Constrained Multi-Objective Optimization

First of all, the paper title is entirely misleading. "Constrained Multi-Objective Optimization" sounds like a book title, or at least a PhD thesis, in which the entire research field is investigated from various perspectives, such as diverse benchmark problems, algorithmic approaches, visualization methods, performance measures, etc. However, this paper is limited to the gradient-based part of that research field and essentially focusses on the introduction of a new algorithm. Within this work it is stated that the goal of MOO is to find a Pareto stationary solution. However, as the optima of MOO problems usually consist of entire solution sets, the goal of algorithms usually is to find a good approximation of the solution set. Claiming that MOO focusses on finding a single solution would downplay the complexity of this research area. The proposed approach has been designed for MOO problems that are (strongly) convex. Yet, it is not clear whether this property relates to the search or objective space. The usefulness of MLM-CMOO remains unclear, as real-world problems likely aren't compositions of purely (strongly) convex single-objective functions. As has been shown in various publications, concatenations of convex problems are a very special case, failing to capture the complexity of most MOO problems. By integrating a single multimodal (single-objective) function, the resulting MOO problem will have several local optima, which serve as traps for gradient-based approaches. In the benchmark study, the proposed algorithm is compared to NSGA-II and PSL. However, NSGA-II is a population-based approach that evaluates multiple solutions per iteration. In consequence, it will likely take more time to converge. Also, population-based approaches such as NSGA-II are designed to find optima in complex black-box problems. If the problem is convex, gradient-based approaches will usually win such a comparison. Therefore, fair benchmark studies should consider state-of-the-art gradient-based approaches. The convergence rate is indicated in relation to T, however, T is not specified/defined within the paper. For citing, in multiple cases the wrong citing command has been used. For instance, it should be "The NSGA-II (Deb et al., 2002)" instead of "The NSGA-II Deb et al. (2002)". The experimental analysis is kept extremely brief, providing hardly any insights.

1. In some parts of the text, "multi-objective optimization" is used, while in others, "multiple-objective optimization" is used. It is recommended that the author standardize the terminology.
2. The authors claim that the proposed method outperforms the state-of-the-art. It might be somewhat controversial to regard NSGA-II as state-of-the-art as it was proposed 20 years ago.
3. L38. The authors believe that gradient-free methods can not provide a convergence guarantee. I think gradient-free methods could also have a convergence guarantee under certain assumptions [1,2].
4. L51, "However, this transformation can not give a stable guarantee for the convergence rate as it may give the farthest Pareto front for the given coefficient." What does "farthest Pareto front" mean? Could you please provide some clarification for this claim or some references?
5. I think the title does not accurately reflect the contributions of this paper. The author could consider using "Gradient-based Constrained Multi-Objective Optimization" or "Constrained Multiple-Gradient Descent".
6. L397. The authors believe that NSGA-II cannot handle constraints. Actually, NSGA-II is originally designed to solve constrained problems [3].
7. The empirical study is not well-designed. The proposed method is compared with two black-box methods on deep learning tasks. I believe it is more reasonable to use gradient-based methods as baselines, for instance, [4, 5]. Some experimental details are missing, e.g., the population size of NSGA-II.
8. From Fig. 1, I did not find that CMOO significantly outperforms the baselines. It is recommended to use some performance indicators to quantitatively measure the difference.
9. L406. Typo: $1e^{-5}$ --> $10^{-5}$.

References

[1] Zheng, Weijie, Yufei Liu, and Benjamin Doerr. "A first mathematical runtime analysis of the Non-Dominated Sorting Genetic Algorithm II (NSGA-II)." AAAI 2022.

[2] Do, Anh Viet, et al. "Rigorous runtime analysis of MOEA/D for solving multi-objective minimum weight base problems." NeurIPS 2023.

[3] K. Deb, A. Pratap, S. Agarwal, and T. Meyarivan. A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans. Evol. Comput., 2002.

[4] Liu et al., Profiling Pareto front with multi-objective stein variational gradient descent, NeurIPS 2021.

[5] Chen et al., Multi-Objective Deep Learning with Adaptive Reference Vectors, NeurIPS 2022.

There are a number of weaknesses.

1. Why do you consider gradient based approaches only in this paper to tackle multi-objective optimisation tasks? Under what conditions, gradient based methods are good?

2. The number of objectives in all the datasets should be explicitly identified and described clearly.

3. In your experiments, you compared the proposed method with NSGA-II and PSL. Why didn't you consider SPEA2 and MOEA/D?

4. The results are far from complete --- only Table 1 and Figure 1 are presented. Many interesting results such as the Pareto-fronts for each dataset, the hyper-volume or IGD comparisons are missing. It is not clear to me the comparison between those algorithms are fair. Only providing the loss and time values is not sufficient to understand the effectiveness of the proposed method.

1. The submission focuses on constrained multi-objective optimization (CMOO), however, no CMOO algorithms are compared in the experiments. The authors should compare several CMOO algorithms in their experiments. For example, the authors reviewed some gradient-free CMOO algorithms in the related work section, which could be included in the comparisons.
2. In Section 2, only gradient-based MOO and gradient-free CMOO are discussed in the related work. The authors should also review some constraint handling techniques (CHTs).
3. The experiments in Section 5 are all multi-task learning (MTL) problems. The authors should add explanations to clarify how these MTL problems are used to evaluate the performance of CMOO algorithms, including what the constraints are in these MTL problems.
4. The presentation of experimental results lacks detail. The authors should provide detailed explanations of Table 1 and Figure 1 to help readers understand experimental results. For example, the caption for Figure 1 is too brief, the authors should clarify the meaning of each subfigure in Figure 1.
5. The symbols in the submission are inconsistent. For instance, the $i^{th}$ objective function is denoted as $f_i(x)$ or $f^i(x)$.