Parameters
Parameters act like fixed variables. Internally, ExaModels keeps track of where parameters appear in the model, making it possible to efficiently modify their value without rebuilding the entire model.
Creating Parametric Models
Let's modify the example in Getting Started to use parameters. Suppose we want to make the penalty coefficient in the objective function adjustable:
First, let's create a core:
using ExaModels, NLPModelsIpopt
c_param = ExaCore()An ExaCore
Float type: ...................... Float64
Array type: ...................... Vector{Float64}
Backend: ......................... Nothing
number of objective patterns: .... 0
number of constraint patterns: ... 0
Adding parameters is very similar to adding variables – just pass a vector of values denoting the initial values.
θ = parameter(c_param, [100.0, 1.0]) # [penalty_coeff, offset]Parameter
θ ∈ R^{2}
Define the variables as before:
N = 10
x_p = variable(c_param, N; start = (mod(i, 2) == 1 ? -1.2 : 1.0 for i = 1:N))Variable
x ∈ R^{10}
Now we can use the parameters in our objective function just like variables:
objective(c_param, θ[1] * (x_p[i-1]^2 - x_p[i])^2 + (x_p[i-1] - θ[2])^2 for i = 2:N)Objective
min (...) + ∑_{p ∈ P} f(x,θ,p)
where |P| = 9
Add the same constraints as before:
constraint(
c_param,
3x_p[i+1]^3 + 2 * x_p[i+2] - 5 + sin(x_p[i+1] - x_p[i+2])sin(x_p[i+1] + x_p[i+2]) + 4x_p[i+1] -
x_p[i]exp(x_p[i] - x_p[i+1]) - 3 for i = 1:(N-2)
)Constraint
s.t. (...)
g♭ ≤ [g(x,θ,p)]_{p ∈ P} ≤ g♯
where |P| = 8
Create the model as before:
m_param = ExaModel(c_param)An ExaModel{Float64, Vector{Float64}, ...}
Problem name: Generic
All variables: ████████████████████ 10 All constraints: ████████████████████ 8
free: ████████████████████ 10 free: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0
lower: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0 lower: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0
upper: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0 upper: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0
low/upp: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0 low/upp: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0
fixed: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0 fixed: ████████████████████ 8
infeas: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0 infeas: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0
nnzh: (-36.36% sparsity) 75 linear: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0
nonlinear: ████████████████████ 8
nnzj: ( 70.00% sparsity) 24
Solve with original parameters:
result1 = ipopt(m_param)
println("Original objective: $(result1.objective)")This is Ipopt version 3.14.17, running with linear solver MUMPS 5.8.0.
Number of nonzeros in equality constraint Jacobian...: 24
Number of nonzeros in inequality constraint Jacobian.: 0
Number of nonzeros in Lagrangian Hessian.............: 75
Total number of variables............................: 10
variables with only lower bounds: 0
variables with lower and upper bounds: 0
variables with only upper bounds: 0
Total number of equality constraints.................: 8
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 2.0570000e+03 2.48e+01 2.73e+01 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 1.0953147e+03 1.49e+01 8.27e+01 -1.0 2.20e+00 - 1.00e+00 1.00e+00f 1
2 3.2865521e+02 4.28e+00 1.36e+02 -1.0 1.43e+00 - 1.00e+00 1.00e+00f 1
3 1.3995370e+01 3.09e-01 2.18e+01 -1.0 5.63e-01 - 1.00e+00 1.00e+00f 1
4 6.2325715e+00 1.73e-02 8.47e-01 -1.0 2.10e-01 - 1.00e+00 1.00e+00f 1
5 6.2324586e+00 1.15e-05 8.16e-04 -1.7 3.35e-03 - 1.00e+00 1.00e+00h 1
6 6.2324586e+00 8.35e-12 7.97e-10 -5.7 2.00e-06 - 1.00e+00 1.00e+00h 1
Number of Iterations....: 6
(scaled) (unscaled)
Objective...............: 7.8692659500473017e-01 6.2324586324374636e+00
Dual infeasibility......: 7.9746955363607132e-10 6.3159588647976857e-09
Constraint violation....: 8.3546503049092280e-12 8.3546503049092280e-12
Variable bound violation: 0.0000000000000000e+00 0.0000000000000000e+00
Complementarity.........: 0.0000000000000000e+00 0.0000000000000000e+00
Overall NLP error.......: 7.9746955363607132e-10 6.3159588647976857e-09
Number of objective function evaluations = 7
Number of objective gradient evaluations = 7
Number of equality constraint evaluations = 7
Number of inequality constraint evaluations = 0
Number of equality constraint Jacobian evaluations = 7
Number of inequality constraint Jacobian evaluations = 0
Number of Lagrangian Hessian evaluations = 6
Total seconds in IPOPT = 0.210
EXIT: Optimal Solution Found.
Original objective: 6.232458632437464
Now change the penalty coefficient and solve again:
set_parameter!(c_param, θ, [200.0, 1.0]) # Double the penalty coefficient
result2 = ipopt(m_param)
println("Modified penalty objective: $(result2.objective)")This is Ipopt version 3.14.17, running with linear solver MUMPS 5.8.0.
