Semidefinite Multiobjective Mathematical Programming Problems with Vanishing Constraints Using Convexificators
, Jitendra Kumar Maurya 3,†and Shashi Kant Mishra 2,†Round 1
Reviewer 1 Report
Please see the attachment.
Comments for author File: 
Comments.pdf
Author Response
We have followed the lines of suggestions given by the Reviewer # 1.
Response to Reviewer # 1
1. Comment: In Lemma 3.1, the authors claim that GCCQ implies GGCQ. I am interested in the inverse inclusion, i.e., under what conditions, GGCQ implies GCCQ. If yes, please prove it, otherwise, please take a counter-example.
Explanation: The converse of the Lemma 3.1 does not hold in general, we provide an explanatory counter-example (Example 3.1), which shows that GGCQ does not imply GCCQ.
2. Comment: In Example 3.1, the authors employ the S-stationary Condition, but it is defined later in Definition 3.3. Please change the order. On the other hand, does the above concept originate from the authors or refer to reference?
Explanation: We have corrected the order according as the reviewer.
3. Comment: I think the Theorem 3.4 is not clear and strict, please take an example to clarify its reasonableness.
Explanation: We have modified the Theorem 3.4 and provided an example to validate the result.
4. Comment: Please check the whole manuscript, say, on Page 13, ‘Now, let a feasible point ?Ì… of S-MMPVC satisfies S-stationary Condition conditions’,…
Explanation: We have checked and revised the whole manuscript thoroughly.
The authors are indebted to the reviewer for their valuable comments and remarks that helped to improve the presentation and quality of the manuscript.
Author Response File: Author Response.pdf
Reviewer 2 Report
Respected Editor
After reading carefully, I suggest that the authors improve the manuscript by using such suggestions
- Improve the title of this paper
- Add few more suitable keywords
- Highlight the novelty
- I could not understand that which technique and method have been used to solve the Examples
- Please provide a suitable methodology
- Expand the conclusion
Author Response
We have followed the lines of suggestions given by the Reviewer # 2.
Response to Reviewer # 2
1. Comment: Improve the title of this paper
Explanation: We have improved shortened the title. The new title is “Semidefinite
Multiobjective Mathematical Programming Problems with Vanishing Constraints
using Convexificators”. We think the new title satisfies the essence of our manuscript.
2. Comment: Add few more suitable keywords
Explanation: We have added two more keywords, ‘Nonsmooth Analysis’ and
‘Constraint Qualifications’
3. Comment: Highlight the novelty
Explanation: Golestani and Nobakhtian [Positivity, 19; 221-236(2015)] established
optimality conditions for nonsmooth semidefinite single optimization problems. We have established the optimality conditions for more interesting class of nonlinear optimization namely mathematical programming problems with vanishing constraints (MPVC), which is more applicable in the topology optimization and many real life problems. We have further extended the single objective semidefinite optimization problems to multiobjective semidefinite optimization problems.
4. Comment: I could not understand that which technique and method have been used to solve the Examples.
Explanation: We have used the technique motivated by [Golestani, M. and Nobakhtian, S.:Optimality conditions for nonsmooth semidefinite programming via convexificators. Positivity, 19; 221-236(2015)], to solve the examples.
5. Comment: Please provide a suitable methodology
Explanation: We have used the constraint qualifications technique motivated by Li [J. Optim. Theory Appl., 106(2), 373-398, 2000] and provide some generalized constraint qualifications for semidefinite optimization problems. We also used the linearization technique inspired by Kanzow et al. [Comput. Optim. Appl. 31, 251-273, 2005]. Further, we used the methodology motivated by Golestani and Nobakhtian [Positivity, 19; 221-236 (2015)] and Miettinen [Nonlinear Multiobjective Optimization, Kluwer Academic Publishers, Boston, 1999] to prove our results.
6. Comment: Expand the conclusion
Explanation: We have extended the Section 4 (Conclusions and future remarks) and added some references and motivation towards variational control problems. The added references are listed below:
(i) Treanta, S.: Well Posedness of New Optimization Problems with Variational
Inequality Constraints. Fractal Fract., 5(3), 123 (2021)
(ii) Mititelu, S., Treanta, S.: Efficiency conditions in vector control problems
governed by multiple integrals. J. Appl. Math. Comput., 57(1), 647–665 (2018)
(iii) Treanta, S.: On modified interval-valued variational control problems with first order PDE constraints. Symmetry, 12(3), 472 (2020)
The authors are indebted to the reviewer for their valuable comments and remarks that helped to improve the presentation and quality of the manuscript.
Author Response File: Author Response.pdf
Reviewer 3 Report
Suggestions:
1.Correct the title, shorten it and implicitly say what the essence of the paper is.
2.Take care of the notation of the equations.
3.Label the essential equations
I suggest to the authors that the research be extended to the results on multiobjective semidefinite optimization problems to variational control problem.
Author Response
We have followed the lines of suggestions given by the Reviewer # 3.
Response to Reviewer # 3
1. Comment: Correct the title, shorten it and implicitly say what the essence of
the paper is.
Explanation: We have shortened the title now the title is “Semidefinite
Multiobjective Mathematical Programming Problems with Vanishing Constraints
using Convexificators”. We think the new title justifies the manuscript.
2. Comment: Take care of the notation of the equations.
Explanation: We have corrected the notation of equations.
3. Comment: Label the essential equations
Explanation: We have made labelling to the essential equations.
4. Comment: I suggest to the authors that the research be extended to the
results on multiobjective semidefinite optimization problems to variational
control problem.
Explanation: The research can be extended to variational control problem. We
have included the some references and future direction in Section 4(Conclusions
and future remarks) in which we will extend our work.
The added references are listed below:
(i) Treanta, S.: Well Posedness of New Optimization Problems with Variational
Inequality Constraints. Fractal Fract., 5(3), 123 (2021)
(ii) Mititelu, S., Treanta, S.: Efficiency conditions in vector control problems
governed by multiple integrals. J. Appl. Math. Comput., 57(1), 647–665 (2018)
(iii) Treanta, S.: On modified interval-valued variational control problems with first order PDE constraints. Symmetry, 12(3), 472 (2020)
The authors are indebted to the reviewer for their valuable comments and remarks that helped to improve the presentation and quality of the manuscript.
Author Response File: Author Response.pdf
Round 2
Reviewer 1 Report
The revision clearly answers my proposed question, and give a correct example to illustrate the relationship between GCCQ and GGCQ. I think it can be accepted in present form.
Reviewer 2 Report
The authors improved the paper as suggested. Hence, I recommend this paper for publication
