Tutorial

Tutorial

NLPModelsIpopt is a thin IPOPT wrapper for NLPModels. In this tutorial we'll show examples of problems created with NLPModels and solved with Ipopt.

Simple problems

The interface for calling Ipopt is very simple:

output = ipopt(nlp)

Solves the NLPModel problem nlp using IpOpt.

Let's create an NLPModel for the Rosenbrock function

$f(x) = (x_1 - 1)^2 + 100 (x_2 - x_1^2)^2$

to test this interface:

using NLPModels, NLPModelsIpopt

nlp = ADNLPModel(x -> (x[1] - 1)^2 + 100 * (x[2] - x[1]^2)^2, [-1.2; 1.0])
stats = ipopt(nlp)
print(stats)

******************************************************************************
This program contains Ipopt, a library for large-scale nonlinear optimization.
Ipopt is released as open source code under the Eclipse Public License (EPL).
******************************************************************************

This is Ipopt version 3.12.10, running with linear solver mumps.
NOTE: Other linear solvers might be more efficient (see Ipopt documentation).

Number of nonzeros in equality constraint Jacobian...:        0
Number of nonzeros in inequality constraint Jacobian.:        0
Number of nonzeros in Lagrangian Hessian.............:        3

Total number of variables............................:        2
variables with only lower bounds:        0
variables with lower and upper bounds:        0
variables with only upper bounds:        0
Total number of equality constraints.................:        0
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  2.4200000e+01 0.00e+00 1.00e+02  -1.0 0.00e+00    -  0.00e+00 0.00e+00   0
1  4.7318843e+00 0.00e+00 2.15e+00  -1.0 3.81e-01    -  1.00e+00 1.00e+00f  1
2  4.0873987e+00 0.00e+00 1.20e+01  -1.0 4.56e+00    -  1.00e+00 1.25e-01f  4
3  3.2286726e+00 0.00e+00 4.94e+00  -1.0 2.21e-01    -  1.00e+00 1.00e+00f  1
4  3.2138981e+00 0.00e+00 1.02e+01  -1.0 4.82e-01    -  1.00e+00 1.00e+00f  1
5  1.9425854e+00 0.00e+00 1.62e+00  -1.0 6.70e-02    -  1.00e+00 1.00e+00f  1
6  1.6001937e+00 0.00e+00 3.44e+00  -1.0 7.35e-01    -  1.00e+00 2.50e-01f  3
7  1.1783896e+00 0.00e+00 1.92e+00  -1.0 1.44e-01    -  1.00e+00 1.00e+00f  1
8  9.2241158e-01 0.00e+00 4.00e+00  -1.0 2.08e-01    -  1.00e+00 1.00e+00f  1
9  5.9748862e-01 0.00e+00 7.36e-01  -1.0 8.91e-02    -  1.00e+00 1.00e+00f  1
iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
10  4.5262510e-01 0.00e+00 2.42e+00  -1.7 2.97e-01    -  1.00e+00 5.00e-01f  2
11  2.8076244e-01 0.00e+00 9.25e-01  -1.7 1.02e-01    -  1.00e+00 1.00e+00f  1
12  2.1139340e-01 0.00e+00 3.34e+00  -1.7 1.77e-01    -  1.00e+00 1.00e+00f  1
13  8.9019501e-02 0.00e+00 2.25e-01  -1.7 9.45e-02    -  1.00e+00 1.00e+00f  1
14  5.1535405e-02 0.00e+00 1.49e+00  -1.7 2.84e-01    -  1.00e+00 5.00e-01f  2
15  1.9992778e-02 0.00e+00 4.64e-01  -1.7 1.09e-01    -  1.00e+00 1.00e+00f  1
16  7.1692436e-03 0.00e+00 1.03e+00  -1.7 1.39e-01    -  1.00e+00 1.00e+00f  1
17  1.0696137e-03 0.00e+00 9.09e-02  -1.7 5.50e-02    -  1.00e+00 1.00e+00f  1
18  7.7768464e-05 0.00e+00 1.44e-01  -2.5 5.53e-02    -  1.00e+00 1.00e+00f  1
19  2.8246695e-07 0.00e+00 1.50e-03  -2.5 7.31e-03    -  1.00e+00 1.00e+00f  1
iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
20  8.5170750e-12 0.00e+00 4.90e-05  -5.7 1.05e-03    -  1.00e+00 1.00e+00f  1
21  3.7439756e-21 0.00e+00 1.73e-10  -5.7 2.49e-06    -  1.00e+00 1.00e+00f  1

