  10 Dpth:  18 Lvs:  20 Obj:  1.35E+00 Gap:      NaN
      5.16,     1.07492,   0.973333,   0.222481
Iter:   20 I:  0 Tm:      1.23 NLPi:    7 Dpth:  19 Lvs:   21 Obj:  1.35E+00 Gap:      NaN
      5.28,      1.0667,   0.976412,    0.19935
Iter:    21 I:  0 Tm:      1.37 NLPi:   8 Dpth:  20 Lvs:   22 Obj:  1.35E+00 Gap:      NaN
        5.4,    1.05936,    0.97913,   0.178262
Iter:    22 I:  0 Tm:     1.37 NLPi:  12 Dpth:  21 Lvs:   23 Obj:  1.35E+00 Gap:      NaN
      5.52,      1.0528,    0.981533,   0.159402
Iter:   23 I:  0 Tm:      1.34 NLPi:  11 Dpth:  22 Lvs:   24 Obj:  1.36E+00 Gap:      NaN
      5.64,    1.04696,   0.983656,    0.142349
Iter:   24 I:  0 Tm:      1.30 NLPi:    8 Dpth:  23 Lvs:   25 Obj:  1.36E+00 Gap:      NaN
      5.76,    1.04175,   0.985532,    0.127041
Iter:    25 I:  0 Tm:      1.33 NLPi:  14 Dpth:  24 Lvs:  26 Obj:  1.36E+00 Gap:      NaN
      5.88,      1.0371,   0.987191,     0.1133
Iter:   26 I:  0 Tm:      1.13 NLPi:    7 Dpth:  25 Lvs:   27 Obj:  1.36E+00 Gap:      NaN
          6,    1.03297,   0.988657,   0.100985
Iter:    27 I:  0 Tm:      0.97 NLPi:    7 Dpth:  26 Lvs:  28 Obj:  1.36E+00 Gap:      NaN
      6.12,    1.02929,   0.989954,   0.0899604
Iter:   28 I:  0 Tm:      0.99 NLPi:    6 Dpth:  27 Lvs:  29 Obj:  1.36E+00 Gap:      NaN
      6.24,    1.02601,   0.991101,   0.0800981
Iter:   29 I:  0 Tm:      0.93 NLPi:    6 Dpth:  28 Lvs:   30 Obj:  1.36E+00 Gap:      NaN
      6.36,     1.0231,    0.992117,   0.0712934
Iter:    30 I:  0 Tm:      0.66 NLPi:    5 Dpth:  29 Lvs:  31 Obj:  1.36E+00 Gap:      NaN
      6.48,    1.02051,   0.993015,   0.0634308
Iter:    31 I:  0 Tm:      0.73 NLPi:    5 Dpth:  30 Lvs:   32 Obj:  1.36E+00 Gap:      NaN
        6.6,    1.01821,   0.993811,   0.0564127
Iter:    32 I:  0 Tm:      0.66 NLPi:    5 Dpth:  31 Lvs:  33 Obj:  1.36E+00 Gap:      NaN
      6.72,    1.01617,   0.994515,   0.0501573
Iter:   33 I:  0 Tm:      0.70 NLPi:    5 Dpth:   32 Lvs:  34 Obj:  1.36E+00 Gap:      NaN
      6.84,    1.01435,   0.995139,  0.0445824
Iter:   34 I:  0 Tm:      0.67 NLPi:   5 Dpth:  33 Lvs:  35 Obj:  1.36E+00 Gap:      NaN
      6.96,    1.01274,   0.995691,   0.0396177
Iter:    35 I:  0 Tm:      0.82 NLPi:    9 Dpth:  34 Lvs:  36 Obj:  1.36E+00 Gap:      NaN
      7.08,    1.01131,    0.99618,   0.0351979
Iter:   36 I:  0 Tm:      0.77 NLPi:    8 Dpth:  35 Lvs:   37 Obj:  1.36E+00 Gap:      NaN
        7.2,    1.01003,   0.996613,   0.0312714
Iter:    37 I:  0 Tm:      0.75 NLPi:    8 Dpth:  36 Lvs:  38 Obj:  1.36E+00 Gap:      NaN
      7.32,      1.0089,   0.996997,   0.027765
Iter:   38 I:  0 Tm:      0.69 NLPi:    6 Dpth:  37 Lvs:   39 Obj:  1.36E+00 Gap:      NaN
      7.44,      1.0079,   0.997337,   0.0246626
Iter:    39 I:  0 Tm:      0.71 NLPi:    6 Dpth:  38 Lvs:  40 Obj:  1.36E+00 Gap:      NaN
      7.56,    1.00701,   0.997639,   0.0218975
Iter:   40 I:  0 Tm:      0.81 NLPi:    9 Dpth:   39 Lvs:  41 Obj:  1.36E+00 Gap:      NaN
      7.68,    1.00622,   0.997905,   0.0194423
Iter:   41 I:  0 Tm:      0.70 NLPi:    8 Dpth:  40 Lvs:   42 Obj:  1.36E+00 Gap:      NaN
        7.8,    1.00552,   0.998142,   0.0172603
Iter:    42 I:  0 Tm:     0.69 NLPi:    6 Dpth:  41 Lvs:  43 Obj:  1.36E+00 Gap:      NaN
      7.92,     1.0049,   0.998351,   0.0153224
Iter:   43 I:  0 Tm:      0.67 NLPi:    5 Dpth:  42 Lvs:   44 Obj:  1.36E+00 Gap:      NaN
      8.04,    1.00435,   0.998537,   0.0136015
Iter:   44 I:  0 Tm:      0.60 NLPi:    4 Dpth:  43 Lvs:   45 Obj:  1.36E+00 Gap:      NaN
      8.16,    1.00386,   0.998701,   0.0120735
Iter:    45 I:  0 Tm:      0.71 NLPi:    6 Dpth:  44 Lvs:  46 Obj:  1.36E+00 Gap:      NaN
      8.28,    1.00342,   0.998847,   0.0107171
Iter:   46 I:  0 Tm:      0.59 NLPi:    4 Dpth:  45 Lvs:   47 Obj:  1.36E+00 Gap:      NaN
        8.4,    1.00303,   0.998976,  0.00951328
Iter:    47 I:  0 Tm:      0.69 NLPi:    6 Dpth:  46 Lvs:  48 Obj:  1.36E+00 Gap:      NaN
      8.52,    1.00269,    0.99909, 0.00844491
Iter:   48 I:  0 Tm:      0.65 NLPi:    5 Dpth:  47 Lvs:   49 Obj:  1.36E+00 Gap:      NaN
      8.64,    1.00239,   0.999191, 0.00749702
--Integer Solution:  1.36E+00 Lowest Leaf:  1.34E+00 Gap:  1.35E-02
      8.76,    1.00212,    0.999281, 0.00665598
Iter:   49 I:  0 Tm:      0.57 NLPi:    3 Dpth:  48 Lvs:  48 Obj:  1.36E+00 Gap:  1.35E-02
      8.88,    1.00188,    0.99936, 0.00590975
Iter:    50 I: 0 Tm:      1.29 NLPi:   8 Dpth:  48 Lvs:  47 Obj:  1.36E+00 Gap: 1.35E-02
          9,    1.00167,    0.99943, 0.00524776
Warning: best integer solution returned after maximum MINLP iterations
      9.12,    1.00148,    0.999492, 0.00466052
  Adjust minlp_max_iter_with_int_sol          50  in apopt.opt to change limit
      9.24,    1.00131,    0.999547,  0.00413914
  Successful solution
      9.36,    1.00117,    0.999596,  0.00367206
   
