# The solution deteriorates if the domain of decisions is improperly set when using Ipopt

**URL:** <https://discourse.julialang.org/t/the-solution-deteriorates-if-the-domain-of-decisions-is-improperly-set-when-using-ipopt/126743>\
**Category:** Optimization (Mathematical)\
**Tags:** jump, ipopt, caveat\
**Created:** [March 10, 2025, 4:47am UTC](https://discourse.julialang.org/t/the-solution-deteriorates-if-the-domain-of-decisions-is-improperly-set-when-using-ipopt/126743 "2025-03-10T04:47:28Z")\
**Posts on this page:** 5\
**Page:** 1

<div class="post-metadata">

**Author:** ![WalterMadelim](https://avatars.discourse-cdn.com/v4/letter/w/3e96dc/32.png) [@WalterMadelim](https://discourse.julialang.org/u/WalterMadelim)\
**Post date:** [March 10, 2025, 4:47am UTC](https://discourse.julialang.org/t/the-solution-deteriorates-if-the-domain-of-decisions-is-improperly-set-when-using-ipopt/126743/1 "2025-03-10T04:47:28Z")

</div>

We expound the title through the following Convex Minimization Problem.

```julia
function convex_f(x) return x * log(x) end # ⚠️ This convex function is NOT defined at 0
function f_gradient(x) return log(x) + 1 end

import JuMP
import Ipopt
function optimise(lb_of_x)
    CP = JuMP.Model(() -> Ipopt.Optimizer()) # CP stands for Convex Program
    JuMP.@variable(CP, x >= lb_of_x)
    JuMP.@objective(CP, Min, convex_f(x))
    JuMP.optimize!(CP)
    @assert JuMP.termination_status(CP) == JuMP.LOCALLY_SOLVED
    return JuMP.value(x)
end
function assess(x)
    @info "x = $x, f = $(convex_f(x)), ∇f = $(f_gradient(x))"
end

# ❌ The following solution returned by Ipopt is unexpected, since the domain of `x` is improperly set
x = optimise(0)
assess(x) # [ Info: x = 0.2627101793829332, f = -0.35116570400222863, ∇f = -0.3367038339628263

# ✅ The following solution is the expected optimal solution.
x = optimise(1e-6)
assess(x) # [ Info: x = 0.36787944368298275, f = -0.36787944117144233, ∇f = 6.827074683357637e-9

```

---

<div class="post-metadata">

**Author:** ![WalterMadelim](https://avatars.discourse-cdn.com/v4/letter/w/3e96dc/32.png) [@WalterMadelim](https://discourse.julialang.org/u/WalterMadelim)\
**Post date:** [March 10, 2025, 4:57am UTC](https://discourse.julialang.org/t/the-solution-deteriorates-if-the-domain-of-decisions-is-improperly-set-when-using-ipopt/126743/2 "2025-03-10T04:57:08Z")

</div>

Does this constitute a bug of Ipopt?  
Since when it reports LOCALLY\_SOLVED, I’m expecting that `∇f = 0` at this solution.  
In this case `∇f = -0.3367038339628263`.  
@odow

---

<div class="post-metadata">

**Author:** ![odow](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/odow/32/28685_2.png) [@odow](https://discourse.julialang.org/u/odow)\
**Post date:** [March 10, 2025, 7:00am UTC](https://discourse.julialang.org/t/the-solution-deteriorates-if-the-domain-of-decisions-is-improperly-set-when-using-ipopt/126743/3 "2025-03-10T07:00:44Z")

</div>

This is related to Ipopt’s automatic scaling that it applies.

If you solve the original model, you’ll see that Ipopt claims success, but reports

```julia
Overall NLP error.......: 3.3680175321150538e-09 3.3680175321150535e-01

```

So Ipopt solved the scaled problem to optimality, but the unscaled error is very high! I don’t know if this is classified as a bug. It happens in LP solvers as well.

