Closest Farthest Widest

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The manuscript is technically sound and it is of a certain interest to a broad section of the operators theory’s community, as well as the approximation theory’s community and related fields. The methods, all reasonable and expected, are introduced and preceded by a suitable discussion in the section. Based on the unifying approach dealing with the definitions and properties of projection operators and distance measures, the content of the paper is of reasonable quality. The goals and general approach are clearly summarized in the abstract. The introduction and preliminaries provide the readers with an adequate background to understand the finding of the manuscript. The author provides good motivations for the study of the specific problem, and defines notions and notation used in the work. In details, the results are clearly explained and established by consolidated and effective arguments in their sections. The author discusses the following issues: design of Frank-Wolfe and projected gradient ascent algorithms. These algorithms are corroborated by a proper discussion mainly on the support function in appropriate convex (cone) sets and a convergence analysis of the algorithm. At the best of my knowledge the results are new and their development is correct. The crucial point of the work is in understanding how projection operators design a good geometry of the problem, hence modifying, or not modifying, the robustness of the approach. The quality of English language is in general good, without any relevant typos to solve. The bibliography collects the salient points in the literature review, but some additional references could be noted, dealing with certain aspects of the general theory and its recent applications. According to the above motivations, the manuscript is worthy of publication in  Algorithms, but before it, if the author agree, I suggest to cite and properly discuss in the introduction (on page 2, lines 50/51), the following contributions (or similar ones):

Calzi, M., Peloso, M.M. Boundedness of Bergman projectors on homogeneous Siegel domains. Rend. Circ. Mat. Palermo, II. Ser 72, 2653–2701 (2023). https://doi.org/10.1007/s12215-022-00798-9

where the authors focus on the boundedness of Bergman projectors on weighted Bergman spaces on homogeneous Siegel domains of Type II. The study is carried out on the scale of mixed-norm weighted Lebesgue spaces. The authors prove that the boundedness of such Bergman projectors is equivalent to various notions of atomic decomposition, duality, and characterization of boundary values of the mixed-norm weighted Bergman spaces. The authors provide a useful characterization which was previously known for tube domains. Maybe these operators could be setup naturally to a variant of Frank-Wolfe  algorithm achieving acceleration.

Ibrahim, A., et al. A Dai-Liao-type projection method for monotone nonlinear equations and signal processing. Dem. Math. 55 (2022). 978-1013. https://doi.org/10.1515/dema-2022-0159

This work presents an application of the main finding to reconstructing sparse signals and images. A conjugate gradient method for solving nonlinear equations with convex constraints is proposed. Under suitable hypotheses, the global convergence of the method is established. This way the authors underline the key role of projected gradients in developing real world applications.

Finally, I suggest the author a minor revision of the abstract, since it should provide in a clear and concise way the goal of the work and the strategy to get it. So, the initial sentences could be deleted.

Based on the previous comments, I will positively recommend the publication of the manuscript in the journal, after the above minor revision is done. 

Author Response

Thanks for your kind review. I have revised the abstract and referenced and commented on the Ibrahim et al paper. The Calzi and Peloso paper does not seem very pertinent. I now give web pages for projection software. Changes to the text appear in red.

Reviewer 2 Report

Comments and Suggestions for Authors

This is interesting and well written paper. The methodology sounds and is supported by evaluation. I have following minor comments:
1) The title of the manuscript is witty however it does not summarize the subject of the paper in the unambiguous way - I suggest to change it, however it is not mandatory.
2) Please do not use equations in Abstract, for example I am not sure what ||x||_{1} means in this context
3) Please number equations (1), (2), etc.
4) The transition from equation in line 108 to line 109 is unclear
5) line 144 - what is a difference between multi-row formula in () and in []? See also line 198
6) Please move source codes outside the paper to the online repository like Github. Also add some example with algorithm's execution to make research reproducible and easy to use by the reader.
7) Please discuss in introduction state-of-the-art of research on projection onto a closed convex sets taking into account most up-to-date achievements from the last three years.

Author Response

Thanks for your thoughtful review. I like my title. I have removed the math symbols from the abstract. I only number equations as needed. In my opinion, unused equation numbers create visual clutter. I have rewritten the paragraphs mentioned in your items 3) and 4) to achieve greater clarity. My Julia code has been placed in a Github repository with an address. The most important recent advance in the projection field is the creation of easy to use software. I now give web pages for projection software. Changes to the text are highlighted in red.

Reviewer 3 Report

Comments and Suggestions for Authors

There are various comments to be considered by the author. Please refer to my comments as follows.
Comment 1. Please follow the format of the journal.
Comment 2. Carefully check if researchers “Heinz Bauschke, Qiang Heng, and Joong-Ho Won” should be co-authors.
Comment 3. Order the references based on their appearance in the main text.
Comment 4. A literature review is missing.
Comment 5. Codes should be attached separately.
Comment 6. Research contributions are limited.
Comment 7. Performance evaluation and comparison are missing.
Comment 8. What are the research implications?

Author Response

Thanks for your review. The references are now collected in a bib file and appear in the journal’s preferred order. My acknowledged colleagues were not actively involved in the research for or construction of this paper.  If you want to query them personally about their role, I have no objection. My Julia code has been placed in a Github repository with an address. I now give two web addresses for projection operator repositories. Four new references have also been added in this revision. Changes to the text are highlighted in red. Given the tight deadline for resubmission, my revisions are targeted to specific suggestions for improvement. The major overhaul you seem to suggest is just not possible.

Let me reply in more detail to your comments # 4, 6, 7, and 8.

4. The literature on how to compute the farthest point and diameter is limited to point clouds. If you can suggest pertinent papers dealing with continuous sets that I have missed, then I would happy to cite them.
5. The assertion that the research contributions are limited is a judgement call. Reviewers 1 and 2 disagree, and I side with them.
6. The assertion that performance evaluation and comparison are missing seems to ignore the evidence of my two tables. I am unaware of competing software for my numerical examples. Please be more specific.
7. The query about further research implications misses the importance of these problems as prototype nonconvex optimization problems. The naturalness of the problems also recommends them. I now cite a classic theorem of Jung relating the diameter of a set to the radius of a minimum enclosing ball. My hope is that the current paper will stimulate more research and further applications. Practical applications to robotics are certainly possible. The moving sofa problem of Moser hints at the subtle challenges of motion problems in general.

Round 2

Reviewer 3 Report

Comments and Suggestions for Authors

Please refer to my follow-up comments.
Follow-up comment 1: A Literature review is expected to cover the methodology, results, and limitations of the latest existing works.
Follow-up comment 2: Elaborate on the analysis for Tables 1 and 2.
Follow-up comment 3: Performance comparison with existing works is needed.
Follow-up comment 4: Provide numbers for the equations.

Author Response

please see attachment

Author Response File: Author Response.pdf

Round 3

Reviewer 3 Report

Comments and Suggestions for Authors

There are some key comments to be further addressed by the author.

Comment 1. More terms can be added to the keywords to increase the chance of the article being searched.
Comment 2. Limited references are within the last five years.
Comment 3. Compare the results between the Frank-Wolfe and Projected Gradient Ascent algorithms.
Comment 4. Refer to the journal’s template; abbreviations of the journal name should be used.
Comment 5. Refer to the journal’s template; please assign a number for the equations. The journal does not use a flexible format.

Author Response

please check attachment

Author Response File: Author Response.pdf