# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2018138

## Continuity of solutions mappings of parametric set optimization problems

 1 School of Mathematics and Statistics, Southwest University, Chongqing 400715, China 2 School of Economics and Management, China University of Geosciences, Wuhan 430074, China 3 College of Management, Chongqing College of Humanities, Science & Technology, Chongqing 401524, China 4 School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu 611130, China

* Corresponding author: Jiawei Chen

Received  December 2016 Revised  November 2017 Published  September 2018

Fund Project: This research is supported by the Natural Science Foundation of China (Nos: 11571055, 11401487, 71471167), the Basic and Advanced Research Project of Chongqing (cstc2016jcyjA0239, cstc2015jcyjBX0131), the China Postdoctoral Science Foundation (No: 2015M582512) and the Fundamental Research Funds for the Central Universities

Set optimization is an indispensable part of theory and method of optimization, and has been received wide attentions due to its extensive applications in group decision and group game problems. This paper focus on the continuity of the strict (weak) minimal solution set mapping of parametric set-valued vector optimization problems with the lower set less order relation. We firstly introduce a concept of strict lower level mapping of parametric set-valued vector optimization problems. Moreover, the upper and lower semicontinuity of the strict lower level mapping are obtained under some suitable conditions. Lastly, the sufficient condition for the continuity of the strict minimal solution set mappings of parametric set optimization problems are established by a new proof method, which is different from that in [27,28].

Citation: Jiawei Chen, Guangmin Wang, Xiaoqing Ou, Wenyan Zhang. Continuity of solutions mappings of parametric set optimization problems. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2018138
##### References:
 [1] M. Alonso and L. Rodríguez-Marín, Optimality conditions for set-valued maps with set optimization, Nonlinear Anal., 70 (2009), 3057-3064. doi: 10.1016/j.na.2008.04.027. [2] J. P. Aubin and I. Ekeland, Applied Nonlinear Analysis, Wiley. New York, 1984. [3] J. W. Chen, Q. H. Ansari and J. C. Yao, Characterizations of set order relations and constrained set optimization problems, Optim., 66 (2017), 1741-1754. doi: 10.1080/02331934.2017.1322082. [4] J. W. Chen, E. Köbis, M. Köbis and J. C. Yao, A new set order relation in set optimization, J. Nonlinear Convex Anal., 18 (2017), 637-649. [5] M. Dhingra and C. S. Lalitha, Well-setness and scalarization in set optimization, Optim. Lett., 10 (2016), 1657-1667. doi: 10.1007/s11590-015-0942-z. [6] A. Göfert, Chr. Tammer, H. Riahi and C. Z$\check{a}$inescu, Variational Methods in Partially Ordered Spaces, Springer-Verlag, New York, 2003. [7] C. Gutiérrez, B. Jiménez, E. Miglierina and E. Molho, Scalarization in set optimization with solid and nonsolid ordering cones, J. Global Optim., 61 (2014), 525-552. doi: 10.1007/s10898-014-0179-x. [8] C. Gutiérrez, E. Miglierina, E. Molho and V. Novo, Pointwise well-posedness in set optimization with cone proper sets, Nonlinear Anal., 75 (2012), 1822-1833. doi: 10.1016/j.na.2011.09.028. [9] A. H. Hamel and A. Löne, Lagrange duality in set optimization, J. Optim. Theory Appl., 161 (2014), 368-397. doi: 10.1007/s10957-013-0431-4. [10] E. Hernández and L. Rodríguez-Marín, Existence theorems for