Solve a problem
We consider the Hanging Chain problem from COPS package as an example. The problem is to find the shape of a chain hanging between two points a and b. The chain is assumed to be a uniform cable with a given length L. The aim is to find the shape of the chain that minimizes the potential energy.
Solving from JuMP model
We need first to import the needed packages and the problem.
using OptimalControlProblems
using JuMP
model = chain(JuMPBackend())A JuMP Model
├ solver: none
├ objective_sense: MIN_SENSE
│ └ objective_function_type: JuMP.VariableRef
├ num_variables: 404
├ num_constraints: 305
│ ├ JuMP.NonlinearExpr in MOI.EqualTo{Float64}: 200
│ └ JuMP.AffExpr in MOI.EqualTo{Float64}: 105
└ Names registered in the model
└ :con_x1, :con_x2, :con_x3, :u, :x1, :x2, :x3Then we can solve the problem using the optimize! function of Ipopt solver.
using Ipopt
# Set the optimizer
set_optimizer(model, Ipopt.Optimizer)
# Set the optimizer attributes
set_optimizer_attribute(model, "tol", 1e-8)
set_optimizer_attribute(model, "constr_viol_tol", 1e-6)
set_optimizer_attribute(model, "mu_strategy", "adaptive")
set_optimizer_attribute(model, "linear_solver", "mumps")
set_optimizer_attribute(model, "sb", "yes")
# Solve the model
optimize!(model)This is Ipopt version 3.14.17, running with linear solver MUMPS 5.7.3.
Number of nonzeros in equality constraint Jacobian...: 1405
Number of nonzeros in inequality constraint Jacobian.: 0
Number of nonzeros in Lagrangian Hessian.............: 800
Total number of variables............................: 404
variables with only lower bounds: 0
variables with lower and upper bounds: 0
variables with only upper bounds: 0
Total number of equality constraints.................: 305
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 1.8607232e+01 2.08e+00 3.06e-04 0.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 5.3977582e+00 2.13e-02 8.41e-03 -11.0 1.32e+01 - 1.00e+00 1.00e+00f 1
2 5.0927541e+00 4.00e-04 4.60e-03 -11.0 4.61e-01 -2.0 1.00e+00 1.00e+00h 1
3 5.0695401e+00 3.01e-03 3.73e-04 -11.0 3.61e+00 - 1.00e+00 1.00e+00h 1
4 5.0687930e+00 1.81e-03 4.12e-04 -11.0 3.01e+00 - 1.00e+00 1.00e+00h 1
5 5.0697325e+00 3.51e-05 3.56e-05 -11.0 3.59e-01 - 1.00e+00 1.00e+00h 1
6 5.0697842e+00 4.43e-07 3.13e-07 -11.0 4.08e-02 - 1.00e+00 1.00e+00h 1
7 5.0697846e+00 3.87e-11 3.99e-11 -11.0 2.31e-04 - 1.00e+00 1.00e+00h 1
Number of Iterations....: 7
(scaled) (unscaled)
Objective...............: 5.0697846107010411e+00 5.0697846107010411e+00
Dual infeasibility......: 3.9922221778398992e-11 3.9922221778398992e-11
Constraint violation....: 3.8657833878463777e-11 3.8657833878463777e-11
Variable bound violation: 0.0000000000000000e+00 0.0000000000000000e+00
Complementarity.........: 0.0000000000000000e+00 0.0000000000000000e+00
Overall NLP error.......: 3.9922221778398992e-11 3.9922221778398992e-11
Number of objective function evaluations = 8
Number of objective gradient evaluations = 8
Number of equality constraint evaluations = 8
Number of inequality constraint evaluations = 0
Number of equality constraint Jacobian evaluations = 8
Number of inequality constraint Jacobian evaluations = 0
Number of Lagrangian Hessian evaluations = 7
Total seconds in IPOPT = 3.025
EXIT: Optimal Solution Found.Solving from OptimalControl model
We need first to import the needed packages and the problem.