Number of Iterations....: 21

(scaled)                 (unscaled)
Objective...............:   1.7365378678754519e-21    3.7439756431394737e-21
Dual infeasibility......:   1.7312156654298279e-10    3.7325009746667082e-10
Constraint violation....:   0.0000000000000000e+00    0.0000000000000000e+00
Complementarity.........:   0.0000000000000000e+00    0.0000000000000000e+00
Overall NLP error.......:   1.7312156654298279e-10    3.7325009746667082e-10

Number of objective function evaluations             = 45
Number of objective gradient evaluations             = 22
Number of equality constraint evaluations            = 0
Number of inequality constraint evaluations          = 0
Number of equality constraint Jacobian evaluations   = 0
Number of inequality constraint Jacobian evaluations = 0
Number of Lagrangian Hessian evaluations             = 21
Total CPU secs in IPOPT (w/o function evaluations)   =      1.077
Total CPU secs in NLP function evaluations           =      0.210

EXIT: Optimal Solution Found.
Generic Execution stats
status: "first-order stationary"
objective value: 3.743975643139474e-21
primal feasibility: 0.0
dual feasibility: 3.732500974666708e-10
primal feasibility: 0.0
solution: [1.0  1.0]
iterations: 21
elapsed time: 1.287
solver specific:
multipliers_U: [0.0  0.0]
multipliers_L: [0.0  0.0]
multipliers_con: ∅
internal_msg: :Solve_Succeeded

For comparison, we present the same problem and output using the JuMP route:

using JuMP, Ipopt

model = Model(with_optimizer(Ipopt.Optimizer))
x0 = [-1.2; 1.0]
@variable(model, x[i=1:2], start=x0[i])
@NLobjective(model, Min, (x[1] - 1)^2 + 100 * (x[2] - x[1]^2)^2)
optimize!(model)
This is Ipopt version 3.12.10, running with linear solver mumps.
NOTE: Other linear solvers might be more efficient (see Ipopt documentation).

Number of nonzeros in equality constraint Jacobian...:        0
Number of nonzeros in inequality constraint Jacobian.:        0
Number of nonzeros in Lagrangian Hessian.............:        3

Total number of variables............................:        2
variables with only lower bounds:        0
variables with lower and upper bounds:        0
variables with only upper bounds:        0
Total number of equality constraints.................:        0
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  2.4200000e+01 0.00e+00 1.00e+02  -1.0 0.00e+00    -  0.00e+00 0.00e+00   0
1  4.7318843e+00 0.00e+00 2.15e+00  -1.0 3.81e-01    -  1.00e+00 1.00e+00f  1
2  4.0873987e+00 0.00e+00 1.20e+01  -1.0 4.56e+00    -  1.00e+00 1.25e-01f  4
3  3.2286726e+00 0.00e+00 4.94e+00  -1.0 2.21e-01    -  1.00e+00 1.00e+00f  1
4  3.2138981e+00 0.00e+00 1.02e+01  -1.0 4.82e-01    -  1.00e+00 1.00e+00f  1
5  1.9425854e+00 0.00e+00 1.62e+00  -1.0 6.70e-02    -  1.00e+00 1.00e+00f  1
6  1.6001937e+00 0.00e+00 3.44e+00  -1.0 7.35e-01    -  1.00e+00 2.50e-01f  3
7  1.1783896e+00 0.00e+00 1.92e+00  -1.0 1.44e-01    -  1.00e+00 1.00e+00f  1
8  9.2241158e-01 0.00e+00 4.00e+00  -1.0 2.08e-01    -  1.00e+00 1.00e+00f  1
9  5.9748862e-01 0.00e+00 7.36e-01  -1.0 8.91e-02    -  1.00e+00 1.00e+00f  1
iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
10  4.5262510e-01 0.00e+00 2.42e+00  -1.7 2.97e-01    -  1.00e+00 5.00e-01f  2
11  2.8076244e-01 0.00e+00 9.25e-01  -1.7 1.02e-01    -  1.00e+00 1.00e+00f  1
12  2.1139340e-01 0.00e+00 3.34e+00  -1.7 1.77e-01    -  1.00e+00 1.00e+00f  1
13  8.9019501e-02 0.00e+00 2.25e-01  -1.7 9.45e-02    -  1.00e+00 1.00e+00f  1
14  5.1535405e-02 0.00e+00 1.49e+00  -1.7 2.84e-01    -  1.00e+00 5.00e-01f  2
15  1.9992778e-02 0.00e+00 4.64e-01  -1.7 1.09e-01    -  1.00e+00 1.00e+00f  1
16  7.1692436e-03 0.00e+00 1.03e+00  -1.7 1.39e-01    -  1.00e+00 1.00e+00f  1
17  1.0696137e-03 0.00e+00 9.09e-02  -1.7 5.50e-02    -  1.00e+00 1.00e+00f  1
18  7.7768464e-05 0.00e+00 1.44e-01  -2.5 5.53e-02    -  1.00e+00 1.00e+00f  1
19  2.8246695e-07 0.00e+00 1.50e-03  -2.5 7.31e-03    -  1.00e+00 1.00e+00f  1
iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
20  8.5170750e-12 0.00e+00 4.90e-05  -5.7 1.05e-03    -  1.00e+00 1.00e+00f  1
21  3.7439756e-21 0.00e+00 1.73e-10  -5.7 2.49e-06    -  1.00e+00 1.00e+00f  1