For this problem, you could turn off scaling:

```julia
julia> using JuMP, Ipopt

julia> function main(; nlp_scaling_method = nothing)
           model = Model(Ipopt.Optimizer)
           if nlp_scaling_method !== nothing
               set_attribute(model, "nlp_scaling_method", nlp_scaling_method)
           end
           @variable(model, x >= 0)
           @objective(model, Min, x * log(x))
           optimize!(model)
           assert_is_solved_and_feasible(model)
           return value(x)
       end
main (generic function with 1 method)

julia> main()

******************************************************************************
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 5.7.3.

Number of nonzeros in equality constraint Jacobian...: 0
Number of nonzeros in inequality constraint Jacobian.: 0
Number of nonzeros in Lagrangian Hessian.............: 1

Total number of variables............................: 1
                     variables with only lower bounds: 1
                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 -4.6051666e-02 0.00e+00 1.00e+00 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0
   1 -2.4280022e-01 0.00e+00 1.00e-02 -1.0 1.00e-01 - 9.90e-01 1.00e+00f 1
   2 -2.5997605e-01 0.00e+00 1.52e-06 -3.8 1.50e-02 - 1.00e+00 1.00e+00f 1
   3 -2.6269769e-01 0.00e+00 1.06e-08 -8.6 2.55e-03 - 9.93e-01 1.00e+00f 1
   4 -3.5116570e-01 0.00e+00 3.37e-09 -12.9 1.35e-01 - 1.00e+00 1.00e+00f 1

Number of Iterations....: 4

                                   (scaled) (unscaled)
Objective...............: -3.5116570400222862e-09 -3.5116570400222863e-01
Dual infeasibility......: 3.3680175321150538e-09 3.3680175321150535e-01
Constraint violation....: 0.0000000000000000e+00 0.0000000000000000e+00
Variable bound violation: 0.0000000000000000e+00 0.0000000000000000e+00
Complementarity.........: 2.5724384364771816e-13 2.5724384364771815e-05
Overall NLP error.......: 3.3680175321150538e-09 3.3680175321150535e-01

Number of objective function evaluations = 5
Number of objective gradient evaluations = 5
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 = 4
Total seconds in IPOPT = 2.535

EXIT: Optimal Solution Found.
0.26271017938293323

julia> main(; nlp_scaling_method = "none")
This is Ipopt version 3.14.17, running with linear solver MUMPS 5.7.3.

Number of nonzeros in equality constraint Jacobian...: 0
Number of nonzeros in inequality constraint Jacobian.: 0
Number of nonzeros in Lagrangian Hessian.............: 1

Total number of variables............................: 1
                     variables with only lower bounds: 1
                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 -4.6051666e-02 0.00e+00 4.61e+00 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0
   1 -1.9902165e-01 0.00e+00 4.75e+00 -1.0 6.80e-02 - 1.00e+00 1.00e+00f 1
   2 -2.6593036e-01 0.00e+00 1.07e+00 -1.0 5.27e-02 - 7.77e-01 1.00e+00f 1
   3 -3.6776246e-01 0.00e+00 7.35e-01 -1.0 2.28e-01 - 1.00e+00 1.00e+00f 1
   4 -3.6779739e-01 0.00e+00 1.09e-03 -1.7 1.70e-02 - 1.00e+00 1.00e+00f 1
   5 -3.6787937e-01 0.00e+00 2.05e-04 -3.8 7.56e-03 - 1.00e+00 1.00e+00f 1
   6 -3.6787944e-01 0.00e+00 2.01e-07 -5.7 2.33e-04 - 1.00e+00 1.00e+00f 1
   7 -3.6787944e-01 0.00e+00 1.43e-11 -8.6 1.96e-06 - 1.00e+00 1.00e+00f 1

Number of Iterations....: 7

                                   (scaled) (unscaled)
Objective...............: -3.6787944117144233e-01 -3.6787944117144233e-01
Dual infeasibility......: 1.4279221332907585e-11 1.4279221332907585e-11
Constraint violation....: 0.0000000000000000e+00 0.0000000000000000e+00
Variable bound violation: 0.0000000000000000e+00 0.0000000000000000e+00
Complementarity.........: 2.5167856134015930e-09 2.5167856134015930e-09
Overall NLP error.......: 2.5167856134015930e-09 2.5167856134015930e-09

Number of objective function evaluations = 8
Number of objective gradient evaluations = 8
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 = 7
Total seconds in IPOPT = 0.002

EXIT: Optimal Solution Found.
0.36787944368297487

