Optimizing small systems of NonLinear equations with Optim.jl

Specific Domains Optimization (Mathematical)
1

Hello there I have a very small system of nonlinear equation and i do not quite understand how I can use this using Optim.jl. So far I have only figured out how to do one equation but I have a system (3 Equations):
The system is defined by:

function fb(x)
    dco, dm, dcr = x*1e3
    
    rco = dco
    rm = dco + dm
    rcr = dco + dm + dcr

    ρco = 13e3 #kg/m^3
    ρm = 4.5e3
    ρcr = 2.9e3
    
    R_planet = 6062e3 #m
    M_planet = 4.88e24 #kg
    C = 0.33*M_planet*R_planet^2

    radius = dco + dcr + dm - R_planet
    mass = 4/3*pi *( dco^3 * ρco + ((dco + dm)^3 - dm^3)*ρm + ρcr*((dco + dm + dcr)^3 - (dco + dm)^3)) - M_planet
    MoI = 8/15*π*(ρco * rco^5 + ρm * (rm^5 - rco^5) + ρcr*(rcr^5 - rm^5)) - C
return [radius, mass, MoI]
end

How can I solve for possible solution vectors x with Optim.jl?
Also anyone know how to implement constraints?
Thank you in advance.

2

What do you mean by minimize a system of equations?

If you’re doing constrained nonlinear optimization, you might want to use JuMP instead: Simple examples · JuMP

3

Sorry to be confusing what i mean with minimizing is searching for one or more solution vectors which let my functions vector output be close to 0 so. fb(xsol) ≈ [0,0,0].

Do I have to do this with JuMP as everything I have seen of JuMP seems to be quite more complicated than Optim.jl. Does Optim.jl not allow for “Solving” systems of equations?
I am sorry if any of what I am saying makes not much sense, I am trying to figure this out as I follow a course at my University.

Thanks for the Answer and the link tho.