      9.48,    1.00103,   0.999637, 0.00326159
  ---------------------------------------------------
        9.6,    1.00092,    0.999675,  0.00290302
Solver        :  APOPT (v1.0)
      9.72,    1.00082,    0.999707,  0.00258504
Solution time  :    111.364099999999      sec
      9.84,    1.00072,    0.999736,  0.00230262
Objective      :   1.36258198934523   
      9.96,    1.00064,    0.999762,  0.00205186
  Successful solution
      10.08,    1.00057,    0.999785,  0.00182951
---------------------------------------------------
      10.2,    1.00051,    0.999805,  0.00163302
      10.32,    1.00045,    0.999822,  0.00146039
      10.44,      1.0004,    0.999838,  0.00131011
      10.56,    1.00036,    0.999851,  0.00118097
      10.68,    1.00032,    0.999863,  0.00107188
      10.8,    1.00028,    0.999872, 0.000981659
      10.92,    1.00025,    0.99988,  0.00090944
      11.04,    1.00022,    0.999886, 0.000854893
      11.16,    1.00019,    0.99989, 0.000818284
      11.28,    1.00017,    0.999892, 0.000801181
      11.4,    1.00014,    0.999891, 0.000807844
      11.52,    1.00012,    0.999886, 0.000848734
      11.64,    1.00009,    0.999878, 0.000951928
      11.76,    1.00006,    0.999864,  0.00121852
      11.88,    1.00002,    0.999838,  0.00267036
        12,    0.999915,    0.999769,          0
</pre>
</pre>
   
   
[[Category:APMonitor]]
[[Category:APMonitor]]

Latest revision as of 22:15, 20 November 2017

This page contains a solution of the MIOCP Lotka Volterra fishing problem in APMonitor Python format. A MATLAB version is also available from the Dynamic Optimization Course as Example 3 (lotka_volterra_fishing.zip).

