Hanging Chain Problem
We will solve a reformulated version of the Hanging Chain Problem from the Constrained Optimization Problem Set.
Problem Statement and Model
In this problem, we seek to minimize the potential energy of a chain of length $L$ suspended between points $a$ and $b$. The potential energy constrained by length is represented by:
\[\begin{gathered} \int_0^1 x(1 + (x')^2)^{1/2} \, dt \end{gathered}\]
Our optimization problem is defined as follows:
\[\begin{aligned} &&\min_{x,u} x_2(t_f)\\ &&\text{s.t.} &&& \quad \dot{x}_1= u\\ &&&&&\dot{x}_2 = x_1(1+u^2)^{1/2}\\ &&&&&\dot{x}_3 = (1+u^2)^{1/2}\\ &&&&&x(t_0) = (a,0,0)^T\\ &&&&&x_1(t_f) = b\\ &&&&&x_3(t_f) = L\\ &&&&&x(t) \in [0,10]\\ &&&&&u(t) \in [-10,20]\\ \end{aligned}\]
Model Definition
First we must import $InfiniteOpt$ and other packages.
using InfiniteOpt, Ipopt, Plots;Next we specify an array of initial conditions as well as problem variables.
a, b, L = 1, 3, 4
x0 = [a, 0, 0]
xtf = [b, NaN, L];We initialize the infinite model and opt to use the Ipopt solver
m = InfiniteModel(Ipopt.Optimizer);t is specified as $\ t \in [0,1]$. The bounds are arbitrary for this problem:
@infinite_parameter(m, t in [0,1], num_supports = 100);Now let's specify variables. $u$ is our controller variable.
@variable(m, 0 <= x[1:3] <= 10, Infinite(t))
@variable(m, -10 <= u <= 20, Infinite(t));Specifying the objective to minimize kinetic energy at the final time:
@objective(m, Min, x[2](1));Define the ODEs which serve as our system model.
@constraint(m, ∂(x[1],t) == u)
@constraint(m, ∂(x[2],t) == x[1] * (1 + u^2)^(1/2))
@constraint(m, ∂(x[3],t) == (1 + u^2)^(1/2));Set our inital and final conditions.
@constraint(m, [i in 1:3], x[i](0) == x0[i])
@constraint(m, x[1](1) == xtf[1])
@constraint(m, x[3](1) == xtf[3]);Problem Solution
Optimize the model:
optimize!(m)This is Ipopt version 3.14.4, running with linear solver MUMPS 5.4.1.
Number of nonzeros in equality constraint Jacobian...: 1596
Number of nonzeros in inequality constraint Jacobian.: 0
Number of nonzeros in Lagrangian Hessian.............: 400
Total number of variables............................: 700
variables with only lower bounds: 0
variables with lower and upper bounds: 400
variables with only upper bounds: 0
Total number of equality constraints.................: 602
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 9.9999900e-03 3.99e+00 1.00e+00 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 8.6845875e-02 1.95e+01 1.61e+04 -1.0 6.57e+01 - 2.56e-03 3.02e-01f 1
2 1.7965356e-01 1.87e+01 2.25e+04 -1.0 5.31e+01 4.0 3.21e-02 4.23e-02h 1
3 1.8137048e-01 1.87e+01 2.33e+04 -1.0 4.73e+01 4.4 6.31e-02 6.68e-04h 1
4 1.8140295e-01 1.87e+01 1.89e+06 -1.0 4.83e+01 4.9 1.26e-01 1.22e-05h 1
