Fishing Optimal Control
We will solve an optimal control problem to maximize profit constrained by fish population.
Problem Statement and Model
In this system, our input is the fishing rate $u$, and the profit $J$ will be defined in the objective to maximize. Our profit objective is represented as:
\[\begin{aligned} &&\max_{u(t)} J(t) \\ &&&&&J = \int_0^{10} \left(E - \frac{c}{x}\right) u U_{max} \, dt \\ &&\text{s.t.} &&& \frac{dx}{dt}= rx(t)\left(1 - \frac{x(t)}{k}\right) - uU_{max}, t \in [0,10] \\ &&&&&x(0) = 70 \\ &&&&&0 \leq u(t) \leq 1 \\ &&&&&E = 1, \; c = 17.5, \; r = 0.71, \; k = 80.5, \; U_{max} = 20 \\ &&&&&J(0) = 0 \\ \end{aligned}\]
Modeling in InfiniteOpt
First we must import $InfiniteOpt$ and other packages.
using InfiniteOpt, Ipopt, Plots;Next we specify our initial conditions and problem variables.
x0 = 70
E, c, r, k, Umax = 1, 17.5, 0.71, 80.5, 20;Model Initialization
We initialize the infinite model with InfiniteModel and select Ipopt as our optimizer that will be used to solve it.
m = InfiniteModel(Ipopt.Optimizer);Infinite Parameter Definition
We now define the infinite parameter $t \in [0, 10]$ to represent time over a 10 year period. We'll also specify 100 equidistant time points.
@infinite_parameter(m, t in [0, 10], num_supports=100)tInfinite Variable Definition
Now that we have our infinite parameter defined, let's specify our infinite variables:
- $1 \leq x(t)$ : fish population at time $t$
- $0 \leq u(t) \leq 1$ : fishing rate
- $J(t)$ : profit over time
@variable(m, 1 <= x, Infinite(t))
@variable(m, 0 <= u <= 1, Infinite(t));
@variable(m, J, Infinite(t));Objective Definition
Now we add the objective using @objective to maximize profit $J$ at the end of the 10 year period:
@objective(m, Max, J(10));Constraint Definition
The last step is to add our constraints. First, define the ODEs which serve as our system model.
@constraint(m, ∂(J,t) == (E-c/x) * u * Umax)
@constraint(m, ∂(x,t) == r * x *(1 - x/k) - u*Umax);We also set our initial conditions for $x$ and $J$.
@constraint(m, x(0) == x0)
@constraint(m, J(0) == 0);Problem Solution
Now we're ready to solve! We can solve the model by invoking optimize!:
optimize!(m)
******************************************************************************
This program contains Ipopt, a library for large-scale nonlinear optimization.
Ipopt is released as open source code under the Eclipse Public License (EPL).
For more information visit https://github.com/coin-or/Ipopt
******************************************************************************
This is Ipopt version 3.14.17, running with linear solver MUMPS 5.8.0.
