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doi: 10.3934/jimo.2021216
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## Sufficient optimality conditions and Mond-Weir duality results for a fractional multiobjective optimization problem

 1 LAMA, FSDM, Sidi Mohamed Ben Abdellah University, Fez, Morocco 2 FSDM, Sidi Mohamed Ben Abdellah University, Fez, Morocco

* Corresponding author: Fatima Zahra Rahou

Received  December 2020 Revised  September 2021 Early access December 2021

In this work, we are concerned with a fractional multiobjective optimization problem $(P)$ involving set-valued maps. Based on necessary optimality conditions given by Gadhi et al. [14], using support functions, we derive sufficient optimality conditions for $\left( P\right) ,$ and we establish various duality results by associating the given problem with its Mond-Weir dual problem $\left( D\right) .$ The main tools we exploit are convexificators and generalized convexities. Examples that illustrates our findings are also given.

Citation: Nazih Abderrazzak Gadhi, Fatima Zahra Rahou. Sufficient optimality conditions and Mond-Weir duality results for a fractional multiobjective optimization problem. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2021216
##### References:
 [1] G. Y. Chen and J. Jahn, Optimality conditions for set-valued optimization problems, Mathematical Methods of Operations Research, 48 (1998), 187-200.  doi: 10.1007/s001860050021. [2] F. H. Clarke, Optimization and nonsmooth analysis, Wiley-Interscience, John Wiley & Sons, Inc., New York, 1983. [3] H. W. Corley, Optimality conditions for maximization of set-valued functions, Journal of Optimization Theory and Applications, 58 (1988), 1-10.  doi: 10.1007/BF00939767. [4] J. Dutta and S. Chandra, Convexifactors, generalized convexity and optimality conditions, Journal of Optimization Theory and Applications, 113 (2002), 41-64.  doi: 10.1023/A:1014853129484. [5] J. Dutta and S. Chandra, Convexifactors, generalized convexity and vector optimization, Optimization, 53 (2004), 77-94.  doi: 10.1080/02331930410001661505. [6] P. H. Dien, Locally Lipschitzian set-valued maps and general extremal problems with inclusion constraints, Acta Mathematica Vietnamica, 1 (1983), 109-122. [7] P. H. Dien, On the regularity condition for the extremal problem under locally Lipschitz inclusion constraints, Applied Mathematics and Optimization, 13 (1985), 151-161.  doi: 10.1007/BF01442204. [8] V. F. Demyanov, Convexification and concavication of a positively homegeneous function by the same family of linear functions, Report 3, 208, 802, Universita di Pisa, 1994. [9] V. F. Demyanov and V. Jeyakumar, Hunting for a smaller convex subdifferential, Journal of Global Optimization, 10 (1997), 305-326.  doi: 10.1023/A:1008246130864. [10] B. El Abdouni and L. Thibault, Conditions d'optimalité pour les problèmes d'optimisation de Pareto dont les Objectifs sont des multiapplications, Thèse de Doctorat d'état, Université Mohamed 5, Faculté des Sciences, Rabat, 1995. [11] N. Gadhi, Optimality conditions for difference of convex set-valued mappings, Positivity, 9 (2005), 687-703.  doi: 10.1007/s11117-005-2786-8. [12] N. Gadhi, K. Hamdaoui and M. El idrissi, Sufficient optimality conditions and duality results for a bilevel multiobjective optimization problem via a $\psi$ reformulation, Optimization, 69 (2020), 681-702.  doi: 10.1080/02331934.2019.1625901. [13] N. Gadhi and A. Jawhar, Necessary optimality conditions for a set-valued fractional extremal programming problem under inclusion constraint, Journal of Global Optimization, 56 (2013), 489-501.  doi: 10.1007/s10898-012-9849-8. [14] N. Gadhi, K. Hamdaoui, M. El idrissi and F. Rahou, Necessary optimality conditions for a fractional multiobjective optimization problem, RAIRO-Operations Research, 55 (2021), S1037-S1049.  doi: 10.1051/ro/2020049. [15] V. Jeyakumar and D. T. Luc, Nonsmooth calculus, minimality, and monotonicity of convexificators, Journal of Optimization Theory and Applications, 101 (1999), 599-621.  doi: 10.1023/A:1021790120780. [16] A. Jourani and L. Thibault, Approximations and metric regularity in mathematical programming in Banach space, Mathematics of Operations Research, 18 (1993), 255-510.  doi: 10.1287/moor.18.2.390. [17] B. Kohli, Optimality conditions for optimistic bilevel programming problem using convexificators, Journal of Optimization Theory and Applications, 152 (2012), 632-651.  doi: 10.1007/s10957-011-9941-0. [18] B. S. Mordukhovich, The extremal principle and its applications to optimization and economics, In Optimization and Related Topics, Applied Optimization, 47, 2001, 343–499. doi: 10.1007/978-1-4757-6099-6_17. [19] B. S. Mordukhovich and Y. Shao, On nonconvex subdifferential calculus in Banach spaces, Journal of Convex Analysis, 2 (1995), 211-227. [20] J.-P. Penot and P. Michel, A generalized derivative for calm and stable functions, Differential and Integral Equations, 5 (1992), 433-454. [21] Y. Sawaragi and T. Tanino, Conjugate maps and duality in multiobjective optimization, Journal of Optimization Theory and Applications, 31 (1980), 473-499.  doi: 10.1007/BF00934473.