5 1.9254888e-01 1.86e+01 2.48e+06 -1.0 4.81e+01 4.4 2.04e-01 4.17e-03h 1
6 2.0905429e-01 1.85e+01 3.64e+06 -1.0 4.86e+01 4.8 1.09e-01 6.07e-03h 1
7 2.2802793e-01 1.84e+01 3.41e+07 -1.0 4.82e+01 5.2 3.59e-01 6.93e-03h 1
8 2.8201307e-01 1.80e+01 3.06e+07 -1.0 4.62e+01 4.7 1.02e-01 2.02e-02h 1
9 3.1317553e-01 1.78e+01 3.47e+07 -1.0 4.47e+01 5.2 2.45e-01 1.13e-02h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
10 3.4723467e-01 1.76e+01 4.07e+07 -1.0 4.39e+01 5.6 5.62e-02 1.22e-02h 1
11 3.8552094e-01 1.73e+01 6.10e+07 -1.0 4.28e+01 6.0 6.34e-02 1.37e-02h 1
12 4.0843146e-01 1.72e+01 6.02e+07 -1.0 3.93e+01 5.6 7.08e-03 8.64e-03h 1
13 4.1028640e-01 1.72e+01 1.02e+08 -1.0 3.99e+01 6.0 6.60e-02 6.60e-04h 1
14 4.3553508e-01 1.70e+01 3.20e+08 -1.0 4.04e+01 6.4 1.31e-01 8.93e-03h 1
15 4.5808211e-01 1.69e+01 3.20e+08 -1.0 3.81e+01 5.9 2.79e-02 8.08e-03h 1
16 4.6870391e-01 1.68e+01 3.89e+08 -1.0 3.84e+01 6.4 7.68e-02 3.74e-03h 1
17 5.1251695e-01 1.66e+01 5.77e+08 -1.0 3.84e+01 6.8 8.31e-02 1.54e-02f 1
18 5.2292005e-01 1.65e+01 5.69e+08 -1.0 3.52e+01 6.3 1.53e-02 3.70e-03h 1
19 5.2697033e-01 1.65e+01 6.03e+08 -1.0 3.63e+01 6.7 6.98e-02 1.42e-03h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
20 6.1393774e-01 1.60e+01 7.16e+08 -1.0 3.66e+01 7.2 8.66e-02 3.03e-02f 1
21 7.3213394e-01 1.53e+01 6.54e+08 -1.0 3.40e+01 7.6 6.77e-03 4.09e-02f 1
22 8.0635655e-01 1.49e+01 6.80e+08 -1.0 3.09e+01 8.0 4.06e-04 2.55e-02f 1
23 8.5443045e-01 1.47e+01 1.27e+09 -1.0 2.92e+01 8.4 1.25e-03 1.65e-02f 1
24 8.5472095e-01 1.47e+01 7.25e+08 -1.0 2.70e+01 8.0 4.02e-02 9.99e-05h 1
25 9.4924730e-01 1.42e+01 1.17e+09 -1.0 2.79e+01 8.4 7.89e-03 3.24e-02f 1
26 9.7274767e-01 1.41e+01 1.75e+09 -1.0 2.61e+01 8.8 1.73e-05 8.07e-03f 1
27 9.7893019e-01 1.41e+01 1.74e+09 -1.0 2.48e+01 8.3 7.44e-02 2.13e-03h 1
28 1.1355863e+00 1.33e+01 2.81e+09 -1.0 2.54e+01 8.8 2.24e-05 5.39e-02f 1
29 1.1851152e+00 1.31e+01 5.26e+09 -1.0 2.26e+01 9.2 1.46e-03 1.71e-02f 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
30 1.1875976e+00 1.31e+01 6.54e+09 -1.0 2.07e+01 8.7 5.75e-02 8.61e-04h 1
31 1.3495034e+00 1.23e+01 1.76e+10 -1.0 2.16e+01 9.1 1.24e-03 5.62e-02f 1
32 1.4016517e+00 1.21e+01 4.27e+10 -1.0 1.92e+01 9.6 1.85e-04 1.84e-02f 1
33 1.7460477e+00 1.06e+01 5.77e+10 -1.0 1.75e+01 9.1 1.05e-02 1.22e-01f 1
34 1.7872628e+00 1.05e+01 9.82e+10 -1.0 1.38e+01 9.5 1.91e-05 1.53e-02f 1
35 1.8094359e+00 1.04e+01 1.72e+11 -1.0 1.38e+01 9.9 4.00e-05 8.30e-03f 1
36 4.4588974e+00 6.90e+00 1.58e+12 -1.0 1.32e+01 9.5 4.08e-06 9.90e-01f 1
37 4.4745421e+00 6.74e+00 1.56e+12 -1.0 6.33e+00 9.9 2.59e-05 2.30e-02f 1
38 5.1417859e+00 6.94e+00 1.26e+12 -1.0 5.94e+00 9.4 4.03e-05 9.90e-01f 1
39 5.1719734e+00 4.63e+00 1.06e+13 -1.0 5.68e+00 8.9 4.41e-06 1.00e+00f 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