APMonitor

The model in Python code for a fixed control discretization grid using orthogonal collocation and a simultaneous optimization method. The APMonitor package is available with pip install APMonitor or from the APMonitor Python Github repository. 

import numpy as np
import matplotlib.pyplot as plt

# retrieve apm.py from
# https://raw.githubusercontent.com/APMonitor/apm_python/master/apm.py
# or
# http://apmonitor.com/wiki/index.php/Main/PythonApp
# from apm import *

# pip install with 'pip install APMonitor'
from APMonitor.apm import *

# local APMonitor servers are available for Windows or Linux
# http://apmonitor.com/wiki/index.php/Main/APMonitorServer
# with clients in Python, MATLAB, and Julia

# write model
model = '''
! apopt MINLP solver options (see apopt.com)
File apopt.opt
 minlp_maximum_iterations 1000     ! minlp iterations
 minlp_max_iter_with_int_sol 50    ! minlp iterations if integer solution is found
 minlp_as_nlp 0                    ! treat minlp as nlp
 nlp_maximum_iterations 200        ! nlp sub-problem max iterations
 minlp_branch_method 1             ! 1 = depth first, 2 = breadth first
 minlp_gap_tol 0.001               ! covergence tolerance
 minlp_integer_tol 0.001           ! maximum deviation from whole number to be considered an integer
 minlp_integer_leaves 0            ! create soft (1) integer leaves or hard (2) integer leaves with branching  
End File

Constants
  c0 = 0.4 
  c1 = 0.2

Parameters
  last

Variables
  x0 = 0.5 , >= 0
  x1 = 0.7 , >= 0
  x2 = 0.0 , >= 0
  int_w = 0 , >= 0 , <= 1

Intermediates
  w = int_w

Equations
  minimize last * x2

  $x0 = x0 - x0*x1 - c0*x0*w 
  $x1 = - x1 + x0*x1 - c1*x1*w
  $x2 = (x0-1)^2 + (x1-1)^2                                                                                       
'''
fid = open('lotka_volterra.apm','w')
fid.write(model)
fid.close()

# write data file
time = np.linspace(0,12,121)
time = np.insert(time, 1, 0.01)
last = np.zeros(122)
last[-1] = 1.0
data = np.vstack((time,last))
np.savetxt('data.csv',data.T,delimiter=',',header='time,last',comments='')

# specify server and application name
s = 'http://byu.apmonitor.com'
#s = 'http://127.0.0.1/'  # for local APMonitor server
a = 'lotka'

apm(s,a,'clear all')
apm_load(s,a,'lotka_volterra.apm')
csv_load(s,a,'data.csv')

apm_option(s,a,'nlc.imode',6)              # Nonlinear control / dynamic optimization
apm_option(s,a,'nlc.nodes',3)

apm_info(s,a,'MV','int_w')                 # M or MV = Manipulated variable - independent variable over time horizon
apm_option(s,a,'int_w.status',1)           # Status: 1=ON, 0=OFF
apm_option(s,a,'int_w.mv_type',0)          # MV Type = Zero Order Hold

apm_option(s,a,'nlc.solver',1)             # 1 = APOPT

# solve
output = apm(s,a,'solve')            
print(output)

# retrieve solution
y = apm_sol(s,a)

plt.figure(1)
plt.step(y['time'],y['int_w'],'r-',label='w (0/1)')
plt.plot(y['time'],y['x0'],'b-',label=r'$x_0$')
plt.plot(y['time'],y['x1'],'k-',label=r'$x_1$')
plt.plot(y['time'],y['x2'],'g-',label=r'$x_2$')
plt.xlabel('Time')
plt.ylabel('Variables')
plt.legend(loc='best')
plt.show()

Results with APOPT (MINLP)

An MINLP solution is calculated with APOPT with an objective function value of x2(tf)=1.36. APOPT requires 50 NLP solutions to find an integer solution (111 seconds of processing time). Each NLP solution in the branch and bound method requires an average of 2.2 seconds to complete with a range between 12.99 and 0.57 seconds. 

