Constrained Parameters on Nonlinear Least Squares fitting

[ChrisRackauckas]

Discussed in https://github.com/orgs/SciML/discussions/21

^{Originally posted by andrewkhardy March 10, 2025}
I am attempting to fit a correlation function. I found a great example of using NonLinearLeastSquaresProblem, which I'll repeat here

using NonlinearSolve, Plots
function curve_fit(model, u0, xdata, ydata, p)
    data = (xdata, ydata, p)

    function lossfn!(du, u, data)
        (xs, ys, p) = data   
        du .= model.(xs, Ref(u), Ref(p)) .- ys
        return nothing
    end

    prob = NonlinearLeastSquaresProblem(
        NonlinearFunction(lossfn!, resid_prototype = similar(ydata)), u0, data)
    sol = solve(prob)
    u = sol.u
    fit = model.(xdata, Ref(u), Ref(p))
    return (;sol, fit)
end

The problem is that I do not know how to give constraints to the allowed parameters to search over ( or fix one of the parameters if I know if for example).

In LsqFit.jl, this is accomplished by a upper and lower keyword. I'm wondering if there's anything similar?

BTW: The form hopefully doesn't matter, but including it for completion. The issue is that there is finite frequency data I am trying to capture and a least-squares fit erases that. If there's a suggestion for a better way to be searching, then I'd be very appreciative.

$$ \langle n(r) n(0)\rangle = \frac{K_\rho}{(\pi r)^2} + A_1 \frac{\cos \left(2 k_F r\right)}{r^{1+K_\rho}} \ln ^{-3 / 2}(r) + A_2 \cos \left(4 k_F r\right) r^{-4 K_\rho} $$