Proposal: Adding Differentiable Multi-Objective Optimization (MOO) Test Functions

[Souza-DR]

Hello,

I'm exploring the possibility of contributing a collection of differentiable multi-objective optimization (MOO) test functions to the OptimizationProblems.jl repository. I have personally implemented these functions and their gradients in Julia.

Motivation

My main motivation is to use these problems as a standardized testbed for experiments with multi-objective algorithms I'm currently developing. This suite of functions has already been used in several published works, such as:

1. Lucambio Pérez, L. R., & Prudente, L. F. (2018). Nonlinear conjugate gradient methods for vector optimization. SIAM Journal on Optimization, 28(3), 2690-2720.

2. Gonçalves, M. L. N., & Prudente, L. F. (2020). On the extension of the Hager–Zhang conjugate gradient method for vector optimization. Computational Optimization and Applications, 76(3), 889-916.

3. Assunção, P. B., Ferreira, O. P., & Prudente, L. F. (2021). Conditional gradient method for multiobjective optimization. Computational Optimization and Applications, 78(3), 741-768.

4. Gonçalves, M. L. N., Lima, F. S., & Prudente, L. F. (2022). A study of Liu-Storey conjugate gradient methods for vector optimization. Applied Mathematics and Computation, 425, 127099.

5. Gonçalves, M. L. N., Lima, F. S., & Prudente, L. F. (2022). Globally convergent Newton-type methods for multiobjective optimization. Computational Optimization and Applications, 83(2), 403-434.

6. Bello-Cruz, Y., Melo, J. G., Prudente, L. F., & Serra, R. V. G. (2024). A Proximal Gradient Method with an Explicit Line search for Multiobjective Optimization. arXiv preprint arXiv:2404.10993.

7. Lapucci, M., & Mansueto, P. (2023). A limited memory Quasi-Newton approach for multi-objective optimization. Computational Optimization and Applications, 85(1), 33-73.

8. Yang, Y. X., Deng, X., & Tang, L. P. (2025). Global Convergence of a Modified BFGS-Type Method Based on Function Information for Nonconvex Multiobjective Optimization Problems. Journal of the Operations Research Society of China.

9. Chen, W., Tang, L., & Yang, X. (2025). Improvements to steepest descent method for multi-objective optimization. Numerical Algorithms.

10. He, Q. R., Li, S. J., Zhang, B. Y., et al. (2024). A family of conjugate gradient methods with guaranteed positiveness and descent for vector optimization. Computational Optimization and Applications, 89(3), 805-842.

11. Prudente, L. F., & Souza, D. R. (2022). A Quasi-Newton Method with Wolfe Line Searches for Multiobjective Optimization. Journal of Optimization Theory and Applications, 194(3), 1107-1140.

12. Prudente, L. F., & Souza, D. R. (2024). Global convergence of a BFGS-type algorithm for nonconvex multiobjective optimization problems. Computational Optimization and Applications, 88(3), 719-757.

I believe these functions could be useful for researchers working on multi-objective algorithms in Julia and would complement the existing scalar test problems available in the repository.

Open Questions

Before proceeding with a more detailed proposal or implementation, I'd like to ask:

1. Would a contribution like this align with the goals of the repository?
2. Would it be acceptable to introduce a new category or model structure for multi-objective problems?
3. Is there any existing effort or recommendation I should be aware of before structuring the code?

Additional Information

I'd be happy to follow the design patterns of OptimizationProblems.jl, and I'm open to feedback on how best to integrate this kind of functionality.

Looking forward to your thoughts!

Best regards,
Danilo R. Souza

[tmigot]

Hey @Souza-DR !

Thank you for the suggestion, that would be awesome! More problems from the literature is good and multiobjective would be interested.

Currently, this repository contains problems in 2 formats:

• JuMP models
• NLPModels models

Are you familiar with those? Minimally, it would be great to have JuMP multi-objective.

The reason to use those models is to be able to rely on automatic differentiation for the problems instead of manually coding the derivatives (also it would be cool to be do that and something else to discussed).

[Souza-DR]

Hi @tmigot

Thank you for your encouraging response!

Yes, I'm familiar with both JuMP and NLPModels. I've already implemented the multi-objective functions and their gradients in Julia. I'm open to adapting these implementations to align with the repository's structure.

Regarding JuMP, I understand that it primarily supports single-objective models. However, I've come across the [MultiObjectiveAlgorithms.jl] package, which extends JuMP's capabilities for multi-objective optimization. I believe this could be a suitable pathway for integrating multi-objective problems.

As for NLPModels, I'm not certain if it currently supports vector-valued objective functions directly. From my understanding, NLPModels is designed for scalar objectives, and extending it to handle multi-objective problems might require significant modifications or the development of a new model type. I would appreciate any guidance or suggestions on this matter.

[tmigot]

Thanks again for your proposition, it is very exciting.

Regarding JuMP, according to this https://jump.dev/JuMP.jl/stable/tutorials/linear/multi_objective_examples/ it seems that JuMP supports natively multiobjective.

For NLPModels, it misses a bit of code indeed. It will be very similar to what we did for nonlinear least squares problem. So you can use this constructor: src/ADNLPProblems/arglina.jl.

This jso.dev/OptimizationProblems.jl/dev/contributing/ from the documentation should help explaining the structure of the package. Essentially each problem has 3 files, one with the JuMP implementation, one with the AD implementation, and one "meta" that contains information. In this case, in the meta file you should have :objtype => :multiobjective.

If that works for you, let's move on to try and add one problem. You can open a pull request on this repo with what you think is right and I will help improve, complete and fix tests. If you are unusre about the NLPModels part, we can do it later on.

Would that be ok for you?