Number of nonzeros in equality constraint Jacobian...: 1296
Number of nonzeros in inequality constraint Jacobian.: 0
Number of nonzeros in Lagrangian Hessian.............: 400
Total number of variables............................: 500
variables with only lower bounds: 100
variables with lower and upper bounds: 100
variables with only upper bounds: 0
Total number of equality constraints.................: 400
Total number of inequality constraints...............: 0
inequality constraints with only lower bounds: 0
inequality constraints with lower and upper bounds: 0
inequality constraints with only upper bounds: 0
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
0 0.0000000e+00 6.90e+01 1.45e+00 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 -1.8548441e+00 6.89e+01 1.12e+01 -1.0 9.50e+03 -2.0 1.44e-04 1.76e-03h 1
2 -2.6101623e+00 6.88e+01 1.12e+01 -1.0 3.84e+03 -1.6 3.43e-04 7.39e-04h 1
3 -8.1519321e+00 6.85e+01 5.40e+01 -1.0 3.34e+03 -2.1 1.00e-03 4.91e-03h 1
4 -1.5382109e+01 6.80e+01 6.91e+01 -1.0 2.26e+03 -2.5 1.14e-03 7.62e-03f 4
5 -2.3521800e+01 6.72e+01 6.64e+01 -1.0 2.27e+03 - 1.32e-02 1.07e-02h 3
6 -3.2524402e+01 6.62e+01 6.32e+01 -1.0 1.67e+03 - 2.84e-02 1.51e-02h 2
7 -3.9860630e+01 6.51e+01 5.06e+01 -1.0 9.09e+02 - 8.43e-02 1.60e-02h 1
8 -4.5178330e+01 6.40e+01 3.21e+01 -1.0 2.71e+03 - 3.44e-02 1.70e-02h 1
9 -4.9707626e+01 6.30e+01 3.08e+01 -1.0 1.38e+03 - 1.50e-02 1.62e-02h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
10 -5.3966386e+01 6.19e+01 3.24e+01 -1.0 2.10e+03 - 1.11e-02 1.72e-02h 1
11 -5.5735727e+01 6.14e+01 3.15e+01 -1.0 3.23e+03 - 1.00e-02 9.14e-03h 1
12 -5.7438744e+01 6.07e+01 3.26e+01 -1.0 2.39e+03 - 7.16e-03 1.01e-02f 1
13 -5.6890891e+01 6.02e+01 3.64e+01 -1.0 4.84e+03 - 2.54e-03 8.82e-03f 1
14 -5.6868789e+01 5.97e+01 3.60e+01 -1.0 1.10e+04 - 2.31e-03 7.95e-03f 2
15 -5.7110966e+01 5.95e+01 3.69e+01 -1.0 1.97e+03 -2.1 3.18e-03 3.64e-03h 1
16 -5.9420322e+01 5.84e+01 4.18e+01 -1.0 1.70e+03 -2.6 3.38e-03 1.82e-02f 1
17 -5.9499934e+01 5.83e+01 3.16e+01 -1.0 1.14e+03 -2.2 1.21e-02 1.23e-03h 1
18 -6.0120020e+01 5.75e+01 4.82e+01 -1.0 1.34e+03 -1.7 8.68e-03 1.48e-02h 1
19 -6.0855779e+01 5.70e+01 4.78e+01 -1.0 1.92e+03 -2.2 5.24e-03 8.16e-03f 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
20 -6.0963105e+01 5.69e+01 3.59e+01 -1.0 2.14e+03 -1.8 6.02e-03 1.80e-03f 1
21 -7.3097707e+01 5.62e+01 4.14e+01 -1.0 2.01e+03 -2.3 1.80e-03 1.27e-02f 1
22 -7.3128657e+01 5.61e+01 1.23e+02 -1.0 1.21e+03 -0.9 8.10e-03 1.26e-03h 1
23 -7.3128597e+01 5.61e+01 1.53e+03 -1.0 2.73e+03 -0.5 8.41e-03 9.58e-05h 1
24 -7.3194925e+01 5.46e+01 7.12e+02 -1.0 3.17e+03 -1.0 7.53e-03 2.73e-02f 1
