Design gear train with specific ratio with a set of fixed gears

Specific Domains Optimization (Mathematical)
8

I am giving it a try with some Julia code too:

g = [8, 12, 16, 20, 24, 36, 40]                     # Available gears.
r = []                                              # Initialize an array of gear ratios.
for g1 in g                                         # Fill in the array of gear ratios with values.
    for g2 in g
        if g1 > g2                                  # If nonnegativity constraints on the xᵢ variables are to be imposed below, remove the condition; 
            push!(r, g1/g2)                         #  otherwise only gear ratios > 1 can be achieved.   
        end
    end
end

a = log.(r)
n = length(a)                                       # Number of (types of) gear ratios.

d = 45/28                                           # Required gear ratio.
b = log(d)

using JuMP
using LinearAlgebra
using Juniper
using Ipopt

optimizer = Juniper.Optimizer
nl_solver = optimizer_with_attributes(Ipopt.Optimizer, "print_level" => 0)

optimizer = Juniper.Optimizer
model = Model(optimizer_with_attributes(optimizer, "nl_solver"=>nl_solver))
@variable(model, x[1:n]>=0,Int)                     # If assumed nonnegative, only gear ratios >=1 can be obtained. 
#@variable(model, x[1:n],Int)                       # But currently fails if the nonnegativity constraint is removed. Dunno why.

γ = 0.001                                           # Regularization coefficient (if small, it does not try to minimize the number of gears ratios)
@NLexpression(model,expr,(sum(a[i]*x[i] for i = 1:n)-b)^2 + γ*sum(abs(x[i]) for i=1:n))
@NLobjective(model, Min, expr)
print(model)

optimize!(model)

It finally (originally I reported troubles here) seems doing what I expected:

julia> println(JuMP.value.(x))
[0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]

julia> println(JuMP.objective_value(model))
0.0020198412370350254

julia> println(JuMP.termination_status(model))

Let’s compare the achieved gear ratio with the demanded one:

julia> r[3]*r[10]
1.5999999999999999

julia> d
1.6071428571428572

Sort of…

Is this satisfactory, @pepijndevos ?