Writing loops in optimization problems in Julia

Specific Domains Optimization (Mathematical)
5

Perhaps something like this? (I didn’t test, so there might be typos, etc).

using JuMP
import Mosek
import MosekTools

function inner_loop(x, y)
    NumVar = 5
    a = [0,2.5,-2.5,2,-2.5]
    b = [0,0.3,0.25,-2.5,-2.25]
    r = [4, 8, 9, 8, 8]
    R = 0.0
    for i in 1:length(a)
        R = max(R, sqrt((a[i] - x)^2 + (b[i] - y)^2) + r[i])
    end
    model = Model(Mosek.Optimizer)
    @variable(model, x >= 0)
    @variable(model, y >= 0)
    @variable(model, z >= 0)
    @objective(model, Min, x^2 + y^2 - z)
    @constraint(
        model, 
        [i in 1:NumVar],
        -2 * x * a[i] - 2 * b[i] * y + z <= (R - r[i])^2 - a[i]^2 - b[i]^2,
    )
    optimize!(model)
    return value(x), value(y), value(z), R
end

function main()
    x, y, z, R = 0.0, 0.0, Inf, 0.0
    while abs(x^2 + y^2 - z) >= 0.01
        x, y, z, R = inner_loop(x, y)
    end
    return x, y, z, R
end