25 -7.3257003e+01 5.42e+01 8.00e+02 -1.0 9.45e+02 -0.6 3.10e-02 7.73e-03f 1
26 -7.3260596e+01 5.41e+01 6.85e+02 -1.0 1.39e+03 -1.0 3.96e-02 5.10e-04h 1
27 -7.3243186e+01 5.33e+01 4.89e+03 -1.0 1.62e+03 -0.6 9.60e-02 1.49e-02h 1
28 -7.3562368e+01 5.20e+01 4.65e+03 -1.0 9.38e+02 -1.1 3.48e-02 2.43e-02f 1
29 -7.3562755e+01 5.19e+01 4.54e+03 -1.0 7.25e+02 -0.7 1.17e-01 1.67e-03h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
30 -7.3601728e+01 5.06e+01 2.75e+03 -1.0 8.08e+02 -1.1 2.21e-01 2.56e-02f 1
31 -7.3564204e+01 4.97e+01 2.75e+03 -1.0 6.83e+02 -0.7 1.07e-01 1.85e-02h 1
32 -7.3354094e+01 4.83e+01 2.33e+03 -1.0 7.15e+02 -1.2 8.51e-02 2.78e-02h 1
33 -7.4140845e+01 4.62e+01 2.38e+03 -1.0 6.32e+02 -1.7 1.86e-02 4.24e-02f 1
34 -7.4005466e+01 4.58e+01 2.31e+03 -1.0 4.26e+02 -1.2 2.16e-02 1.05e-02h 1
35 -7.3322754e+01 4.52e+01 2.34e+03 -1.0 6.08e+02 -1.7 1.06e-04 1.22e-02h 2
36 -7.3322243e+01 4.52e+01 2.34e+03 -1.0 3.86e+02 -0.4 5.12e-02 9.73e-04h 1
37 -7.3320164e+01 4.51e+01 2.34e+03 -1.0 3.79e+02 -0.9 1.03e-01 2.85e-04h 1
38 -7.2740602e+01 4.39e+01 2.02e+03 -1.0 5.38e+02 -1.3 1.35e-01 2.71e-02h 1
39 -7.1495073e+01 4.25e+01 1.90e+03 -1.0 7.55e+02 -1.8 4.60e-02 3.14e-02f 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
40 -7.1303911e+01 4.22e+01 1.70e+03 -1.0 4.84e+02 -1.4 1.68e-01 7.67e-03h 1
41 -6.9076921e+01 4.05e+01 1.72e+03 -1.0 7.80e+02 -1.9 8.19e-03 4.05e-02f 1
42 -6.8670722e+01 4.00e+01 1.71e+03 -1.0 5.03e+02 -1.4 2.03e-03 1.20e-02h 1
43 -6.8670014e+01 3.99e+01 1.64e+03 -1.0 4.17e+02 -1.0 4.93e-02 1.68e-03h 1
44 -6.8217569e+01 3.94e+01 1.64e+03 -1.0 5.28e+02 -1.5 9.93e-04 1.35e-02h 1
45 -6.8217573e+01 3.94e+01 1.06e+03 -1.0 4.19e+02 -1.1 4.44e-01 5.86e-04h 1
46 -6.7829649e+01 3.88e+01 1.04e+03 -1.0 5.65e+02 -1.5 2.23e-02 1.55e-02h 1
47 -6.7829414e+01 3.88e+01 1.09e+03 -1.0 2.40e+02 -1.1 9.39e-02 4.31e-05h 1
48 -6.7135068e+01 3.79e+01 1.03e+03 -1.0 5.82e+02 -1.6 6.14e-02 2.20e-02h 1
49 -6.5035447e+01 3.67e+01 9.60e+02 -1.0 8.39e+02 -2.1 4.68e-02 3.15e-02f 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
50 -6.4843797e+01 3.65e+01 9.09e+02 -1.0 4.02e+02 -1.6 6.57e-02 5.60e-03h 1
51 -6.3479506e+01 3.59e+01 9.31e+02 -1.0 1.20e+03 -2.1 4.71e-05 1.66e-02f 1
52 -6.3548816e+01 3.59e+01 9.27e+02 -1.0 6.76e+02 -1.7 3.76e-03 1.92e-03h 1
53 -6.3548598e+01 3.59e+01 8.86e+02 -1.0 2.83e+02 -1.3 1.16e-01 4.52e-05h 1
54 -6.3285066e+01 3.56e+01 8.70e+02 -1.0 7.53e+02 -1.7 6.38e-02 5.79e-03h 1
55 -6.2051582e+01 3.50e+01 8.07e+02 -1.0 1.27e+03 -2.2 4.25e-02 1.71e-02f 1
56 -6.1758101e+01 3.48e+01 8.47e+02 -1.0 9.05e+02 -1.8 1.03e-01 5.92e-03h 1
57 -6.0328811e+01 3.42e+01 7.88e+02 -1.0 1.52e+03 -2.3 4.52e-02 1.83e-02f 1