```

---

<div class="post-metadata">

**Author:** ![odow](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/odow/32/28685_2.png) [@odow](https://discourse.julialang.org/u/odow)\
**Post date:** [March 10, 2025, 7:56pm UTC](https://discourse.julialang.org/t/the-solution-deteriorates-if-the-domain-of-decisions-is-improperly-set-when-using-ipopt/126743/4 "2025-03-10T19:56:48Z")

</div>

The underlying issue is that the gradient of the objective is not defied at the starting point. I think we can do something to warn uses if this is the case: [Tools to test and debug JuMP models · Issue #3664 · jump-dev/JuMP.jl · GitHub](https://github.com/jump-dev/JuMP.jl/issues/3664)

```Julia
julia> using JuMP, Ipopt

julia> function main(; start)
           model = Model(Ipopt.Optimizer)
           @variable(model, x >= 0, start = start)
           @objective(model, Min, x * log(x))
           optimize!(model)
           assert_is_solved_and_feasible(model)
           return value(x)
       end
main (generic function with 1 method)

julia> main(; start = 0)
This is Ipopt version 3.14.17, running with linear solver MUMPS 5.7.3.

Number of nonzeros in equality constraint Jacobian...: 0
Number of nonzeros in inequality constraint Jacobian.: 0
Number of nonzeros in Lagrangian Hessian.............: 1

Total number of variables............................: 1
                     variables with only lower bounds: 1
                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 -4.6051666e-02 0.00e+00 1.00e+00 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0
   1 -2.4280022e-01 0.00e+00 1.00e-02 -1.0 1.00e-01 - 9.90e-01 1.00e+00f 1
   2 -2.5997605e-01 0.00e+00 1.52e-06 -3.8 1.50e-02 - 1.00e+00 1.00e+00f 1
   3 -2.6269769e-01 0.00e+00 1.06e-08 -8.6 2.55e-03 - 9.93e-01 1.00e+00f 1
   4 -3.5116570e-01 0.00e+00 3.37e-09 -12.9 1.35e-01 - 1.00e+00 1.00e+00f 1

Number of Iterations....: 4

                                   (scaled) (unscaled)
Objective...............: -3.5116570400222862e-09 -3.5116570400222863e-01
Dual infeasibility......: 3.3680175321150538e-09 3.3680175321150535e-01
Constraint violation....: 0.0000000000000000e+00 0.0000000000000000e+00
Variable bound violation: 0.0000000000000000e+00 0.0000000000000000e+00
Complementarity.........: 2.5724384364771816e-13 2.5724384364771815e-05
Overall NLP error.......: 3.3680175321150538e-09 3.3680175321150535e-01

Number of objective function evaluations = 5
Number of objective gradient evaluations = 5
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 = 4
Total seconds in IPOPT = 0.003

EXIT: Optimal Solution Found.
0.26271017938293323

julia> main(; start = 1)
This is Ipopt version 3.14.17, running with linear solver MUMPS 5.7.3.

Number of nonzeros in equality constraint Jacobian...: 0
Number of nonzeros in inequality constraint Jacobian.: 0
Number of nonzeros in Lagrangian Hessian.............: 1

Total number of variables............................: 1
                     variables with only lower bounds: 1
                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 0.0000000e+00 0.00e+00 0.00e+00 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0
   1 -3.4340576e-01 0.00e+00 1.83e-01 -1.7 4.90e-01 - 1.00e+00 1.00e+00f 1
   2 -3.6522860e-01 0.00e+00 2.08e-02 -1.7 9.71e-02 - 1.00e+00 1.00e+00f 1
   3 -3.6783766e-01 0.00e+00 4.88e-03 -2.5 3.95e-02 - 1.00e+00 1.00e+00f 1
   4 -3.6787938e-01 0.00e+00 1.03e-04 -3.8 5.34e-03 - 1.00e+00 1.00e+00f 1
   5 -3.6787944e-01 0.00e+00 1.67e-07 -5.7 2.13e-04 - 1.00e+00 1.00e+00f 1
   6 -3.6787944e-01 0.00e+00 1.37e-11 -8.6 1.93e-06 - 1.00e+00 1.00e+00f 1

Number of Iterations....: 6

                                   (scaled) (unscaled)
Objective...............: -3.6787944117144233e-01 -3.6787944117144233e-01
Dual infeasibility......: 1.3730039409520724e-11 1.3730039409520724e-11
Constraint violation....: 0.0000000000000000e+00 0.0000000000000000e+00
Variable bound violation: 0.0000000000000000e+00 0.0000000000000000e+00
Complementarity.........: 2.5163095253048878e-09 2.5163095253048878e-09
Overall NLP error.......: 2.5163095253048878e-09 2.5163095253048878e-09

Number of objective function evaluations = 7
Number of objective gradient evaluations = 7
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 = 6
Total seconds in IPOPT = 0.003

EXIT: Optimal Solution Found.
0.3678794436827008

