Unclear complementarity in example

[blegat]

The P variable has no bound

JuMP.jl/docs/src/tutorials/nonlinear/complementarity.jl

Line 186 in afa5214

@variable(model, P[ω = 1:5], start = 1) # Electricity price

so what does the complementarity means ?

JuMP.jl/docs/src/tutorials/nonlinear/complementarity.jl

Lines 194 to 195 in afa5214

	## Production is equal to consumption
	@constraint(model, [ω = 1:5], Y[ω] - Q[ω] ⟂ P[ω])

According to @frapac , in AMPL, since it has no bounds, it will just add equalities to the lhs

[odow]

The relevant lines are:

@variable(model, P[ω = 1:5], start = 1) 
@constraint(model, [ω = 1:5], Y[ω] - Q[ω] ⟂ P[ω])

Complements is defined as:

• If 𝑙𝑖<𝑥𝑖<𝑢𝑖, then 𝐹𝑖(𝑥)=0
• If 𝑙𝑖=𝑥𝑖, then 𝐹𝑖(𝑥)≥0
• If 𝑥𝑖=𝑢𝑖, then 𝐹𝑖(𝑥)≤0

Since li = -Inf and ui = Inf, neither can apply. Therefore the only situation is 𝐹𝑖(𝑥)=0.

The constraint is equivalent to:

@constraint(model, [ω = 1:5], Y[ω] - Q[ω] == 0)

[WalterMadelim]

Complements is defined as:

I'm curious about that 3-line definition and wonder where does it come from?
I searched some web and could not find it.

I think the cause should be always "loose" (strict <) and the effect should always be "tight" (==).
Okay I know its meaning.

[WalterMadelim]

@blegat Electricity price is a dual variable (Lagrangian multiplier), whose range should be implicitly decided by the primal objective function (and constraints) if you impose optimality. The P may not have a a priori specified bound, but it's range might be implicitly confined if your primal formulation has finite optimality.
@odow In your definition, the x_i should be a primal decision variable (i.e. a physical one), whereas the F(x) should be a dual variable. I don't think you associate them correctly (I see you are associating P with x, which was probably not correct, I think).

[odow]

https://en.m.wikipedia.org/wiki/Mixed_complementarity_problem

I’m going to close this issue because there’s nothing to be done here.