58 -6.0061572e+01 3.40e+01 8.69e+02 -1.0 1.19e+03 -1.8 6.22e-02 4.73e-03h 1
59 -5.8602031e+01 3.34e+01 8.14e+02 -1.0 1.48e+03 -2.3 4.38e-02 1.88e-02f 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
60 -5.8336469e+01 3.32e+01 8.57e+02 -1.0 1.03e+03 -1.9 4.30e-02 5.04e-03h 1
61 -5.7966898e+01 3.30e+01 1.11e+03 -1.0 1.08e+03 -1.5 3.96e-02 7.77e-03h 1
62 -5.7078590e+01 3.26e+01 1.09e+03 -1.0 1.81e+03 -1.9 4.27e-02 1.18e-02f 1
63 -5.6841293e+01 3.25e+01 1.40e+03 -1.0 1.62e+03 -1.5 3.40e-02 3.42e-03h 1
64 -5.5578931e+01 3.20e+01 1.31e+03 -1.0 2.14e+03 -2.0 6.65e-02 1.41e-02f 1
65 -5.5291009e+01 3.19e+01 1.68e+03 -1.0 1.88e+03 -1.6 3.86e-02 3.46e-03h 1
66 -5.3911861e+01 3.14e+01 1.44e+03 -1.0 2.20e+03 -2.1 9.88e-02 1.45e-02f 1
67 -5.3332039e+01 3.12e+01 1.74e+03 -1.0 2.04e+03 -1.6 3.91e-02 6.27e-03f 1
68 -5.2104830e+01 3.08e+01 1.44e+03 -1.0 1.95e+03 -2.1 8.01e-02 1.33e-02f 1
69 -5.1086135e+01 3.05e+01 1.67e+03 -1.0 2.04e+03 -1.7 4.62e-02 1.08e-02f 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
70 -4.9970887e+01 3.01e+01 1.47e+03 -1.0 1.51e+03 -2.2 5.01e-02 1.22e-02f 1
71 -4.8771076e+01 2.97e+01 1.55e+03 -1.0 1.78e+03 -1.7 5.53e-02 1.43e-02f 1
72 -4.8642430e+01 2.96e+01 2.59e+03 -1.0 1.56e+03 -1.3 5.23e-02 1.61e-03h 1
73 -4.6434003e+01 2.90e+01 2.63e+03 -1.0 2.25e+03 -1.8 9.89e-02 2.02e-02f 1
74 -4.6217701e+01 2.90e+01 3.65e+03 -1.0 1.68e+03 -1.4 7.31e-02 2.40e-03h 1
75 -4.4329808e+01 2.84e+01 3.41e+03 -1.0 1.93e+03 -1.8 6.00e-02 1.89e-02f 1
76 -4.3910729e+01 2.83e+01 4.24e+03 -1.0 1.68e+03 -1.4 9.38e-02 4.64e-03f 1
77 -4.2230654e+01 2.78e+01 3.77e+03 -1.0 1.58e+03 -1.9 7.54e-02 1.84e-02f 1
78 -4.1016603e+01 2.74e+01 4.03e+03 -1.0 1.52e+03 -1.5 8.90e-02 1.46e-02f 1
79 -3.9161710e+01 2.68e+01 3.58e+03 -1.0 1.61e+03 -1.9 6.91e-02 2.08e-02f 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
80 -3.8916774e+01 2.67e+01 3.65e+03 -1.0 8.55e+02 -1.5 8.70e-02 3.47e-03h 1
81 -3.7317470e+01 2.62e+01 3.22e+03 -1.0 1.76e+03 -2.0 7.60e-02 1.98e-02f 1
82 -3.6862207e+01 2.60e+01 3.41e+03 -1.0 1.61e+03 -1.6 5.49e-02 5.59e-03f 1
83 -3.5750281e+01 2.56e+01 4.02e+03 -1.0 1.25e+03 -1.1 5.98e-02 1.73e-02f 1
84 -3.4943221e+01 2.53e+01 3.81e+03 -1.0 1.80e+03 -1.6 6.55e-02 8.81e-03f 1
85 -3.3888511e+01 2.49e+01 4.24e+03 -1.0 1.22e+03 -1.2 9.04e-02 1.59e-02f 1
86 -3.2359512e+01 2.45e+01 3.93e+03 -1.0 1.72e+03 -1.7 4.97e-02 1.73e-02f 1
87 -3.1972051e+01 2.43e+01 3.77e+03 -1.0 8.77e+02 -1.2 1.38e-01 6.96e-03h 1
88 -3.0047044e+01 2.38e+01 3.48e+03 -1.0 1.64e+03 -1.7 5.26e-02 2.31e-02f 1
89 -2.9908135e+01 2.37e+01 4.30e+03 -1.0 1.25e+03 -1.3 1.02e-01 1.88e-03h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
