JohnHedengren: /* Results with APOPT (MINLP) */

2017-11-20T22:15:45Z

Results with APOPT (MINLP)

← Older revision		Revision as of 22:15, 20 November 2017
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	== Results with APOPT (MINLP) ==		== Results with APOPT (MINLP) ==

	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 52 NLP solutions to find an integer solution (111.0 seconds of processing time).		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.

	[[File:Volterra_fishing_APMonitor.png]]		[[File:Volterra_fishing_APMonitor.png]]

JohnHedengren: Results section

2017-11-20T22:12:26Z

Results section

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JohnHedengren: APMontor solution of Volterra fishing problem

2017-11-20T22:03:56Z

APMontor solution of Volterra fishing problem

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This page contains a solution of the MIOCP [[Lotka Volterra fishing problem]] in [http://www.apmonitor.com APMonitor] Python format. A MATLAB version is also available from the [http://apmonitor.com/do/index.php/Main/DiscreteVariables Dynamic Optimization Course] as [http://apmonitor.com/do/uploads/Main/lotka_volterra_fishing.zip 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 [https://github.com/APMonitor/apm_python/blob/master/apm.py APMonitor Python Github repository].

<source lang="Python">
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()
</source>

== Results ==

=== IPOPT ===
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:

<pre>
# t, x_0, x_1, u
0, 0.5, 0.7, 5.18436e-09
0.12, 0.519594, 0.659989, 5.42459e-09
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
0.72, 0.66843, 0.515028, 8.90236e-09
0.84, 0.709312, 0.496134, 1.02545e-08
0.96, 0.75426, 0.480401, 1.19611e-08
1.08, 0.803421, 0.4678, 1.41346e-08
1.2, 0.856919, 0.458344, 1.69417e-08
1.32, 0.914842, 0.452091, 2.06377e-08
1.44, 0.97722, 0.449154, 2.56285e-08
1.56, 1.044, 0.449708, 3.2592e-08
1.68, 1.11503, 0.454003, 4.27322e-08
1.8, 1.18998, 0.462372, 5.83658e-08
1.92, 1.26837, 0.475251, 8.44574e-08
2.04, 1.34942, 0.493189, 1.33425e-07
2.16, 1.43209, 0.516862, 2.44977e-07
2.28, 1.51492, 0.547087, 6.24464e-07
2.4, 1.59605, 0.584816, 0.715281
2.52, 1.61776, 0.61833, 0.999999
2.64, 1.61116, 0.6499, 0.999999
2.76, 1.59844, 0.68229, 1
2.88, 1.57963, 0.714939, 1
3, 1.55497, 0.747196, 1
3.12, 1.52487, 0.778343, 0.999999
3.24, 1.48993, 0.807625, 0.999999
3.36, 1.4509, 0.83429, 0.999999
3.48, 1.40865, 0.857635, 0.999998
3.6, 1.36412, 0.877047, 0.999997
3.72, 1.31827, 0.892039, 0.999991
3.84, 1.27203, 0.90228, 0.999646
3.96, 1.22628, 0.907616, 0.536112
4.08, 1.20755, 0.919653, 0.581667
4.2, 1.18502, 0.928525, 0.503883
4.32, 1.16606, 0.936899, 0.464153
4.44, 1.14851, 0.944178, 0.418428
4.56, 1.13278, 0.950658, 0.378617
4.68, 1.11859, 0.956378, 0.341493
4.8, 1.10582, 0.961433, 0.3073
4.92, 1.09438, 0.965904, 0.276462
5.04, 1.08411, 0.969849, 0.248219
5.16, 1.07492, 0.973333, 0.222481
5.28, 1.0667, 0.976412, 0.19935
5.4, 1.05936, 0.97913, 0.178262
5.52, 1.0528, 0.981533, 0.159402
5.64, 1.04696, 0.983656, 0.142349
5.76, 1.04175, 0.985532, 0.127041
5.88, 1.0371, 0.987191, 0.1133
6, 1.03297, 0.988657, 0.100985
6.12, 1.02929, 0.989954, 0.0899604
6.24, 1.02601, 0.991101, 0.0800981
6.36, 1.0231, 0.992117, 0.0712934
6.48, 1.02051, 0.993015, 0.0634308
6.6, 1.01821, 0.993811, 0.0564127
6.72, 1.01617, 0.994515, 0.0501573
6.84, 1.01435, 0.995139, 0.0445824
6.96, 1.01274, 0.995691, 0.0396177
7.08, 1.01131, 0.99618, 0.0351979
7.2, 1.01003, 0.996613, 0.0312714
7.32, 1.0089, 0.996997, 0.027765
7.44, 1.0079, 0.997337, 0.0246626
7.56, 1.00701, 0.997639, 0.0218975
7.68, 1.00622, 0.997905, 0.0194423
7.8, 1.00552, 0.998142, 0.0172603
7.92, 1.0049, 0.998351, 0.0153224
8.04, 1.00435, 0.998537, 0.0136015
8.16, 1.00386, 0.998701, 0.0120735
8.28, 1.00342, 0.998847, 0.0107171
8.4, 1.00303, 0.998976, 0.00951328
8.52, 1.00269, 0.99909, 0.00844491
8.64, 1.00239, 0.999191, 0.00749702
8.76, 1.00212, 0.999281, 0.00665598
8.88, 1.00188, 0.99936, 0.00590975
9, 1.00167, 0.99943, 0.00524776
9.12, 1.00148, 0.999492, 0.00466052
9.24, 1.00131, 0.999547, 0.00413914
9.36, 1.00117, 0.999596, 0.00367206
9.48, 1.00103, 0.999637, 0.00326159
9.6, 1.00092, 0.999675, 0.00290302
9.72, 1.00082, 0.999707, 0.00258504
9.84, 1.00072, 0.999736, 0.00230262
9.96, 1.00064, 0.999762, 0.00205186
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>

[[Category:APMonitor]]

Lotka Volterra fishing problem (APMonitor) - Revision history

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