40 5.1832741e+00 3.21e-02 4.22e+09 -1.0 4.15e+00 - 1.28e-02 1.00e+00f 1
41 5.1788574e+00 5.04e-02 3.25e+07 -1.0 3.22e-01 - 9.90e-01 1.00e+00h 1
42 5.1780818e+00 2.22e+00 4.66e+05 -1.0 4.11e+00 - 9.86e-01 1.00e+00h 1
43 5.1783273e+00 8.55e-03 3.40e+08 -1.0 1.14e+00 8.4 9.89e-01 1.00e+00h 1
44 5.1783273e+00 2.01e-07 1.24e+06 -1.0 4.63e-03 8.0 9.90e-01 1.00e+00h 1
45 5.1783273e+00 3.55e-15 3.25e+00 -1.0 1.05e-07 7.5 9.98e-01 1.00e+00h 1
46 5.1783273e+00 3.55e-15 2.63e-04 -3.8 2.54e-11 7.0 1.00e+00 1.00e+00h 1
47 5.1783273e+00 3.55e-15 3.15e-04 -5.7 9.13e-11 6.5 1.00e+00 1.00e+00h 1
48 5.1783273e+00 3.55e-15 3.15e-04 -5.7 2.74e-10 6.1 1.00e+00 1.00e+00h 1
49 5.1783273e+00 3.55e-15 3.15e-04 -5.7 8.22e-10 5.6 1.00e+00 1.00e+00h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
50 5.1783273e+00 3.55e-15 3.15e-04 -5.7 2.47e-09 5.1 1.00e+00 1.00e+00h 1
51 5.1783273e+00 3.55e-15 3.15e-04 -5.7 7.40e-09 4.6 1.00e+00 1.00e+00h 1
52 5.1783273e+00 1.07e-14 3.15e-04 -5.7 2.22e-08 4.2 1.00e+00 1.00e+00h 1
53 5.1783273e+00 3.55e-15 3.15e-04 -5.7 6.66e-08 3.7 1.00e+00 1.00e+00h 1
54 5.1783273e+00 3.55e-15 3.15e-04 -5.7 2.00e-07 3.2 1.00e+00 1.00e+00h 1
55 5.1783273e+00 1.44e-14 3.15e-04 -5.7 5.99e-07 2.7 1.00e+00 1.00e+00h 1
56 5.1783273e+00 1.30e-13 3.15e-04 -5.7 1.80e-06 2.2 1.00e+00 1.00e+00h 1
57 5.1783272e+00 1.17e-12 3.15e-04 -5.7 5.39e-06 1.8 1.00e+00 1.00e+00h 1
58 5.1783270e+00 1.05e-11 3.15e-04 -5.7 1.62e-05 1.3 1.00e+00 1.00e+00h 1
59 5.1783264e+00 9.48e-11 3.15e-04 -5.7 4.85e-05 0.8 1.00e+00 1.00e+00h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
60 5.1783247e+00 8.53e-10 3.15e-04 -5.7 1.46e-04 0.3 1.00e+00 1.00e+00h 1
61 5.1783196e+00 7.67e-09 3.15e-04 -5.7 4.37e-04 -0.1 1.00e+00 1.00e+00h 1
62 5.1783044e+00 6.90e-08 3.15e-04 -5.7 1.31e-03 -0.6 1.00e+00 1.00e+00h 1
63 5.1782586e+00 6.21e-07 3.15e-04 -5.7 3.93e-03 -1.1 1.00e+00 1.00e+00h 1
64 5.1781215e+00 5.56e-06 3.15e-04 -5.7 1.18e-02 -1.6 1.00e+00 1.00e+00h 1
65 5.1777132e+00 4.95e-05 3.14e-04 -5.7 3.52e-02 -2.1 1.00e+00 1.00e+00h 1
66 5.1765147e+00 4.30e-04 3.10e-04 -5.7 1.04e-01 -2.5 1.00e+00 1.00e+00h 1
67 5.1731373e+00 3.50e-03 3.01e-04 -5.7 3.04e-01 -3.0 1.00e+00 1.00e+00h 1
68 5.1645970e+00 2.37e-02 2.76e-04 -5.7 8.36e-01 -3.5 1.00e+00 1.00e+00h 1
69 5.1475112e+00 4.74e-01 4.77e-03 -5.7 2.05e+00 -4.0 1.00e+00 1.00e+00h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
70 5.1452517e+00 4.51e-01 4.54e-03 -5.7 2.48e+01 -4.4 2.28e-01 4.96e-02h 1
71 5.1380143e+00 4.88e-01 4.90e-03 -5.7 2.06e+00 -4.0 1.00e+00 8.22e-01h 1
72 5.1360582e+00 4.58e-01 4.60e-03 -5.7 2.08e+01 -4.5 9.45e-01 6.32e-02h 1
73 5.1331707e+00 1.85e-01 1.86e-03 -5.7 2.21e+00 -4.1 1.00e+00 6.12e-01h 1
74 5.1326171e+00 1.84e-01 1.85e-03 -5.7 2.35e+02 -4.5 6.85e-02 6.33e-03h 1