 ----------------------------------------------------------------
 APMonitor, Version 0.7.9
 APMonitor Optimization Suite
 ----------------------------------------------------------------
 
 
 --------- APM Model Size ------------
 Each time step contains
   Objects      :            0
   Constants    :            2
   Variables    :            5
   Intermediates:            1
   Connections  :            0
   Equations    :            5
   Residuals    :            4
 
 Number of state variables:           2178
 Number of total equations: -         2057
 Number of slack variables: -            0
 ---------------------------------------
 Degrees of freedom       :            121
 
 ----------------------------------------------
 Dynamic Control with APOPT Solver
 ----------------------------------------------
Iter:     1 I:  0 Tm:     12.99 NLPi:   93 Dpth:    0 Lvs:    2 Obj:  1.34E+00 Gap:       NaN
Iter:     2 I:  0 Tm:      2.82 NLPi:   18 Dpth:    1 Lvs:    3 Obj:  1.34E+00 Gap:       NaN
Iter:     3 I:  0 Tm:      3.39 NLPi:   30 Dpth:    2 Lvs:    4 Obj:  1.34E+00 Gap:       NaN
Iter:     4 I:  0 Tm:     10.37 NLPi:  131 Dpth:    3 Lvs:    5 Obj:  1.34E+00 Gap:       NaN
Iter:     5 I:  0 Tm:      1.81 NLPi:   10 Dpth:    4 Lvs:    6 Obj:  1.34E+00 Gap:       NaN
Iter:     6 I:  0 Tm:      7.93 NLPi:  102 Dpth:    5 Lvs:    7 Obj:  1.34E+00 Gap:       NaN
Iter:     7 I:  0 Tm:      3.22 NLPi:   28 Dpth:    6 Lvs:    8 Obj:  1.34E+00 Gap:       NaN
Iter:     8 I:  0 Tm:      1.85 NLPi:   12 Dpth:    7 Lvs:    9 Obj:  1.34E+00 Gap:       NaN
Iter:     9 I:  0 Tm:     10.24 NLPi:  140 Dpth:    8 Lvs:   10 Obj:  1.34E+00 Gap:       NaN
Iter:    10 I:  0 Tm:      2.04 NLPi:    9 Dpth:    9 Lvs:   11 Obj:  1.35E+00 Gap:       NaN
Iter:    11 I:  0 Tm:      6.50 NLPi:   86 Dpth:   10 Lvs:   12 Obj:  1.35E+00 Gap:       NaN
Iter:    12 I:  0 Tm:      1.94 NLPi:   17 Dpth:   11 Lvs:   13 Obj:  1.35E+00 Gap:       NaN
Iter:    13 I:  0 Tm:      2.17 NLPi:   18 Dpth:   12 Lvs:   14 Obj:  1.35E+00 Gap:       NaN
Iter:    14 I:  0 Tm:      3.56 NLPi:   44 Dpth:   13 Lvs:   15 Obj:  1.35E+00 Gap:       NaN
Iter:    15 I:  0 Tm:      2.34 NLPi:   23 Dpth:   14 Lvs:   16 Obj:  1.35E+00 Gap:       NaN
Iter:    16 I:  0 Tm:      2.46 NLPi:   26 Dpth:   15 Lvs:   17 Obj:  1.35E+00 Gap:       NaN
Iter:    17 I:  0 Tm:      2.72 NLPi:   26 Dpth:   16 Lvs:   18 Obj:  1.35E+00 Gap:       NaN
Iter:    18 I:  0 Tm:      3.99 NLPi:   60 Dpth:   17 Lvs:   19 Obj:  1.35E+00 Gap:       NaN
Iter:    19 I:  0 Tm:      1.91 NLPi:   10 Dpth:   18 Lvs:   20 Obj:  1.35E+00 Gap:       NaN
Iter:    20 I:  0 Tm:      1.23 NLPi:    7 Dpth:   19 Lvs:   21 Obj:  1.35E+00 Gap:       NaN
Iter:    21 I:  0 Tm:      1.37 NLPi:    8 Dpth:   20 Lvs:   22 Obj:  1.35E+00 Gap:       NaN
Iter:    22 I:  0 Tm:      1.37 NLPi:   12 Dpth:   21 Lvs:   23 Obj:  1.35E+00 Gap:       NaN
Iter:    23 I:  0 Tm:      1.34 NLPi:   11 Dpth:   22 Lvs:   24 Obj:  1.36E+00 Gap:       NaN
Iter:    24 I:  0 Tm:      1.30 NLPi:    8 Dpth:   23 Lvs:   25 Obj:  1.36E+00 Gap:       NaN