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##### References:
 [1] G. Y. Chen and J. Jahn, Optimality conditions for set-valued optimization problems, Mathematical Methods of Operations Research, 48 (1998), 187-200.  doi: 10.1007/s001860050021. [2] F. H. Clarke, Optimization and nonsmooth analysis, Wiley-Interscience, John Wiley & Sons, Inc., New York, 1983. [3] H. W. Corley, Optimality conditions for maximization of set-valued functions, Journal of Optimization Theory and Applications, 58 (1988), 1-10.  doi: 10.1007/BF00939767. [4] J. Dutta and S. Chandra, Convexifactors, generalized convexity and optimality conditions, Journal of Optimization Theory and Applications, 113 (2002), 41-64.  doi: 10.1023/A:1014853129484. [5] J. Dutta and S. Chandra, Convexifactors, generalized convexity and vector optimization, Optimization, 53 (2004), 77-94.  doi: 10.1080/02331930410001661505. [6] P. H. Dien, Locally Lipschitzian set-valued maps and general extremal problems with inclusion constraints, Acta Mathematica Vietnamica, 1 (1983), 109-122. [7] P. H. Dien, On the regularity condition for the extremal problem under locally Lipschitz inclusion constraints, Applied Mathematics and Optimization, 13 (1985), 151-161.  doi: 10.1007/BF01442204. [8] V. F. Demyanov, Convexification and concavication of a positively homegeneous function by the same family of linear functions, Report 3, 208, 802, Universita di Pisa, 1994. [9] V. F. Demyanov and V. Jeyakumar, Hunting for a smaller convex subdifferential, Journal of Global Optimization, 10 (1997), 305-326.  doi: 10.1023/A:1008246130864. [10] B. El Abdouni and L. Thibault, Conditions d'optimalité pour les problèmes d'optimisation de Pareto dont les Objectifs sont des multiapplications, Thèse de Doctorat d'état, Université Mohamed 5, Faculté des Sciences, Rabat, 1995. [11] N. Gadhi, Optimality conditions for difference of convex set-valued mappings, Positivity, 9 (2005), 687-703.  doi: 10.1007/s11117-005-2786-8. [12] N. Gadhi, K. Hamdaoui and M. El idrissi, Sufficient optimality conditions and duality results for a bilevel multiobjective optimization problem via a $\psi$ reformulation, Optimization, 69 (2020), 681-702.  doi: 10.1080/02331934.2019.1625901. [13] N. Gadhi and A. Jawhar, Necessary optimality conditions for a set-valued fractional extremal programming problem under inclusion constraint, Journal of Global Optimization, 56 (2013), 489-501.  doi: 10.1007/s10898-012-9849-8. [14] N. Gadhi, K. Hamdaoui, M. El idrissi and F. Rahou, Necessary optimality conditions for a fractional multiobjective optimization problem, RAIRO-Operations Research, 55 (2021), S1037-S1049.  doi: 10.1051/ro/2020049. [15] V. Jeyakumar and D. T. Luc, Nonsmooth calculus, minimality, and monotonicity of convexificators, Journal of Optimization Theory and Applications, 101 (1999), 599-621.  doi: 10.1023/A:1021790120780. [16] A. Jourani and L. Thibault, Approximations and metric regularity in mathematical programming in Banach space, Mathematics of Operations Research, 18 (1993), 255-510.  doi: 10.1287/moor.18.2.390. [17] B. Kohli, Optimality conditions for optimistic bilevel programming problem using convexificators, Journal of Optimization Theory and Applications, 152 (2012), 632-651.  doi: 10.1007/s10957-011-9941-0. [18] B. S. Mordukhovich, The extremal principle and its applications to optimization and economics, In Optimization and Related Topics, Applied Optimization, 47, 2001, 343–499. doi: 10.1007/978-1-4757-6099-6_17. [19] B. S. Mordukhovich and Y. Shao, On nonconvex subdifferential calculus in Banach spaces, Journal of Convex Analysis, 2 (1995), 211-227. [20] J.-P. Penot and P. Michel, A generalized derivative for calm and stable functions, Differential and Integral Equations, 5 (1992), 433-454. [21] Y. Sawaragi and T. Tanino, Conjugate maps and duality in multiobjective optimization, Journal of Optimization Theory and Applications, 31 (1980), 473-499.  doi: 10.1007/BF00934473.
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