```

---

<div class="post-metadata">

**Author:** ![WalterMadelim](https://avatars.discourse-cdn.com/v4/letter/w/3e96dc/32.png) [@WalterMadelim](https://discourse.julialang.org/u/WalterMadelim)\
**Post date:** [March 15, 2025, 5:36am UTC](https://discourse.julialang.org/t/the-solution-deteriorates-if-the-domain-of-decisions-is-improperly-set-when-using-ipopt/126743/5 "2025-03-15T05:36:06Z")

</div>

We can also employ a global optimizer to do the same task, the correct code should be like this

```julia
import JuMP, MosekTools
CP = JuMP.Model(() -> MosekTools.Optimizer())
JuMP.@variable(CP, t)
JuMP.@variable(CP, x >= 1e-6)
JuMP.@constraint(CP, [-t, x, 1] in JuMP.MOI.ExponentialCone())
JuMP.@objective(CP, Min, t) # minimize `x -> x * log(x)` over positive reals
JuMP.optimize!(CP)
JuMP.assert_is_solved_and_feasible(CP; allow_local = false)

```

A direct extension of this is to solve a (nonconvex) problem that has a “linear times convex” objective. e.g. this nonconvex problem (NC):  
\min\_{x\_1 \> 0, x\_2} \{x\_1 x\_2 \log x\_1 \; | \; 0 \le x\_2 \le 2 \}   
The code example of a convex relaxation (CR) that can solve (NC) to global optimality is as follows:

```julia
import JuMP, MosekTools
CR = JuMP.Model(() -> MosekTools.Optimizer()); JuMP.set_silent(CR);
JuMP.@variable(CR, t);
JuMP.@variable(CR, x[1:2]);
JuMP.set_lower_bound(x[1], 1e-6);
JuMP.set_lower_bound(x[2], 0); JuMP.set_upper_bound(x[2], 2);
JuMP.@variable(CR, X[eachindex(x), eachindex(x)], Symmetric);
# JuMP.set_lower_bound(X[1, 2], 0)
JuMP.@constraint(CR, [-t, X[1, 2], x[2]] in JuMP.MOI.ExponentialCone());
JuMP.@constraint(CR, X[1, 2] <= 2 * x[1]); # VI
JuMP.@objective(CR, Min, t);
JuMP.optimize!(CR)
JuMP.assert_is_solved_and_feasible(CR; allow_local = false)
x = JuMP.value.(x) # [0.3678829664527772, 2.0]
ub = x[2] * x[1] * log(x[1]) # -0.735758882309103
lb = JuMP.objective_bound(CR) # -0.735758881911808

```

In this example, we again discover the fact that:  
although the lb might become tighter after adding the SDP cut, the primal feasible solution **could nonetheless deteriorate**. Therefore it is surely advantageous to collect primal candidate solutions **along the way**.

A related caveat about the function `x -> x * log(x)` is [here](https://discourse.julialang.org/t/reformulation-perspectification-technique-a-code-example-using-jump/126729/3).