Iter:    25 I:  0 Tm:      1.33 NLPi:   14 Dpth:   24 Lvs:   26 Obj:  1.36E+00 Gap:       NaN
Iter:    26 I:  0 Tm:      1.13 NLPi:    7 Dpth:   25 Lvs:   27 Obj:  1.36E+00 Gap:       NaN
Iter:    27 I:  0 Tm:      0.97 NLPi:    7 Dpth:   26 Lvs:   28 Obj:  1.36E+00 Gap:       NaN
Iter:    28 I:  0 Tm:      0.99 NLPi:    6 Dpth:   27 Lvs:   29 Obj:  1.36E+00 Gap:       NaN
Iter:    29 I:  0 Tm:      0.93 NLPi:    6 Dpth:   28 Lvs:   30 Obj:  1.36E+00 Gap:       NaN
Iter:    30 I:  0 Tm:      0.66 NLPi:    5 Dpth:   29 Lvs:   31 Obj:  1.36E+00 Gap:       NaN
Iter:    31 I:  0 Tm:      0.73 NLPi:    5 Dpth:   30 Lvs:   32 Obj:  1.36E+00 Gap:       NaN
Iter:    32 I:  0 Tm:      0.66 NLPi:    5 Dpth:   31 Lvs:   33 Obj:  1.36E+00 Gap:       NaN
Iter:    33 I:  0 Tm:      0.70 NLPi:    5 Dpth:   32 Lvs:   34 Obj:  1.36E+00 Gap:       NaN
Iter:    34 I:  0 Tm:      0.67 NLPi:    5 Dpth:   33 Lvs:   35 Obj:  1.36E+00 Gap:       NaN
Iter:    35 I:  0 Tm:      0.82 NLPi:    9 Dpth:   34 Lvs:   36 Obj:  1.36E+00 Gap:       NaN
Iter:    36 I:  0 Tm:      0.77 NLPi:    8 Dpth:   35 Lvs:   37 Obj:  1.36E+00 Gap:       NaN
Iter:    37 I:  0 Tm:      0.75 NLPi:    8 Dpth:   36 Lvs:   38 Obj:  1.36E+00 Gap:       NaN
Iter:    38 I:  0 Tm:      0.69 NLPi:    6 Dpth:   37 Lvs:   39 Obj:  1.36E+00 Gap:       NaN
Iter:    39 I:  0 Tm:      0.71 NLPi:    6 Dpth:   38 Lvs:   40 Obj:  1.36E+00 Gap:       NaN
Iter:    40 I:  0 Tm:      0.81 NLPi:    9 Dpth:   39 Lvs:   41 Obj:  1.36E+00 Gap:       NaN
Iter:    41 I:  0 Tm:      0.70 NLPi:    8 Dpth:   40 Lvs:   42 Obj:  1.36E+00 Gap:       NaN
Iter:    42 I:  0 Tm:      0.69 NLPi:    6 Dpth:   41 Lvs:   43 Obj:  1.36E+00 Gap:       NaN
Iter:    43 I:  0 Tm:      0.67 NLPi:    5 Dpth:   42 Lvs:   44 Obj:  1.36E+00 Gap:       NaN
Iter:    44 I:  0 Tm:      0.60 NLPi:    4 Dpth:   43 Lvs:   45 Obj:  1.36E+00 Gap:       NaN
Iter:    45 I:  0 Tm:      0.71 NLPi:    6 Dpth:   44 Lvs:   46 Obj:  1.36E+00 Gap:       NaN
Iter:    46 I:  0 Tm:      0.59 NLPi:    4 Dpth:   45 Lvs:   47 Obj:  1.36E+00 Gap:       NaN
Iter:    47 I:  0 Tm:      0.69 NLPi:    6 Dpth:   46 Lvs:   48 Obj:  1.36E+00 Gap:       NaN
Iter:    48 I:  0 Tm:      0.65 NLPi:    5 Dpth:   47 Lvs:   49 Obj:  1.36E+00 Gap:       NaN
--Integer Solution:   1.36E+00 Lowest Leaf:   1.34E+00 Gap:   1.35E-02
Iter:    49 I:  0 Tm:      0.57 NLPi:    3 Dpth:   48 Lvs:   48 Obj:  1.36E+00 Gap:  1.35E-02
Iter:    50 I:  0 Tm:      1.29 NLPi:    8 Dpth:   48 Lvs:   47 Obj:  1.36E+00 Gap:  1.35E-02
 Warning: best integer solution returned after maximum MINLP iterations
 Adjust minlp_max_iter_with_int_sol           50  in apopt.opt to change limit
 Successful solution
 
 ---------------------------------------------------
 Solver         :  APOPT (v1.0)
 Solution time  :    111.364099999999      sec
 Objective      :    1.36258198934523     
 Successful solution
 ---------------------------------------------------
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