I have been trying to find a way to fit two functions simultaneously using nonlinear least squares (I have to find the optimum 3 variables, common for both models, that fits best both of them). I typically use Python's scipy.optimize.least_squares module for NLLS work, which uses the Levenberg–Marquardt algorithm.

I tried some specialised multi-objective optimization packages (like pymoo), but they don't seem suitable for my problem as they rely on evolutionary algorithms that output a set of solutions (I only need one optimum solution per variable) and they are made to work for conflicting objectives.

I also tried to take the sum of the norms of the residuals of the two functions (making it into a single objective problem) and to minimize that by various gradient and non-gradient based algorithms from Python's scipy.minimize package, but I found this norm becomes so huge (even with parameter bounds!) that I get oveflow error (34, results too large), crashing the programme sooner or later. It didn't crash using Truncated Newton's Method, but the results produced were rubbish (and from running an optimization on this same data on a simpler model, I know they shouldn't be!)

I have to perform this fit for a few thousand data sets per experiment, so it has to be quite robust.

Surprisingly, I can not find a way to do multiobjective NLLS (only for linear regression). I have found some papers on this, but I'm not a mathematician so it's quite out of my depth to understand them and apply them in Python..

Has anyone had a similar problem to solve?

Many thanks!