90 -2.8185232e+01 2.32e+01 3.53e+03 -1.0 1.53e+03 -1.8 1.11e-01 2.29e-02f 1
91 -2.7789946e+01 2.31e+01 4.09e+03 -1.0 1.37e+03 -1.3 8.16e-02 5.38e-03h 1
92 -2.6406264e+01 2.26e+01 9.61e+02 -1.0 1.32e+03 -1.8 5.21e-01 2.17e-02f 1
93 -2.5907022e+01 2.23e+01 7.19e+02 -1.0 9.48e+02 -1.4 7.05e-02 1.06e-02h 1
94 -2.4814415e+01 2.17e+01 1.03e+03 -1.0 6.97e+02 -1.0 8.21e-02 3.00e-02h 1
95 -2.2729667e+01 2.10e+01 9.91e+02 -1.0 1.42e+03 -1.4 6.88e-02 2.85e-02f 1
96 -2.2612328e+01 2.10e+01 2.04e+03 -1.0 5.83e+02 -1.0 8.98e-02 2.31e-03h 1
97 -2.0613290e+01 2.04e+01 1.75e+03 -1.0 1.30e+03 -1.5 9.99e-02 2.97e-02f 1
98 -2.0476518e+01 2.03e+01 2.98e+03 -1.0 9.70e+02 -1.1 8.89e-02 2.39e-03h 1
99 -1.8680567e+01 1.97e+01 2.20e+03 -1.0 1.18e+03 -1.5 1.55e-01 2.98e-02f 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
100 -1.8426781e+01 1.96e+01 3.23e+03 -1.0 1.00e+03 -1.1 9.40e-02 4.53e-03h 1
101 -1.6882140e+01 1.90e+01 1.58e+03 -1.0 1.04e+03 -1.6 2.59e-01 2.92e-02f 1
102 -1.6236404e+01 1.88e+01 2.29e+03 -1.0 9.94e+02 -1.2 8.74e-02 1.22e-02h 1
103 -1.5089463e+01 1.83e+01 9.94e+02 -1.0 8.25e+02 -1.6 3.29e-01 2.60e-02f 1
104 -1.3445032e+01 1.76e+01 8.79e+02 -1.0 8.54e+02 -1.2 9.09e-02 3.73e-02h 1
105 -1.2546694e+01 1.73e+01 9.50e+02 -1.0 1.03e+03 -1.7 9.05e-02 1.96e-02f 1
106 -1.1886532e+01 1.70e+01 9.13e+02 -1.0 6.28e+02 -1.3 1.07e-01 1.96e-02f 1
107 -1.0379088e+01 1.63e+01 1.08e+03 -1.0 1.01e+03 -1.7 9.44e-02 3.69e-02f 1
108 -1.0310464e+01 1.63e+01 8.07e+02 -1.0 6.43e+02 -1.3 6.57e-02 1.86e-03h 1
109 -9.6672191e+00 1.60e+01 1.24e+03 -1.0 7.16e+02 -0.9 6.49e-02 1.95e-02h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
110 -8.5729304e+00 1.55e+01 8.90e+02 -1.0 7.18e+02 -1.4 1.36e-01 3.13e-02f 1
111 -8.4995146e+00 1.55e+01 1.91e+03 -1.0 6.59e+02 -0.9 8.53e-02 1.98e-03h 1
112 -6.4508726e+00 1.47e+01 1.51e+03 -1.0 8.75e+02 -1.4 2.01e-01 4.78e-02f 1
113 -6.3816741e+00 1.47e+01 2.51e+03 -1.0 5.48e+02 -1.0 1.08e-01 2.05e-03h 1
114 -4.4117376e+00 1.39e+01 1.48e+03 -1.0 7.90e+02 -1.5 3.84e-01 5.09e-02f 1
115 -4.2732133e+00 1.39e+01 2.14e+03 -1.0 5.96e+02 -1.0 1.09e-01 4.21e-03h 1
116 -2.4458467e+00 1.31e+01 1.25e+03 -1.0 6.94e+02 -1.5 3.96e-01 5.37e-02f 1
117 -2.2052570e+00 1.30e+01 1.49e+03 -1.0 5.73e+02 -1.1 1.11e-01 7.99e-03h 1
118 -5.6997467e-01 1.23e+01 9.76e+02 -1.0 5.94e+02 -1.6 4.47e-01 5.61e-02f 1
119 -2.0428884e-01 1.21e+01 9.24e+02 -1.0 5.00e+02 -1.1 1.18e-01 1.46e-02h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
120 1.2183722e+00 1.14e+01 8.64e+02 -1.0 4.95e+02 -1.6 2.11e-01 5.81e-02f 1
121 1.2941454e+00 1.14e+01 8.12e+02 -1.0 4.36e+02 -1.2 1.33e-01 3.44e-03h 1
122 3.0173269e+00 1.13e+01 7.29e+02 -1.0 4.95e+02 -1.7 2.47e-01 7.28e-02f 1
