Bridge from complement to nonlinear

[blegat]

We discussed this with @frapac this week. Given 0 <= f ⟂ g >= 0, there are two possibilities: add a constraint that the product is zero or add one that the product is nonpositive. According to @frapac , the latter is more appropriate if the solver is interior point. So it's unclear which one should be added by default but we can start by creating a bridge (probably parametrized by a Bool indicating which approach is employed).

[odow]

Note that we actually model F(x) ⟂ l <= x <= u.

See

• https://www.gams.com/latest/docs/S_NLPEC.html#NLPEC_PRODUCT_REFORMULATIONS
• https://www.gams.com/latest/docs/S_NLPEC.html#NLPEC_SLACKS_DOUBLY_BOUNDED_VARIABLES

Rather than the w' (x - l) <= 0 type constraint, we should add a disaggregated [i in 1:n], w[i] * (x[i] - l[i]) <= 0

Something like this:

F(x) perp l <= x <= u

y == F(x)
if isfinite(l) && isfinite(u)
    (x - l) * y <= 0
    (x - u) * y <= 0
elseif isfinite(l)
    (x - l) * y <= 0
    y >= 0
elseif isfinite(u)
    (x - u) * y <= 0
    y <= 0
end

[odow]

Conclusion from developer call is that this should be a MOI extension a la MultiObjectiveAlgorithms.jl