Number of Iterations....: 21

(scaled)                 (unscaled)
Objective...............:   1.7365378678754519e-21    3.7439756431394737e-21
Dual infeasibility......:   1.7312156654298279e-10    3.7325009746667082e-10
Constraint violation....:   0.0000000000000000e+00    0.0000000000000000e+00
Complementarity.........:   0.0000000000000000e+00    0.0000000000000000e+00
Overall NLP error.......:   1.7312156654298279e-10    3.7325009746667082e-10

Number of objective function evaluations             = 45
Number of objective gradient evaluations             = 22
Number of equality constraint evaluations            = 0
Number of inequality constraint evaluations          = 0
Number of equality constraint Jacobian evaluations   = 0
Number of inequality constraint Jacobian evaluations = 0
Number of Lagrangian Hessian evaluations             = 21
Total CPU secs in IPOPT (w/o function evaluations)   =      2.871
Total CPU secs in NLP function evaluations           =      1.561

EXIT: Optimal Solution Found.

Another example, using a constrained problem

n = 10
x0 = ones(n)
x0[1:2:end] .= -1.2
nlp = ADNLPModel(x -> sum((x[i] - 1)^2 + 100 * (x[i+1] - x[i]^2)^2 for i = 1:n-1), x0,
c=x -> [3 * x[k+1]^3 + 2 * x[k+2] - 5 + sin(x[k+1] - x[k+2]) * sin(x[k+1] + x[k+2]) +
4 * x[k+1] - x[k] * exp(x[k] - x[k+1]) - 3 for k = 1:n-2],
lcon=zeros(n-2), ucon=zeros(n-2))
stats = ipopt(nlp, print_level=0)
print(stats)
Generic Execution stats
status: "first-order stationary"
objective value: 6.232458632437464
primal feasibility: 8.354206215699378e-12
dual feasibility: 6.315907100018699e-9
primal feasibility: 8.354206215699378e-12
solution: [-0.950556  0.913901  0.989091  0.998559 ⋯ 0.9999999300706429]
iterations: 6
elapsed time: 2.917
solver specific:
multipliers_U: [0.0  0.0  0.0  0.0 ⋯ 0.0]
multipliers_L: [0.0  0.0  0.0  0.0 ⋯ 0.0]
multipliers_con: [4.13586  -1.87649  -0.0655633  -0.0219319 ⋯ -7.37659216376762e-6]
internal_msg: :Solve_Succeeded

Output

The output of ipopt is a GenericExecutionStats from SolverTools. It contains basic information from the solver. In addition to the built-in fields of GenericExecutionStats, we also store in solver_specific the following fields:

• multipliers_con: Constraints multipliers;
• multipliers_L: Variables lower-bound multipliers;
• multipliers_U: Variables upper-bound multipliers;
• internal_msg: Detailed Ipopt output message.
stats.solver_specific[:internal_msg]
:Solve_Succeeded

Manual input

This is an example where we specify the problem and its derivatives manually. For this, we create an NLPModel, and we need to define the following API functions:

• obj(nlp, x): objective
• grad!(nlp, x, g): gradient
• cons!(nlp, x, c): constraints, if any
• jac_structure!(nlp, rows, cols): structure of the Jacobian, if constrained;
• jac_coord!(nlp, x, rows, cols, vals): Jacobian values (the user should not attempt to access rows and cols, as Ipopt doesn't actually pass them);
• hess_structure!(nlp, rows, cols): structure of the lower triangle of the Hessian of the Lagrangian;
• hess_coord!(nlp, x, rows, cols, vals; obj_weight=1.0, y=[]): Hessian of the Lagrangian, where obj_weight is the weight assigned to the objective, and y is the multipliers vector (the user should not attempt to access rows and cols, as Ipopt doesn't actually pass them).