123 3.1855863e+00 1.10e+01 6.60e+02 -1.0 3.71e+02 -1.3 1.34e-01 8.99e-03h 1
124 4.7735325e+00 1.24e+01 5.97e+02 -1.0 4.18e+02 -1.7 2.06e-01 8.31e-02f 1
125 4.8266776e+00 1.24e+01 5.27e+02 -1.0 2.73e+02 -1.3 1.53e-01 3.82e-03h 1
126 6.2951867e+00 1.13e+01 4.92e+02 -1.0 3.62e+02 -1.8 2.12e-01 8.98e-02f 1
127 6.3766365e+00 1.12e+01 4.37e+02 -1.0 2.48e+02 -1.4 1.74e-01 6.59e-03h 1
128 7.8907991e+00 1.02e+01 4.01e+02 -1.0 3.20e+02 -1.8 2.61e-01 1.01e-01f 1
129 7.9299761e+00 1.02e+01 3.45e+02 -1.0 2.11e+02 -1.4 2.28e-01 3.80e-03h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
130 9.8845242e+00 1.01e+01 2.60e+02 -1.0 2.90e+02 -1.9 2.22e-01 1.29e-01f 1
131 9.9524752e+00 1.00e+01 2.78e+02 -1.0 1.68e+02 -1.5 2.75e-01 7.56e-03h 1
132 1.2535831e+01 9.23e+00 2.73e+02 -1.0 2.33e+02 -1.9 8.06e-02 1.68e-01f 1
133 1.2584030e+01 9.17e+00 2.18e+02 -1.0 1.05e+02 -1.5 3.85e-01 6.64e-03h 1
134 1.3714978e+01 6.14e+00 4.27e+02 -1.0 1.28e+02 -1.1 3.93e-01 1.76e-01h 1
135 1.4878662e+01 5.85e+00 2.61e+02 -1.0 7.56e+01 -1.6 2.09e-01 1.24e-01f 1
136 1.4933646e+01 5.79e+00 5.40e+02 -1.0 7.91e+01 -1.1 1.00e+00 9.25e-03h 1
137 1.8811346e+01 6.48e+00 3.79e+02 -1.0 9.37e+01 -1.6 2.94e-01 3.74e-01f 1
138 1.8869967e+01 6.37e+00 1.16e+02 -1.0 2.98e+01 -1.2 1.00e+00 1.73e-02h 1
139 2.4618068e+01 6.46e+00 4.86e+01 -1.0 4.78e+01 -1.7 5.83e-01 6.61e-01f 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
140 5.6088394e+01 1.07e+01 7.43e+01 -1.0 3.15e+01 -2.1 3.17e-01 1.00e+00f 1
141 6.8401142e+01 7.82e+00 3.47e+01 -1.0 1.23e+01 -1.7 4.81e-01 1.00e+00f 1
142 6.9125304e+01 2.17e+00 6.83e+00 -1.0 4.85e+00 -1.3 8.46e-01 1.00e+00f 1
143 7.2449832e+01 9.46e-01 7.08e-01 -1.0 5.64e+00 -1.8 8.82e-01 1.00e+00f 1
144 8.0073894e+01 2.19e+00 3.92e+00 -1.7 7.62e+00 -2.2 5.18e-01 1.00e+00f 1
145 8.5378975e+01 3.73e+00 4.08e+00 -1.7 1.48e+03 - 4.00e-03 8.65e-03f 1
146 9.0984727e+01 5.23e+00 4.01e+00 -1.7 1.57e+02 - 4.87e-02 8.88e-02f 1
147 9.4941272e+01 4.36e+00 2.71e+00 -1.7 2.66e+01 - 3.23e-01 1.69e-01f 1
148 1.0121743e+02 3.29e+00 2.08e+00 -1.7 5.32e+01 - 1.60e-01 2.43e-01f 1
149 1.0462559e+02 2.32e+00 1.48e+00 -1.7 2.98e+01 - 3.48e-01 2.84e-01f 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
150 1.0631723e+02 1.52e+00 1.00e+00 -1.7 3.36e+01 - 3.44e-01 3.21e-01h 1
151 1.0690588e+02 1.00e+00 6.65e-01 -1.7 3.81e+01 - 3.03e-01 3.35e-01h 1
152 1.0698620e+02 7.43e-01 4.79e-01 -1.7 4.61e+01 - 1.00e+00 2.66e-01h 1
153 1.0657507e+02 1.31e-01 3.90e-03 -1.7 3.85e+00 - 1.00e+00 1.00e+00h 1
154 1.0664006e+02 2.99e-02 2.23e-03 -2.5 1.57e+00 - 1.00e+00 1.00e+00h 1
155 1.0678538e+02 7.37e-03 4.11e-03 -3.8 1.12e+00 - 8.96e-01 1.00e+00h 1
