How to implement a piece-wise defined function in JuMP

Specific Domains Optimization (Mathematical)
4

Option 1:

using JuMP, Ipopt
α_0 = 1.0
Lmin = -1.0
Lnom = 0.0
Lmax = 1.0
a = 0.0019727939
b = 0.0078887

model = Model(Ipopt.Optimizer)
@variable(model, Lmin <= x[1:2] <= Lmax)
@constraint(model, sum(x) == 1)
@NLobjective(
    model, 
    Min,
    ifelse(
        x[1] < Lnom,
        a * (x[1]-Lnom)^2 + α_0,
        b * (x[1]-Lnom)^2 + α_0,
    ),
)
optimize!(model)

Option 2:

using JuMP, Ipopt
Lmin = -1.0
Lmax = 1.0
function α(x...)
    α_0 = 1.0
    Lnom = 0.0
    a = 0.0019727939
    b = 0.0078887
    if Lmin <= x[1] < Lnom
        return a * (x[1]-Lnom)^2 + α_0
    else
        return (b * (x[1]-Lnom)^2 + α_0)
    end
end        
model = Model(Ipopt.Optimizer)
@variable(model, Lmin <= x[1:2] <= Lmax)
@constraint(model, sum(x) == 1)
register(model, :α, 2, α; autodiff = true)
@NLobjective(model, Min, α(x...))
optimize!(model)
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