A Novel Self-Adaptive Cooperative Coevolution Algorithm for Solving Continuous Large-Scale Global Optimization Problems^†

Round 1

Reviewer 1 Report

This paper proposed an algorithm to solve continuous large-scale global optimization problem using self-adaptive cooperative coevolution, with self-selecting the number of subcomponents and the population size during the optimization. The results of proposed algorithm in general outperformed the algorithms with the static number of subcomponents and the static population size. The paper is good for publication at its current form.

Author Response

Thank you so much for your review, evaluating and feedback.

We have done corrections, which are based on all reviewers’ comments.

Author Response File: Author Response.pdf

Reviewer 2 Report

The paper is generally well-written and presents some interesting results on numerical investiagtion of metaheuristics to address large-scale global optimization problems (LSGO). The goal is challenging and, in my opinion, the paper should be revised to address the following issues:

1) The problem should be formally stated. The corresponding literature should be better described: it is not limited to metaheuristics and CEC-conferences; please consult GO experts to correctly present the state-of-the-art in this area.

2) The authors propose a method trying to find the global solution but do not provide any theoretical analysis to study the global convergence of their method. This analysis should be performed otherwise the authors should not claim to address GO problems.

3) The same regards numerical part. Strictly speaking, not all considered methods are GO while a comparison with some efficient method widely recognized by the GO community is expected (for example, DIRECT-based algorithms). I would suggest consulting the recent open-access paper https://www.nature.com/articles/s41598-017-18940-4 to obtain more information about methods and benchmarking in GO (including LSGO).

Author Response

Thank you so much for your review and feedback.

We don’t discuss GO as a class of mathematical programming in general. But focus on the developing bio-inspired metaheuristics for solving global black-box optimization problems.

It’s clear, that there exist many scientific societies, which deals with GO, but CEC conferences have proposed a popular and widely used benchmark for the fair comparison of corresponding approaches, and today there exist many dozens of databases with source data of the experimental results. CEC benchmarks are used not only for bio-inspired methods and are implemented in many software frameworks.

We know many test problem generators for GO, but have never found the completed experimental results for state-of-the-art approaches for making fair comparison such as provided by CEC competitions.

We develop bio-inspired metaheuristics, which have no good theoretical background for today. Some results are obtained only for the simplest artificial problems and are not applicable for an arbitrary BB optimization problem. At the same time, it is a common practice for researchers in this field to provide the numerical and/or statistical analysis of the experimental results. The same we do.

Of course, there exist a huge number of methods that address GO. Many papers do not give all details on experimental settings, do not provide the complete experimental results, and do computational experiments in different conditions. Within the CEC benchmark, we can make complete and fair comparison with many dozens of approaches (many of them are not bio-inspired) based on the data proposed by authors.

Author Response File: Author Response.pdf

Reviewer 3 Report

Please see the attachment for the review

Comments for author File: Comments.pdf

Author Response

Please find our comments and answers in the attached pdf file.

With respect, authors

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

Unfortunately, the authors ignored my comments except one (they formulated the problem formally). All my criticisms were related to the claim to deal with global optimization (GO) problems (moreover, with large-scale GO problems), a claim not substantiated by theoretical or experimental results. The authors continue to point out that there are many approaches both to address the stated problem and to test the corresponding numerical methods. But they did not present any result on the state-of-the-art GO method limiting their experimental research to the usage of CEC benchmark problems and metaheuristic algorithms. I repeat that this is a good paper on benchmarking metaheuristic algorithm(s) but not a paper on Continuous Large-scale Global Optimization. Therefore, I cannot do anything else but reject the paper. 

Author Response

We would like to thank the reviewer once again for his opinion and comments.
We want to clarify our position on the answer we gave earlier.

1. The problems we use as test problems and the methods we apply and compare belong to the global optimization field as we can see from many hundred papers in peer-reviewed journals and global scientific societies. We should point out that these societies include scientific groups that investigate both mathematical and heuristic methods.
Nevertheless, we have changed the title of the paper to specify the content from "A Self-Adaptive Cooperative Coevolution Approach for Solving Continuous Large-scale Global Optimization Problems" to "A Novel Self-Adaptive Cooperative Coevolution Algorithm for Solving Continuous Large-scale Global Optimization Problems". We want to highlight that we are focused on the improvement of the Cooperative Coevolution Algorithm and don’t assume a universal approach for all global optimization problems. In the abstract we have added the following text: “The main goal of the study is the improvement of coevolutionary decomposition-based algorithms for solving LSGO problems.”
We have also added additional information about features of the problem being solved in the section with the problem statement. Namely, “It is assumed that the fitness function is continuous. The satisfaction of the Lipschitz condition is not assumed; therefore no operations are performed to estimate the Lipschitz constant. In this case, the convergence of an algorithm to the global optimum cannot be guaranteed. In the case of a huge number of decision variables, it is not possible to adequately explore the high-dimensional search space using a limited fitness budget. And we can clarify the goal of the stated problem as proposed in [4]: “the goal of global optimization methods is often to obtain a better estimate of f^* and x^* given a fixed limited budget of evaluations of f(x ̅ )”.”
[4] Sergeyev, Y.D., Kvasov, D.E. & Mukhametzhanov, M.S. On the efficiency of nature-inspired metaheuristics in expensive global optimization with limited budget. Sci Rep 8, 453 (2018). https://doi.org/10.1038/s41598-017-18940-4

2. As we can conclude from the reviewer’s comments and the reference provided, there is a scientific society with its preferable conferences, journals and traditions in using test problems and research methods. We are grateful for the information provided, we will definitely get acquainted with their work and take it into account into future studies.
Our paper has been submitted to the special issue that includes the Bio-inspired and swarm intelligence topic, thus we cite papers, methods, and test problems and use terminology, that are commonly and globally used in the field.
We cannot fulfill the reviewer's recommendation to compare our approaches with their methods on their problems within the framework of this paper. To do that, we need to make absolutely new research and present new studies.
And we are not sure that the proposed comparison will be fair. The numerical results in the paper proposed by the reviewer are limited by the dimensionality equal to 5 (the same in many other papers on the DIRECT and some DIRECT-similar methods). Our approaches deal with 1000 of variables and, of course, will yield to approaches that are specialized in low-dimensional problems. For example, in a recent paper, there is the following statement: “the algorithm was simple, easy to implement, and usually performed well on low-dimensional problems (up to six variables).” (Jones, D.R., Martins, J.R.R.A. The DIRECT algorithm: 25 years Later. J Glob Optim 79, 521–566 (2021). https://doi.org/10.1007/s10898-020-00952-6)
Nevertheless, we will review information on methods proposed by the reviewer and will definitely carry-out a comparison with global optimization state-of-the-art algorithms, such as DIRECT and ADC, and will test our algorithms using test problems from the GKLS generator in our further works.

Reviewer 3 Report

The authors have addressed all of the comments. Therefore the paper can be accepted in its present form.

Round 3

Reviewer 2 Report

As I said in my previous reports, objectively, this is a good paper (which also fits the purpose of the special issue) on testing a metaheuristic optimization algorithm but not a paper on a (large-scale) global optimization method (the theoretical analysis is completely absent while the numerical results suffer from the absence of comparison with modern efficient global optimization algorithms). In my opinion, the authors should shift their emphasis in the paper (and in the title too) on the direction of numerical experimentation with their method in the framework of metaheuristics rather than raising unnecessary and time-wasting discussions about the different optimization communities, journals and so on.

Author Response

no reply