set optimization problem, Nonlinear Anal., 67 (2007), 1726-1736. doi: 10.1016/j.na.2006.08.013. [11] E. Hernández and L. Rodríguez-Marín, Nonconvex scalarization in set optimization with set-valued maps, J. Math. Anal. Appl., 325 (2007), 1-18. doi: 10.1016/j.jmaa.2006.01.033. [12] N. J. Huang, J. Li and H. B. Thompson, Stability for parametric implicit vector equilibrium problems, Math. Comput. Model., 43 (2006), 1267-1274. doi: 10.1016/j.mcm.2005.06.010. [13] J. Jahn, A derivative-free descent method in set optimization, Comput. Optim. Appl., 60 (2015), 393-411. doi: 10.1007/s10589-014-9674-8. [14] J. Jahn and T. X. D. Ha, New set relations in set optimization, J. Optim. Theory Appl., 148 (2011), 209-236. doi: 10.1007/s10957-010-9752-8. [15] A. A. Khan, C. Tammer and C. Z$\check{a}$linescu, Set-Valued Optimization: An Introduction with Applications, Springer, New York, 2015. doi: 10.1007/978-3-642-54265-7. [16] S. Khoshkhabar-amiranloo and E. Khorram, Pointwise well-posedness and scalarization in set optimization, Math. Meth. Oper. Res., 82 (2015), 195-210. doi: 10.1007/s00186-015-0509-x. [17] B. T. Kien, On the lower semicontinuity of optimal solution sets, Optim., 54 (2005), 123-130. doi: 10.1080/02331930412331330379. [18] D. Klatte, A sufficient condition for lower semicontinuity of solution sets of systems of convex inequalities, Math. Program. Stud., 21 (1984), 139-149. [19] D. Kuroiwa, Some duality theorems of set-valued optimization with natural critera, In: Proceedings of the International Conference on Nonlinear Analysis and Convex Analysis, 221-228. World Scientific, RiverEdge, 1999. [20] D. Kuroiwa, On set-valued optimization, Nonlinear Anal., 47 (2001), 1395-1400. doi: 10.1016/S0362-546X(01)00274-7. [21] S. J. Li, G. Y. Chen and K. L. Teo, On the stability of generalized vector quasivariational inequality problems, J. Optim. Theory Appl., 113 (2002), 283-295. doi: 10.1023/A:1014830925232. [22] S. J. Li and C. R. Chen, Stability of weak vector variational inequality, Nonlinear Anal., 70 (2009), 1528-1535. doi: 10.1016/j.na.2008.02.032. [23] S. J. Li and Z. M. Fang, Lower semicontinuity of the solution mappings to a parametric generalized Ky Fan inequality, J. Optim. Theory Appl., 147 (2010), 507-515. doi: 10.1007/s10957-010-9736-8. [24] X. J. Long and J. W. Peng, Generalized B-well-posedness for set optimization problems, J. Optim. Theory Appl., 157 (2013), 612-623. doi: 10.1007/s10957-012-0205-4. [25] J. Liu, J. W. Chen, W. Y. Zhang and C. F. Wen, Scalarization and pointwise Levitin-Polyak well-posedness for set optimization problems, J. Nonlinear Convex Anal., 18 (2017), 1023-1040. [26] Q. L. Wang and S. J. Li, Lower semicontinuity of the solution mapping to a parametric generalized vector equilibrium problem, J. Ind. Manag. Optim., 10 (2014), 1225-1234. doi: 10.3934/jimo.2014.10.1225. [27] Y. D. Xu and S. J. Li, Continuity of the solution set mappings to a parametric set optimization problem, Optim. Lett., 8 (2014), 2315-2327. doi: 10.1007/s11590-014-0738-6. [28] Y. D. Xu and S. J. Li, On the solution continuity of parametric set optimization problems, Math. Meth. Oper. Res., 84 (2016), 223-237. doi: 10.1007/s00186-016-0541-5. [29] W. Y. Zhang, S. J. Li and K. L. Teo, Well-posedness for set optimization problems, Nonlinear Anal., 71 (2009), 3769-3778. doi: 10.1016/j.na.2009.02.036. [30] J. Zhao, The lower semicontinuity of optimal solution sets, J. Math. Anal. Appl., 207 (1997), 240-254. doi: 10.1006/jmaa.1997.5288.

