# How to model an implication

**URL:** <https://discourse.julialang.org/t/how-to-model-an-implication/132617>\
**Category:** Optimization (Mathematical)\
**Created:** [September 24, 2025, 10:09am UTC](https://discourse.julialang.org/t/how-to-model-an-implication/132617 "2025-09-24T10:09:36Z")\
**Posts on this page:** 5\
**Page:** 2

<div class="post-metadata">

**Author:** ![rocco\_sprmnt21](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/rocco_sprmnt21/32/20127_2.png) [@rocco\_sprmnt21](https://discourse.julialang.org/u/rocco_sprmnt21)\
**Post date:** [September 25, 2025, 2:12pm UTC](https://discourse.julialang.org/t/how-to-model-an-implication/132617/21 "2025-09-25T14:12:55Z")

</div>

This very simplified form also gives similar results.

```julia-auto
julia> main() do n, N, model, x
                   @variable(model, z, Int)
                   @constraint(model, z == sum(x[1, :]) - N)   
                   @constraint(model, !z --> {sum(x[2, :]) == 2})
               end
Test Summary: | Pass Total Time
main | 4070 4070 0.8s

```

Is it just a coincidence? Or are the formulations(\*) actually equivalent?

(\*)I’m having trouble following the logic of the second script.  
For example, regarding the statement  
@constraint(model, z[3] == z[1] + z[2])

I understand that if z[3]==0 then z[1]==z[2]==0.

I don’t understand how it works in different cases.  
For example, if it makes sense, since z is Bin, what happens in the case z[1]==z[2]==1?

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<div class="post-metadata">

**Author:** ![rocco\_sprmnt21](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/rocco_sprmnt21/32/20127_2.png) [@rocco\_sprmnt21](https://discourse.julialang.org/u/rocco_sprmnt21)\
**Post date:** [September 25, 2025, 3:07pm UTC](https://discourse.julialang.org/t/how-to-model-an-implication/132617/22 "2025-09-25T15:07:29Z")

</div>

| | | | | |
| --- | --- | --- | --- | --- |
| Clarabel.jl | | | Apache | LP, QP, SOCP, SDP |
| COSMO.jl | | | Apache | LP, QP, SOCP, SDP |
| DAQP | DAQP.jl | | MIT | (Mixed-binary) QP |
| HiGHS | HiGHS.jl | | MIT | (MI)LP, QP |
| Ipopt | Ipopt.jl | | EPL | LP, QP, NLP |
| MadNLP.jl | | | MIT | LP, QP, NLP |
| NLopt | NLopt.jl | | GPL | LP, QP, NLP |
| OSQP | OSQP.jl | | Apache | LP, QP |
| SCS | SCS.jl | | MIT | LP, QP, SOCP, SDP |
| StatusSwitchingQP.jl | | | MIT | LP, QP |

These are the ones that deal with quadratic problems.  
Which one could I use for my model, with a quadratic objective function?

---

<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:** [September 27, 2025, 12:55am UTC](https://discourse.julialang.org/t/how-to-model-an-implication/132617/23 "2025-09-27T00:55:51Z")

</div>

> Is it just a coincidence?

I need to take a better look at this. I would have expected an error. The `z` variable in an indicator must be binary.

> For example, if it makes sense, since z is Bin, what happens in the case z[1]==z[2]==1?

This can’t happen. The problem would be infeasible. In other words, you can’t have both `s[1]` and `s[2]` be non-zero at the same time.

> Which one could I use for my model, with a quadratic objective function?

If you want a quadratic objective, you should choose SCIP or Gurobi (or maybe DAQP?).

Alternatively, if all the variables in the quadratic, you can linearise the terms as follows:

```julia-auto
model = Model()
@variable(model, x[1:2], Bin)
# @objective(model, Max, x[1] * x[2]) ?
@variable(model, z, Bin)
@constraints(model, begin
    z <= x[1]
    z <= x[2]
    z >= x[1] + x[2] - 1
end)
@objective(model, Max, z)

```

---

<div class="post-metadata">

**Author:** ![rocco\_sprmnt21](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/rocco_sprmnt21/32/20127_2.png) [@rocco\_sprmnt21](https://discourse.julialang.org/u/rocco_sprmnt21)\
**Post date:** [September 27, 2025, 2:48pm UTC](https://discourse.julialang.org/t/how-to-model-an-implication/132617/24 "2025-09-27T14:48:03Z")