156 1.0679941e+02 2.16e-03 4.46e-05 -3.8 9.67e-01 - 1.00e+00 1.00e+00h 1
157 1.0680822e+02 8.37e-04 6.41e-04 -5.7 6.66e-01 - 8.54e-01 1.00e+00h 1
158 1.0680856e+02 1.69e-04 3.04e-05 -5.7 3.32e-01 - 9.83e-01 1.00e+00h 1
159 1.0680861e+02 3.53e-05 1.64e-07 -5.7 1.96e-01 - 1.00e+00 1.00e+00h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
160 1.0680870e+02 4.30e-06 2.32e-06 -8.6 6.96e-02 - 9.88e-01 1.00e+00h 1
161 1.0680871e+02 4.99e-08 2.10e-10 -8.6 8.15e-03 - 1.00e+00 1.00e+00h 1
162 1.0680871e+02 1.39e-11 6.76e-14 -9.0 2.52e-03 - 1.00e+00 1.00e+00h 1
Number of Iterations....: 162
(scaled) (unscaled)
Objective...............: -1.0680870543718251e+02 1.0680870543718251e+02
Dual infeasibility......: 6.7566822382063500e-14 6.7566822382063500e-14
Constraint violation....: 4.2045828098652596e-12 1.3875123272555356e-11
Variable bound violation: 8.8945395493311707e-09 8.8945395493311707e-09
Complementarity.........: 9.1127086454719753e-10 9.1127086454719753e-10
Overall NLP error.......: 9.1127086454719753e-10 9.1127086454719753e-10
Number of objective function evaluations = 177
Number of objective gradient evaluations = 163
Number of equality constraint evaluations = 177
Number of inequality constraint evaluations = 0
Number of equality constraint Jacobian evaluations = 163
Number of inequality constraint Jacobian evaluations = 0
Number of Lagrangian Hessian evaluations = 162
Total seconds in IPOPT = 4.026
EXIT: Optimal Solution Found.
Extract and Plot the Results
Now we can extract the optimal results and plot them to visualize how the fish population compares to the profit. Note that the values of infinite variables are returned as arrays corresponding to how the supports were used to discretize our model. We can use the value function to extract the values of the infinite variables.
ts = value(t)
u_opt = value(u)
x_opt = value(x)
J_opt = value(J);Create the plot for the fish population and profit over time.
p1 = plot(ts, [x_opt, J_opt] ,
label=["Fish Pop" "Profit"],
title="State Variables");Create the plot for the fishing rate over time.
p2 = plot(ts, u_opt,
label = "Fishing Rate",
title = "Fishing Rate vs Time");Visualize the two plots on one figure.
plot(p1,p2 ,layout=(2,1), size=(800,600))
Maintenance Tests
These are here to ensure this example stays up to date.
using Test
tol = 1E-6
@test termination_status(m) == MOI.LOCALLY_SOLVED
@test has_values(m)
@test u_opt isa Vector{<:Real}
@test J_opt isa Vector{<:Real}
@test isapprox(u_opt[end], 1.0000000088945395, atol=tol)
@test isapprox(x_opt[end], 31.441105707837544, atol=tol)
@test isapprox(J_opt[end], 106.80870543718251, atol=tol)Test PassedThis page was generated using Literate.jl.