75 5.1308394e+00 4.02e-02 3.92e-04 -5.7 1.77e+00 -4.1 1.00e+00 9.71e-01h 1
76 5.1302551e+00 5.42e-03 1.46e-04 -5.7 7.07e-01 -3.7 1.00e+00 1.00e+00h 1
77 5.1298845e+00 6.97e-03 1.58e-04 -5.7 3.31e+00 -4.2 1.00e+00 1.48e-01h 1
78 5.1295821e+00 3.08e-03 9.68e-05 -5.7 5.25e-01 -3.7 1.00e+00 1.00e+00h 1
79 5.1284729e+00 5.51e-02 3.23e-04 -5.7 2.67e+00 -4.2 1.00e+00 8.74e-01h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
80 5.1286779e+00 1.42e-02 2.37e-04 -5.7 1.08e+00 -3.8 1.00e+00 9.08e-01h 1
81 5.1281565e+00 2.69e-02 2.80e-04 -5.7 1.19e+00 -4.3 1.00e+00 1.00e+00h 1
82 5.1278347e+00 3.61e-03 6.35e-05 -5.7 4.37e-01 -3.8 1.00e+00 1.00e+00h 1
83 5.1273746e+00 2.72e-02 1.52e-04 -5.7 1.92e+00 -4.3 1.00e+00 8.04e-01h 1
84 5.1271029e+00 3.44e-02 2.83e-04 -5.7 1.41e+01 -4.8 3.52e-01 6.76e-02h 1
85 5.1271295e+00 3.43e-02 2.15e-03 -5.7 3.49e+01 -4.4 5.23e-01 2.51e-03h 3
86 5.1270478e+00 2.13e-02 2.84e-06 -5.7 1.60e+00 - 1.00e+00 1.00e+00h 1
87 5.1270446e+00 4.85e-05 1.49e-07 -5.7 9.34e-02 - 1.00e+00 1.00e+00h 1
88 5.1270447e+00 2.86e-09 1.48e-11 -5.7 8.71e-04 - 1.00e+00 1.00e+00h 1
89 5.1270303e+00 1.32e-05 2.99e-02 -8.6 8.46e-03 - 1.00e+00 1.00e+00h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
90 5.1270301e+00 1.66e-09 1.67e-11 -8.6 1.11e-04 - 1.00e+00 1.00e+00h 1
Number of Iterations....: 90
(scaled) (unscaled)
Objective...............: 5.1270301228511315e+00 5.1270301228511315e+00
Dual infeasibility......: 1.6724968069142987e-11 1.6724968069142987e-11
Constraint violation....: 1.6562617943804980e-09 1.6562617943804980e-09
Variable bound violation: 1.1523690231707695e-41 1.1523690231707695e-41
Complementarity.........: 2.5154832478972195e-09 2.5154832478972195e-09
Overall NLP error.......: 2.5154832478972195e-09 2.5154832478972195e-09
Number of objective function evaluations = 94
Number of objective gradient evaluations = 91
Number of equality constraint evaluations = 94
Number of inequality constraint evaluations = 0
Number of equality constraint Jacobian evaluations = 91
Number of inequality constraint Jacobian evaluations = 0
Number of Lagrangian Hessian evaluations = 90
Total seconds in IPOPT = 0.491
EXIT: Optimal Solution Found.
Extract the results.
ts = value(t)
u_opt = value(u)
x1_opt = value(x[1])
x2_opt = value(x[2])
x3_opt = value(x[3])
@show(objective_value(m))
p1 = plot(ts, [x1_opt, x2_opt, x3_opt],
label=["x1" "x2" "x3"],
title="State Variables")
p2 = plot(ts, u_opt,
label="u(t)",
title="Input")
plot(p1, p2, layout=(2,1), size=(800,600));objective_value(m) = 5.1270301228511315
Maintenance Tests
These are here to ensure this example stays up to date.
using Test
@test termination_status(m) == MOI.LOCALLY_SOLVED
@test has_values(m)
@test u_opt isa Vector{<:Real}
@test x1_opt isa Vector{<:Real}Test PassedThis page was generated using Literate.jl.