# Complementarity.jl with Path

**URL:** <https://discourse.julialang.org/t/complementarity-jl-with-path/103081>\
**Category:** Optimization (Mathematical)\
**Tags:** jump, error, complementarity\
**Created:** [August 22, 2023, 4:22pm UTC](https://discourse.julialang.org/t/complementarity-jl-with-path/103081 "2023-08-22T16:22:53Z")\
**Posts on this page:** 1\
**Showing post:** 5

<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:** [August 23, 2023, 2:56am UTC](https://discourse.julialang.org/t/complementarity-jl-with-path/103081/5 "2023-08-23T02:56:12Z")

</div>

I took another look at this problem, since I thought it’d be a useful test case for [https://github.com/jump-dev/JuMP.jl/pull/3106](https://github.com/jump-dev/JuMP.jl/pull/3106) and some of the complementarity tutorials I’m writing.

The code below won’t run by default, you’ll need to convert it to the Complementarity.jl syntax, or live dangerously and install the development branch via `] add JuMP#od/nlp-expr`.

The example in the textbook is a little tricky to formulate explicitly as an MCP, so I see how you got stuck. This is what I came up with. It finds the same solution (rounded to the nearest MW):

```julia
A = [300, 350, 400, 450, 500]
B = 1                        
I = 90_000 # Given in the text in terms of $/kWh!
C = 60                       
τ = 8_760                    
θ = [0.2, 0.2, 0.2, 0.2, 0.2]
model = Model(PATHSolver.Optimizer)
@variables(model, begin
    x >= 0, (start = 1)
    Q[1:5] >= 0, (start = 1)
    Y[1:5] >= 0, (start = 1)
    P[1:5], (start = 1)
    μ[1:5] >= 0, (start = 1)
end)
@constraints(model, begin
    I - τ * θ' * μ ⟂ x
    [i=1:5], C - P[i] + μ[i] ⟂ Y[i]
    [i=1:5], A[i] - B * Q[i] - P[i] ⟂ Q[i]
    [i=1:5], Y[i] - Q[i] ⟂ P[i]
    [i=1:5], x - Y[i] ⟂ μ[i]
end)
optimize!(model)

julia> value(x)
389.31506849315065

julia> hcat(
           value.(Q),
           value.(P),
           (τ * value.(μ) .- I) / 1_000,
       )'
3×5 adjoint(::Matrix{Float64}) with eltype Float64:
 240.0 290.0 340.0 389.315 389.315
  60.0 60.0 60.0 60.6849 110.685
 -90.0 -90.0 -90.0 -84.0 354.0

```

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