using OptimalControlProblems
using OptimalControl
_, model = chain(OptimalControlBackend())ADNLPModel - Model with automatic differentiation backend ADModelBackend{
ReverseDiffADGradient,
ReverseDiffADHvprod,
ForwardDiffADJprod,
ReverseDiffADJtprod,
SparseADJacobian,
SparseReverseADHessian,
ForwardDiffADGHjvprod,
}
Problem name: Generic
All variables: ████████████████████ 404 All constraints: ████████████████████ 305
free: ████████████████████ 404 free: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0
lower: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0 lower: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0
upper: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0 upper: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0
low/upp: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0 low/upp: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0
fixed: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0 fixed: ████████████████████ 305
infeas: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0 infeas: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0
nnzh: ( 99.75% sparsity) 202 linear: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0
nonlinear: ████████████████████ 305
nnzj: ( 98.86% sparsity) 1405
Counters:
obj: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0 grad: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0 cons: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0
cons_lin: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0 cons_nln: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0 jcon: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0
jgrad: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0 jac: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0 jac_lin: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0
jac_nln: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0 jprod: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0 jprod_lin: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0
jprod_nln: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0 jtprod: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0 jtprod_lin: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0
jtprod_nln: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0 hess: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0 hprod: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0
jhess: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0 jhprod: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0
Then we can solve the problem using the ipopt function of NLPModelsIpopt package.
using NLPModelsIpopt
# Solve the model
sol = NLPModelsIpopt.ipopt(
model;
print_level=5,
tol=1e-8,
mu_strategy="adaptive",
sb="yes",
constr_viol_tol=1e-6,
)This is Ipopt version 3.14.17, running with linear solver MUMPS 5.7.3.
Number of nonzeros in equality constraint Jacobian...: 1405
Number of nonzeros in inequality constraint Jacobian.: 0
Number of nonzeros in Lagrangian Hessian.............: 202
Total number of variables............................: 404
variables with only lower bounds: 0
variables with lower and upper bounds: 0
variables with only upper bounds: 0
Total number of equality constraints.................: 305
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 1.8000000e+01 2.00e+00 3.03e-04 0.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 5.4019494e+00 1.91e-02 6.74e-03 -11.0 1.26e+01 - 1.00e+00 1.00e+00f 1
2 5.0865615e+00 2.68e-04 3.91e-03 -11.0 3.94e-01 -2.0 1.00e+00 1.00e+00h 1
3 5.0686974e+00 5.01e-03 5.21e-04 -11.0 4.46e+00 - 1.00e+00 1.00e+00h 1
4 5.0690826e+00 9.63e-04 3.77e-04 -11.0 2.46e+00 - 1.00e+00 1.00e+00h 1
5 5.0697541e+00 2.90e-05 2.95e-05 -11.0 2.07e-01 - 1.00e+00 1.00e+00h 1
6 5.0697845e+00 1.34e-07 1.37e-07 -11.0 2.25e-02 - 1.00e+00 1.00e+00h 1
7 5.0697846e+00 5.88e-12 6.34e-12 -11.0 6.96e-05 - 1.00e+00 1.00e+00h 1
Number of Iterations....: 7
(scaled) (unscaled)
Objective...............: 5.0697846107338593e+00 5.0697846107338593e+00
Dual infeasibility......: 6.3382910031606343e-12 6.3382910031606343e-12
Constraint violation....: 5.8824056736739294e-12 5.8824056736739294e-12
Variable bound violation: 0.0000000000000000e+00 0.0000000000000000e+00
Complementarity.........: 0.0000000000000000e+00 0.0000000000000000e+00
Overall NLP error.......: 6.3382910031606343e-12 6.3382910031606343e-12
Number of objective function evaluations = 8
Number of objective gradient evaluations = 8
Number of equality constraint evaluations = 8
Number of inequality constraint evaluations = 0
Number of equality constraint Jacobian evaluations = 8
Number of inequality constraint Jacobian evaluations = 0
Number of Lagrangian Hessian evaluations = 7
Total seconds in IPOPT = 5.264
EXIT: Optimal Solution Found.