From an Optimal Point to an Optimal Region: A Novel Methodology for Optimization of Multimodal Constrained Problems and a Novel Constrained Sliding Particle Swarm Optimization Strategy

Round 1

Reviewer 1 Report

The paper tackles an interesting method to obtain an optimal region i optimization problems, using a variant of PSO algorithm. While interesting, it has some minor flaws. The attached file describes most of them. Aside of it, however, it would be nice to also see Himmelblau's  and Rosebrock's functions as a bechmark. They have a slightly different properties that the presented benchmarks,  e.g. minimal regions are more flat, and it would be interesting to see if used PSO does not manifest larger errors or oscilations within the optimization procedure.  Also languare requires moderation. I engourage Authors to perform a proofreading with accordance to the comments provided in attached file (I did not pointed out all errors, just marked some repeating ones). Exluding the above the paper is good and may be published after some revision.

Comments for author File: Comments.pdf

Author Response

Thank you for your comments. Please see attached file.

Author Response File: Author Response.pdf

Reviewer 2 Report

Authors must be congratulated by an interesting,  innovative and useful paper.

This being said, what follows is a list of aspects that require some action, aiming the understanding and  text readability.

1 Abstract– Must be reformulated in order to convey to the Reader a more precise idea of paper content.

2- Meaning of population, population distribution, sampling and sampling distribution  must be clarified in the context deductions on pages 4 and 5.  3- Page 2 – A numerical example of confidence region construction would be useful.

4- The steps of the new proposed algorithm should be expressed in algorithmic form (pseudo-code or other) showing the diferences between the new variant and the original PSO algorithm.

5- Figure 1- eliminate errors (* is missing ) and a more informative legend legend is needed.

6-  Page 4. The meanings of ε and e²  and  their distributions must be explained. Sentence next to expression 6 must be corrected. Also, expressions 7,11,12,13  must be rewriten using the usual conventions of  statistics.

7- Page 4, correct meaning of expression 13 according with statistics language conventions.

8- Page 8. Results. In addition to results of benchmarks,  a real example of application of new algorithm to some important domain would be useful for understanding the new method and support its credibility.

9-References. The references titles were eliminated but the links provided are not enough to eliminate ambiguities and are a source of time waste both to reviewer and reader.

10- Supplementary Materials: The provided link https://www.mdpi.com/xxx/s1

produced systematically the following message: “Error 404 - File not found”, meaning that useful information about the content – such as software- was not available.

Author Response

Thank you for your comments. Please see attached file.

Author Response File: Author Response.pdf

Reviewer 3 Report

The authors propose a new approach to robust design optimization based on the Particle Swarm optimization method which they call Constraints-based Sliding Particle Swarm Optimizer. Population-based confidence regions are derived and topological characterization of the objective function proposed thanks to a statistical analysis of the population.

Overall the work may be interesting however there are also several issues. First and foremost, it is very hard to judge whether this work presents some major contributions with respect to the existing literature. The author should consider strengthening the literature review. In particular, a clear review of reliability-based design optimization (RBDO) and robust design optimization (RDO) would be beneficial.

Note that RDO focuses on minimizing the uncertainty and the sensitivity to uncertain factors of a design/decision. Hence, are not limited to finding optimal operating conditions (introduction lines 32-37) but rather on identifying optimal operational conditions/designs/processes/actions/policies with minimum variance or insensitive to environmental-operational uncertainties.

However, only deterministic objective functions are considered in the example. Thus, it is not clear where the uncertainty arises from. A clear definition of the space of uncertain factors and space of design variables must include.

Also, note that only 2-dimensional design spaces are considered. This is clearly useful to present graphically the results. However, this makes the reviewed suspicious about the applicability of the method to more complex (and thus realistic) problems. A new example with more design variables should be considered and compared to existing methods.

Finally, not all robust design methods are limited by estimation of mean and variance nor distributional assumptions. See for instance given-data approaches:

https://www.sciencedirect.com/science/article/pii/S0951832019309639

Interval methods,

https://www.sciencedirect.com/science/article/pii/S0045794921000286

Distribution free approaches, and many others.

Based on my assessment of the work I would discourage its publication in the current state.

The authors should consider improving the presentation of the methodology and link better their work to the existing literature on RDO.

Author Response

Thank you for your comments. Please see attached file.

Author Response File: Author Response.pdf

Round 2

Reviewer 3 Report

The authors took into considerations all my previous comments and substantially improved the manuscript. The revised version is much clearer. Now I understand they are not dealing with Robust optimization under uncertainty, and the focus is on deterministic optimizations. The idea is to prescribe confidence regions for the optimums using the information from the population-based optimization algorithm (a novel Constrained PSO specifically). The algorithmic implementation of the method is also proesented, which is highly appreciated. Two additional optimization problems are proposed: (1) the 2-dimensional Himmelblau Function with four optimal points and low gradient regions in the design space; (2) An optimization of a Moving Bed Reactor with five design variables. 

• I couldn’t visualize some of the equations in the PDF of the manuscript. For instance, In Eq (33) and the new algorithm, some of the subscripts/superscripts are displayed as black rectangles. 

• The authors should present an itemized list of the main novelty and contributions of this work. I would suggest just before Section "2. Materials and Methods".

• Are Equations (2)-(18) are proposed by the authors of this paper or are well-known from the literature? Add relevant references.

• My understanding of PSO is that the optimal particles influence each other, e.g., based on local and global memory information. Furthermore, at each iteration k, the potion and velocity are updated from the previous iteration k-1. Nonetheless, equation (6) assumes ?^2 is the squared residual error of uncorrelated experiments. Are the squared difference between the analytical optimum and the approximated optimum found by the CSPSO assumed uncorrelated for  k=1,2,…,n_k? Are these differences assumed to be standard random variables?
• After Eq(12), “Knowing that these distributions are independent…”. Include a reference or elaborate further on this statement.  

• The difference between the analytical optimum and a given point (ℎ) is assumed to become a chi-square distribution. Is this a well-known assumption verified in practice? Is this assumption good/bad? Consider presenting the distribution of ℎ, e and b for their examples and test the validity of the chi-square distributional assumptions.

• • The dependency of the objective function in Eq. (33) should be discussed (at least implicitly). For instance, is Purx a function of Qf or Qx or all five design variables in x?

Author Response

Thank you for your comments. Please, see attached file.

Author Response File: Author Response.pdf