Let's implement a logistic regression model. We consider the model $h(\beta; x) = (1 + e^{-\beta^Tx})^{-1}$, and the loss function

$\ell(\beta) = -\sum_{i = 1}^m y_i \ln h(\beta; x_i) + (1 - y_i) \ln(1 - h(\beta; x_i))$

with regularization $\lambda \|\beta\|^2 / 2$.

using DataFrames, LinearAlgebra, NLPModels, NLPModelsIpopt, Random

mutable struct LogisticRegression <: AbstractNLPModel
X :: Matrix
y :: Vector
λ :: Real
meta :: NLPModelMeta # required by AbstractNLPModel
counters :: Counters # required by AbstractNLPModel
end

function LogisticRegression(X, y, λ = 0.0)
m, n = size(X)
meta = NLPModelMeta(n, name="LogisticRegression", nnzh=div(n * (n+1), 2) + n) # nnzh is the length of the coordinates vectors
return LogisticRegression(X, y, λ, meta, Counters())
end

function NLPModels.obj(nlp :: LogisticRegression, β::AbstractVector)
hβ = 1 ./ (1 .+ exp.(-nlp.X * β))
return -sum(nlp.y .* log.(hβ .+ 1e-8) .+ (1 .- nlp.y) .* log.(1 .- hβ .+ 1e-8)) + nlp.λ * dot(β, β) / 2
end

function NLPModels.grad!(nlp :: LogisticRegression, β::AbstractVector, g::AbstractVector)
hβ = 1 ./ (1 .+ exp.(-nlp.X * β))
g .= nlp.X' * (hβ .- nlp.y) + nlp.λ * β
end

function NLPModels.hess_structure!(nlp :: LogisticRegression, rows :: AbstractVector{<:Integer}, cols :: AbstractVector{<:Integer})
n = nlp.meta.nvar
I = ((i,j) for i = 1:n, j = 1:n if i ≥ j)
rows[1 : nlp.meta.nnzh] .= [getindex.(I, 1); 1:n]
cols[1 : nlp.meta.nnzh] .= [getindex.(I, 2); 1:n]
return rows, cols
end

function NLPModels.hess_coord!(nlp :: LogisticRegression, β::AbstractVector, rows::AbstractVector{<: Integer}, cols::AbstractVector{<: Integer}, vals::AbstractVector; obj_weight=1.0, y=Float64[])
n, m = nlp.meta.nvar, length(nlp.y)
hβ = 1 ./ (1 .+ exp.(-nlp.X * β))
fill!(vals, 0.0)
for k = 1:m
hk = hβ[k]
p = 1
for j = 1:n, i = j:n
vals[p] += obj_weight * hk * (1 - hk) * nlp.X[k,i] * nlp.X[k,j]
p += 1
end
end
vals[nlp.meta.nnzh+1:end] .= nlp.λ * obj_weight
return rows, cols, vals
end

Random.seed!(0)

# Training set
m = 1000
df = DataFrame(:age => rand(18:60, m), :salary => rand(40:180, m) * 1000)
df[:buy] = (df.age .> 40 .+ randn(m) * 5) .| (df.salary .> 120_000 .+ randn(m) * 10_000)

X = [ones(m) df.age df.age.^2 df.salary df.salary.^2 df.age .* df.salary]

λ = 1.0e-2
nlp = LogisticRegression(X, y, λ)
stats = ipopt(nlp, print_level=0)
β = stats.solution

# Test set - same generation method
m = 100
df = DataFrame(:age => rand(18:60, m), :salary => rand(40:180, m) * 1000)
df[:buy] = (df.age .> 40 .+ randn(m) * 5) .| (df.salary .> 120_000 .+ randn(m) * 10_000)

X = [ones(m) df.age df.age.^2 df.salary df.salary.^2 df.age .* df.salary]
hβ = 1 ./ (1 .+ exp.(-X * β))
ypred = hβ .> 0.5

acc = count(df.buy .== ypred) / m
println("acc = \$acc")
┌ Warning: setindex!(df::DataFrame, v::AbstractVector, col_ind::ColumnIndex) is deprecated, use begin
│     df[!, col_ind] = v
│     df
│   caller = top-level scope at none:0
└ @ Core none:0
┌ Warning: setindex!(df::DataFrame, v::AbstractVector, col_ind::ColumnIndex) is deprecated, use begin
│     df[!, col_ind] = v
│     df
acc = 0.93
using Plots
contour!(range(18, 60, step=0.1), range(40_000, 180_000, step=1.0), f, levels=[0.5])