# Does BilevelJuMP linearize terms (if possible) after reformulation?

**URL:** https://discourse.julialang.org/t/does-bileveljump-linearize-terms-if-possible-after-reformulation/80981
**Category:** Optimization (Mathematical)
**Tags:** question, package
**Created:** [May 12, 2022, 11:03pm UTC](https://discourse.julialang.org/t/does-bileveljump-linearize-terms-if-possible-after-reformulation/80981 "2022-05-12T23:03:43Z")
**Posts on this page:** 3
**Page:** 1

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### Author: ![math\_opt](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/math_opt/32/25317_2.png) [@math\_opt](https://discourse.julialang.org/u/math_opt)
#### Post date: [May 12, 2022, 11:03pm UTC](https://discourse.julialang.org/t/does-bileveljump-linearize-terms-if-possible-after-reformulation/80981/1 "2022-05-12T23:03:43Z")

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How does `BilevelJuMP.jl` deal with non-linear terms in the complementarity conditions arising from the product of dual variables and the upper-level variables, especially when the upper-level variables are binary?  
For example, suppose we have a model as follows:

```julia
using JuMP, BilevelJuMP, Gurobi

model = BilevelModel(Gurobi.Optimizer, mode = BilevelJuMP.SOS1Mode())

@variable(Lower(model), x)
@variable(Upper(model), y, Bin)

@objective(Upper(model), Min, 3x + y)
@constraints(Upper(model), begin
    x <= 5
    y <= 8
    y >= 0
end)

@objective(Lower(model), Min, -x)
@constraints(Lower(model), begin
     x + y <= 8
    4x + y >= 8
    2x + y <= 13
    2x - 7y <= 0
end)

optimize!(model)

```

Now, in theory, if we write the KKT of the lower level problem, we will have terms where the Lagrange multipliers are multiplied to the binary variable `y` , leading to non-linearity. However, since this is a product of a binary variable and a continuous variable, in practice, we can linearize such terms, eventually leading to a single-level LP problem. Does `BilevelJuMP.jl` do it automatically, or does it treat it as a non-linear problem? Additionally, if `BilevelJuMP.jl` does not eliminate these bilinear terms, does a solver like Gurobi detect it and linearize it?

Thanks.

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<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: [May 12, 2022, 11:10pm UTC](https://discourse.julialang.org/t/does-bileveljump-linearize-terms-if-possible-after-reformulation/80981/2 "2022-05-12T23:10:30Z")

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> [@math\_opt](#):
>
> if `BilevelJuMP.jl` does not eliminate these bilinear terms, does a solver like Gurobi detect it and linearize it?

This is what happens. Gurobi automatically linearizes `x * y` if `x` is binary.

(It can even solve if `x` is not binary, but that’s a different problem.)

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<div class="post-metadata">

### Author: ![math\_opt](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/math_opt/32/25317_2.png) [@math\_opt](https://discourse.julialang.org/u/math_opt)
#### Post date: [May 12, 2022, 11:20pm UTC](https://discourse.julialang.org/t/does-bileveljump-linearize-terms-if-possible-after-reformulation/80981/3 "2022-05-12T23:20:20Z")

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Thanks @odow. I had another question: during the KKT reformulation of the lower-level problem, `BilevelJuMP` treats `y` as a parameter (since `y` is an upper-level variable)? Is that right?