Number of nonzeros in equality constraint Jacobian...: 24
Number of nonzeros in inequality constraint Jacobian.: 0
Number of nonzeros in Lagrangian Hessian.............: 75
Total number of variables............................: 10
variables with only lower bounds: 0
variables with lower and upper bounds: 0
variables with only upper bounds: 0
Total number of equality constraints.................: 8
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 4.0898000e+03 2.48e+01 2.70e+01 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 2.1810502e+03 1.49e+01 8.27e+01 -1.0 2.20e+00 - 1.00e+00 1.00e+00f 1
2 6.5137192e+02 4.27e+00 1.36e+02 -1.0 1.43e+00 - 1.00e+00 1.00e+00f 1
3 2.4064340e+01 3.08e-01 2.18e+01 -1.0 5.62e-01 - 1.00e+00 1.00e+00f 1
4 8.6476680e+00 1.72e-02 8.45e-01 -1.0 2.10e-01 - 1.00e+00 1.00e+00f 1
5 8.6474398e+00 1.15e-05 8.07e-04 -1.7 3.39e-03 - 1.00e+00 1.00e+00h 1
6 8.6474398e+00 8.42e-12 7.91e-10 -5.7 2.03e-06 - 1.00e+00 1.00e+00h 1
Number of Iterations....: 6
(scaled) (unscaled)
Objective...............: 5.4592422674820063e-01 8.6474397516914987e+00
Dual infeasibility......: 7.9051456536755353e-10 1.2521750715422049e-08
Constraint violation....: 8.4190432403374871e-12 8.4190432403374871e-12
Variable bound violation: 0.0000000000000000e+00 0.0000000000000000e+00
Complementarity.........: 0.0000000000000000e+00 0.0000000000000000e+00
Overall NLP error.......: 7.9051456536755353e-10 1.2521750715422049e-08
Number of objective function evaluations = 7
Number of objective gradient evaluations = 7
Number of equality constraint evaluations = 7
Number of inequality constraint evaluations = 0
Number of equality constraint Jacobian evaluations = 7
Number of inequality constraint Jacobian evaluations = 0
Number of Lagrangian Hessian evaluations = 6
Total seconds in IPOPT = 0.002
EXIT: Optimal Solution Found.
Modified penalty objective: 8.647439751691499
Try a different offset parameter:
set_parameter!(c_param, θ, [200.0, 0.5]) # Change the offset in the objective
result3 = ipopt(m_param)
println("Modified offset objective: $(result3.objective)")This is Ipopt version 3.14.17, running with linear solver MUMPS 5.8.0.
Number of nonzeros in equality constraint Jacobian...: 24
Number of nonzeros in inequality constraint Jacobian.: 0
Number of nonzeros in Lagrangian Hessian.............: 75
Total number of variables............................: 10
variables with only lower bounds: 0
variables with lower and upper bounds: 0
variables with only upper bounds: 0
Total number of equality constraints.................: 8
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 4.0810500e+03 2.48e+01 2.69e+01 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 2.1767809e+03 1.49e+01 8.26e+01 -1.0 2.20e+00 - 1.00e+00 1.00e+00f 1
2 6.5050886e+02 4.27e+00 1.36e+02 -1.0 1.43e+00 - 1.00e+00 1.00e+00f 1
3 2.4276149e+01 3.07e-01 2.18e+01 -1.0 5.61e-01 - 1.00e+00 1.00e+00f 1
4 8.8465512e+00 1.72e-02 8.43e-01 -1.0 2.09e-01 - 1.00e+00 1.00e+00f 1
5 8.8451636e+00 1.15e-05 8.04e-04 -1.7 3.40e-03 - 1.00e+00 1.00e+00h 1
6 8.8451630e+00 8.47e-12 7.88e-10 -5.7 2.05e-06 - 1.00e+00 1.00e+00h 1
Number of Iterations....: 6
(scaled) (unscaled)
Objective...............: 5.5805444714793528e-01 8.8451629872947741e+00
Dual infeasibility......: 7.8812124187921384e-10 1.2491721683785540e-08
Constraint violation....: 8.4678930534209940e-12 8.4678930534209940e-12
Variable bound violation: 0.0000000000000000e+00 0.0000000000000000e+00
Complementarity.........: 0.0000000000000000e+00 0.0000000000000000e+00
Overall NLP error.......: 7.8812124187921384e-10 1.2491721683785540e-08
Number of objective function evaluations = 7
Number of objective gradient evaluations = 7
Number of equality constraint evaluations = 7
Number of inequality constraint evaluations = 0
Number of equality constraint Jacobian evaluations = 7
Number of inequality constraint Jacobian evaluations = 0
Number of Lagrangian Hessian evaluations = 6
Total seconds in IPOPT = 0.002
EXIT: Optimal Solution Found.
Modified offset objective: 8.845162987294774
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