</div>

If this helps your analysis, consider the following fact (\*).

```julia-auto

model = Model(HiGHS.Optimizer)
# set_silent(model)
@variable(model, x[1:nr, 1:ng, turni], Bin)
@variable(model,0<=s[1:nr]<=nr, Int)
@constraints(model, begin

    # each day must have exactly one Ti or one Fi
    fer1[k in union(t_feriali,t_card_urg), g in g_feriali], sum(x[:,g,k]) == 1
    fer2[k in t_festivi, g in g_feriali], sum(x[:,g,k]) == 0

    fest1[k in t_festivi, g in g_festivi], sum(x[:,g,k]) == 1
    fest2[k in union(t_feriali, t_card_urg), g in g_festivi], sum(x[:,g,k]) == 0
    
    # range of N and NF for row for month
    [r in ops_lim], sum(x[r,j,"N"]+x[r,j,"NF"] for j in 1:ng) <= 4
    [r in ops_lim], sum(x[r,j,"N"]+x[r,j,"NF"] for j in 1:ng) >= 3
    
    [r in ops_lim], sum(x[r,j,"N"] for j in 1:ng) <= 3
    [r in ops_lim], sum(x[r,j,"N"] for j in 1:ng) >= 1
   
    # # NF==2 ==> N<=1
 
      [r in ops_lim], s[r] == sum(x[r,:,"NF"])-2
      [r in ops_lim], !s[r] --> {sum(x[r,:,"N"]) <= 1}
   ...
# many other constraints
end

...
solution_summary(; result = 1, verbose = false)
├ solver_name : HiGHS
├ Termination
│ ├ termination_status : INFEASIBLE
│ ├ result_count : 1
│ ├ raw_status : kHighsModelStatusInfeasible
│ └ objective_bound : 0.00000e+00
├ Solution (result = 1)
│ ├ primal_status : NO_SOLUTION
│ ├ dual_status : INFEASIBILITY_CERTIFICATE
│ ├ objective_value : 0.00000e+00
│ ├ dual_objective_value : 6.00000e+00
│ └ relative_gap : Inf

```

```julia-auto
model = Model(HiGHS.Optimizer)
# set_silent(model)
@variable(model, x[1:nr, 1:ng, turni], Bin)
@variable(model, z[1:nr, 1:ng], Bin)
@constraints(model, begin

    # each day must have exactly one Ti or one Fi
    fer1[k in union(t_feriali,t_card_urg), g in g_feriali], sum(x[:,g,k]) == 1
    fer2[k in t_festivi, g in g_feriali], sum(x[:,g,k]) == 0

    fest1[k in t_festivi, g in g_festivi], sum(x[:,g,k]) == 1
    fest2[k in union(t_feriali, t_card_urg), g in g_festivi], sum(x[:,g,k]) == 0
  
    
     # range of N and NF for row for month
    [r in ops_lim], sum(x[r,j,"N"]+x[r,j,"NF"] for j in 1:ng) <= 4
    [r in ops_lim], sum(x[r,j,"N"]+x[r,j,"NF"] for j in 1:ng) >= 3
    
    [r in ops_lim], sum(x[r,j,"N"] for j in 1:ng) <= 3
    [r in ops_lim], sum(x[r,j,"N"] for j in 1:ng) >= 1
   
    # # NF==2 ==> N<=1
      [r in ops_lim], sum(x[r,:,"NF"]) == sum(i * z[r, i] for i in 1:ng)
      [r in ops_lim], z[r,2] --> {sum(x[r, :, "N"]) <= 1}
      [r in ops_lim], sum(z[r,:]) <= 1

...

Solving report
  Status Optimal
  Primal bound 0
  Dual bound 0
  Gap 0% (tolerance: 0.01%)
  P-D integral 0
  Solution status feasible        
                    0 (objective)   
                    0 (bound viol.) 
                    0 (int. viol.)  
                    0 (row viol.)   

```

(\*)  
I tried to apply one of your suggestions to my case, but I’m not entirely sure if I did it right

PS

The following form also passes the parser checks and provides results consistent with the intended purpose of the constraint

```julia-auto
    # # NF==2 ==> N<=1 for r in ops_lim
 
    s[ops_lim] .== sum(x[ops_lim,:,"NF"])-2

    [r in ops_lim], !s[r] --> {sum(x[r,:,"N"]) <= 1}

```

The following form, however, gives an unfeasible result

```julia-auto
# NF==2 ==> N<=1 for r in ops_lim
 
   [r in ops_lim], s[r] == sum(x[r,:,"NF"])-2

    [r in ops_lim], !s[r] --> {sum(x[r,:,"N"]) <= 1}

```

---

<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:** [September 28, 2025, 9:17pm UTC](https://discourse.julialang.org/t/how-to-model-an-implication/132617/25 "2025-09-28T21:17:31Z")

</div>

Indicator constraints are not valid when the indicator variable is not binary. We should be throwing an error here: [[Bridges] no error thrown when IndicatorToMILPBridge does not imply first variable is binary · Issue #2856 · jump-dev/MathOptInterface.jl · GitHub](https://github.com/jump-dev/MathOptInterface.jl/issues/2856)

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