American Institute of Mathematical Sciences

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January  2020, 16(1): 25-36. doi: 10.3934/jimo.2018138

Continuity of solutions mappings of parametric set optimization problems

Jiawei Chen 1,, , Guangmin Wang 2, , Xiaoqing Ou 3, and  Wenyan Zhang 4,

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].

Keywords: Continuity, parametric set optimization problem, strict lower lever mapping, lower set less order relation.
Mathematics Subject Classification: Primary: 49J53; Secondary: 49K40, 90C30.
Citation: Jiawei Chen, Guangmin Wang, Xiaoqing Ou, Wenyan Zhang. Continuity of solutions mappings of parametric set optimization problems. Journal of Industrial & Management Optimization, 2020, 16 (1) : 25-36. 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.  Google Scholar

[2]

J. P. Aubin and I. Ekeland, Applied Nonlinear Analysis, Wiley. New York, 1984.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[6]

A. Göfert, Chr. Tammer, H. Riahi and C. Z$\check{a}$inescu, Variational Methods in Partially Ordered Spaces, Springer-Verlag, New York, 2003.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

B. T. Kien, On the lower semicontinuity of optimal solution sets, Optim., 54 (2005), 123-130.  doi: 10.1080/02331930412331330379.  Google Scholar

[18]

D. Klatte, A sufficient condition for lower semicontinuity of solution sets of systems of convex inequalities, Math. Program. Stud., 21 (1984), 139-149.   Google Scholar

[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.  Google Scholar

[20]

D. Kuroiwa, On set-valued optimization, Nonlinear Anal., 47 (2001), 1395-1400.  doi: 10.1016/S0362-546X(01)00274-7.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

J. Zhao, The lower semicontinuity of optimal solution sets, J. Math. Anal. Appl., 207 (1997), 240-254.  doi: 10.1006/jmaa.1997.5288.  Google Scholar

show all references

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.  Google Scholar

[2]

J. P. Aubin and I. Ekeland, Applied Nonlinear Analysis, Wiley. New York, 1984.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[6]

A. Göfert, Chr. Tammer, H. Riahi and C. Z$\check{a}$inescu, Variational Methods in Partially Ordered Spaces, Springer-Verlag, New York, 2003.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

B. T. Kien, On the lower semicontinuity of optimal solution sets, Optim., 54 (2005), 123-130.  doi: 10.1080/02331930412331330379.  Google Scholar

[18]

D. Klatte, A sufficient condition for lower semicontinuity of solution sets of systems of convex inequalities, Math. Program. Stud., 21 (1984), 139-149.   Google Scholar

[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.  Google Scholar

[20]

D. Kuroiwa, On set-valued optimization, Nonlinear Anal., 47 (2001), 1395-1400.  doi: 10.1016/S0362-546X(01)00274-7.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

J. Zhao, The lower semicontinuity of optimal solution sets, J. Math. Anal. Appl., 207 (1997), 240-254.  doi: 10.1006/jmaa.1997.5288.  Google Scholar

Figure 1.  The strict $K$-quasiconvexity of $F(\cdot,\lambda)$ with $\lambda = 0.1$
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