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##### References:
 [1] M. Alonso and L. Rodríguez-Marín, Optimality conditions for set-valued maps with set optimization, Nonlinear Anal., 70 (2009), 3057-3064. doi: 10.1016/j.na.2008.04.027. [2] J. P. Aubin and I. Ekeland, Applied Nonlinear Analysis, Wiley. New York, 1984. [3] J. W. Chen, Q. H. Ansari and J. C. Yao, Characterizations of set order relations and constrained set optimization problems, Optim., 66 (2017), 1741-1754. doi: 10.1080/02331934.2017.1322082. [4] J. W. Chen, E. Köbis, M. Köbis and J. C. Yao, A new set order relation in set optimization, J. Nonlinear Convex Anal., 18 (2017), 637-649. [5] M. Dhingra and C. S. Lalitha, Well-setness and scalarization in set optimization, Optim. Lett., 10 (2016), 1657-1667. doi: 10.1007/s11590-015-0942-z. [6] A. Göfert, Chr. Tammer, H. Riahi and C. Z$\check{a}$inescu, Variational Methods in Partially Ordered Spaces, Springer-Verlag, New York, 2003. [7] C. Gutiérrez, B. Jiménez, E. Miglierina and E. Molho, Scalarization in set optimization with solid and nonsolid ordering cones, J. Global Optim., 61 (2014), 525-552. doi: 10.1007/s10898-014-0179-x. [8] C. Gutiérrez, E. Miglierina, E. Molho and V. Novo, Pointwise well-posedness in set optimization with cone proper sets, Nonlinear Anal., 75 (2012), 1822-1833. doi: 10.1016/j.na.2011.09.028. [9] A. H. Hamel and A. Löne, Lagrange duality in set optimization, J. Optim. Theory Appl., 161 (2014), 368-397. doi: 10.1007/s10957-013-0431-4. [10] E. Hernández and L. Rodríguez-Marín, Existence theorems for set optimization problem, Nonlinear Anal., 67 (2007), 1726-1736. doi: 10.1016/j.na.2006.08.013. [11] E. Hernández and L. Rodríguez-Marín, Nonconvex scalarization in set optimization with set-valued maps, J. Math. Anal. Appl., 325 (2007), 1-18. doi: 10.1016/j.jmaa.2006.01.033. [12] N. J. Huang, J. Li and H. B. Thompson, Stability for parametric implicit vector equilibrium problems, Math. Comput. Model., 43 (2006), 1267-1274. doi: 10.1016/j.mcm.2005.06.010. [13] J. Jahn, A derivative-free descent method in set optimization, Comput. Optim. Appl., 60 (2015), 393-411. doi: 10.1007/s10589-014-9674-8. [14] J. Jahn and T. X. D. Ha, New set relations in set optimization, J. Optim. Theory Appl., 148 (2011), 209-236. doi: 10.1007/s10957-010-9752-8. [15] A. A. Khan, C. Tammer and C. Z$\check{a}$linescu, Set-Valued Optimization: An Introduction with Applications, Springer, New York, 2015. doi: 10.1007/978-3-642-54265-7. [16] S. Khoshkhabar-amiranloo and E. Khorram, Pointwise well-posedness and scalarization in set optimization, Math. Meth. Oper. Res., 82 (2015), 195-210. doi: 10.1007/s00186-015-0509-x. [17] B. T. Kien, On the lower semicontinuity of optimal solution sets, Optim., 54 (2005), 123-130. doi: 10.1080/02331930412331330379. [18] D. Klatte, A sufficient condition for lower semicontinuity of solution sets of systems of convex inequalities, Math. Program. Stud., 21 (1984), 139-149. [19] D. Kuroiwa, Some duality theorems of set-valued optimization with natural critera, In: Proceedings of the International Conference on Nonlinear Analysis and Convex Analysis, 221-228. World Scientific, RiverEdge, 1999. [20] D. Kuroiwa, On set-valued optimization, Nonlinear Anal., 47 (2001), 1395-1400. doi: 10.1016/S0362-546X(01)00274-7. [21] S. J. Li, G. Y. Chen and K. L. Teo, On the stability of generalized vector quasivariational inequality problems, J. Optim. Theory Appl., 113 (2002), 283-295. doi: 10.1023/A:1014830925232. [22] S. J. Li and C. R. Chen, Stability of weak vector variational inequality, Nonlinear Anal., 70 (2009), 1528-1535. doi: 10.1016/j.na.2008.02.032. [23] S. J. Li and Z. M. Fang, Lower semicontinuity of the solution mappings to a parametric generalized Ky Fan inequality, J. Optim. Theory Appl., 147 (2010), 507-515. doi: 10.1007/s10957-010-9736-8. [24] X. J. Long and J. W. Peng, Generalized B-well-posedness for set optimization problems, J. Optim. Theory Appl., 157 (2013), 612-623. doi: 10.1007/s10957-012-0205-4. [25] J. Liu, J. W. Chen, W. Y. Zhang and C. F. Wen, Scalarization and pointwise Levitin-Polyak well-posedness for set optimization problems, J. Nonlinear Convex Anal., 18 (2017), 1023-1040. [26] Q. L. Wang and S. J. Li, Lower semicontinuity of the solution mapping to a parametric generalized vector equilibrium problem, J. Ind. Manag. Optim., 10 (2014), 1225-1234. doi: 10.3934/jimo.2014.10.1225. [27] Y. D. Xu and S. J. Li, Continuity of the solution set mappings to a parametric set optimization problem, Optim. Lett., 8 (2014), 2315-2327. doi: 10.1007/s11590-014-0738-6. [28] Y. D. Xu and S. J. Li, On the solution continuity of parametric set optimization problems, Math. Meth. Oper. Res., 84 (2016), 223-237. doi: 10.1007/s00186-016-0541-5. [29] W. Y. Zhang, S. J. Li and K. L. Teo, Well-posedness for set optimization problems, Nonlinear Anal., 71 (2009), 3769-3778. doi: 10.1016/j.na.2009.02.036. [30] J. Zhao, The lower semicontinuity of optimal solution sets, J. Math. Anal. Appl., 207 (1997), 240-254. doi: 10.1006/jmaa.1997.5288.
The strict $K$-quasiconvexity of $F(\cdot,\lambda)$ with $\lambda = 0.1$
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