×
Han, Yu

Connectedness of weak minimal solution set for set optimization problems. (English) Zbl 1525.49027

Oper. Res. Lett. 48, No. 6, 820-826 (2020).
Summary: This paper aims at investigating connectedness of weak \(p\)-minimal solution set for set optimization problems by using the scalarization method. We make a new attempt to derive a scalarization result for weak \(p\)-minimal solution set for set optimization problems. By using the scalarization result, we establish connectedness of weak \(p\)-minimal solution set for set optimization problems.

Cited in 12 Documents

MSC:

49K40 Sensitivity, stability, well-posedness
49J53 Set-valued and variational analysis
90C29 Multi-objective and goal programming
90C31 Sensitivity, stability, parametric optimization
90C48 Programming in abstract spaces

Keywords:

set optimization problem; connectedness; scalarization; weak minimal solution
Cite Review PDF
Full Text: DOI

References:

[1] Alonso, M.; Rodríguez-Marín, L., Set-relations and optimality conditions in set-valued maps, Nonlinear Anal., 63, 1167-1179 (2005) · Zbl 1091.90090
[2] Crespi, G. P.; Dhingra, M.; Lalitha, C. S., Pointwise and global well-posedness in set optimization: a direct approach, Ann. Oper. Res., 269, 149-166 (2018) · Zbl 1476.90296
[3] Gong, X. H., Connectedness of efficient solution sets for set-valued maps in normed spaces, J. Optim. Theory Appl., 83, 83-96 (1994) · Zbl 0845.90104
[4] Gong, X. H., Connectedness of the solution sets and scalarization for vector equilibrium problems, J. Optim. Theory Appl., 133, 151-161 (2007) · Zbl 1155.90018
[5] Gong, X. H.; Yao, J. C., Connectedness of the set of efficient solutions for generalized systems, J. Optim. Theory Appl., 138, 189-196 (2008) · Zbl 1302.49011
[6] Göpfert, A.; Riahi, H.; Tammer, C.; Zălinescu, C., Variational Methods in Partially Ordered Spaces (2003), Springer: Springer Berlin Heidelberg/New York · Zbl 1140.90007
[7] Gutiérrez, C.; Miglierina, E.; Molho, E.; Novo, V., Pointwise well-posedness in set optimization with cone proper sets, Nonlinear Anal., 75, 1822-1833 (2012) · Zbl 1237.49024
[8] Han, Y., Nonlinear scalarizing functions in set optimization problems, Optimization, 68, 1685-1718 (2019) · Zbl 1431.90142
[9] Han, Y.; Huang, N. J., The connectedness of the solutions set for generalized vector equilibrium problems, Optimization, 65, 357-367 (2016) · Zbl 1334.49029
[10] Han, Y.; Huang, N. J., Some characterizations of the approximate solutions to generalized vector equilibrium problems, J. Ind. Manag. Optim., 12, 1135-1151 (2016) · Zbl 1328.90147
[11] Han, Y.; Huang, N. J., Well-posedness and stability of solutions for set optimization problems, Optimization, 66, 17-33 (2017) · Zbl 1365.90233
[12] Han, Y.; Huang, N. J., Continuity and convexity of a nonlinear scalarizing function in set optimization problems with applications, J. Optim. Theory Appl., 177, 679-695 (2018) · Zbl 1409.90170
[13] Han, Y.; Huang, N. J., Existence and connectedness of solutions for generalized vector quasi-equilibrium problems, J. Optim. Theory Appl., 179, 65-85 (2018) · Zbl 1409.49009
[14] Han, Y.; Huang, N. J.; Yao, J. C., Connectedness and stability of the approximate solutions to generalized vector quasi-equilibrium problems, J. Nonlinear Convex Anal., 18, 1079-1101 (2017) · Zbl 1383.49018
[15] Han, Y.; Wang, S. H.; Huang, N. J., Arcwise connectedness of the solution sets for set optimization problems, Oper. Res. Lett., 47, 168-172 (2019) · Zbl 1476.90349
[16] Han, Y.; Zhang, K.; Huang, N. J., The stability and extended well-posedness of the solution sets for set optimization problems via the Painlevé-Kuratowski convergence, Math. Methods Oper. Res., 91, 175-196 (2020) · Zbl 1433.49038
[17] Hernández, E.; Rodríguez-Marín, L., Existence theorems for set optimization problems, Nonlinear Anal., 67, 1276-1736 (2007) · Zbl 1278.90379
[18] Hernández, E.; Rodríguez-Marín, L., Nonconvex scalarization in set optimization with set-valued maps, J. Math. Anal. Appl., 325, 1-18 (2007) · Zbl 1110.90080
[19] Hiriart-Urruty, J. B., Images of connected sets by semicontinuous multifunctions, J. Math. Anal. Appl., 111, 407-422 (1985) · Zbl 0578.54013
[20] Karuna; Lalitha, C. S., External and internal stability in set optimization, Optimization, 68, 833-852 (2019) · Zbl 1411.49010
[21] Khan, A. A.; Tammer, C.; Zălinescu, C., Set-Valued Optimization (2015), Springer: Springer Heidelberg · Zbl 1308.49004
[22] Khoshkhabar-amiranloo, S., Characterizations of generalized Levitin-Polyak well-posed set optimization problems, Optim. Lett., 13, 147-161 (2019) · Zbl 1448.90099
[23] Kuroiwa, D., Existence of efficient points of set optimization with weighted criteria, J. Nonlinear Convex Anal., 4, 117-123 (2003) · Zbl 1038.90070
[24] Kuroiwa, D., Existence theorems of set optimization with set-valued maps, J. Inf. Optim. Sci., 24, 73-84 (2003) · Zbl 1176.90545
[25] Kuroiwa, D.; Tanaka, T.; Ha, T. X.D., On cone convexity of set-valued maps, Nonlinear Anal., 30, 1487-1496 (1997) · Zbl 0895.26010
[26] Kuwano, I.; Tanaka, T.; Yamada, S., Characterization of nonlinear scalarizing functions for set-valued maps, (Akashi, S.; Takahashi, W.; Tanaka, T., Nonlinear Analysis and Optimization (2008), Yokohama Publishers: Yokohama Publishers Yokohama), 193-204
[27] Li, Z. F.; Wang, S. Y., Connectedness of super efficient sets in vector optimization of set-valued maps, Math. Methods Oper. Res., 48, 207-217 (1998) · Zbl 0928.90081
[28] Long, X. J.; Peng, J. W.; Peng, Z. Y., Scalarization and pointwise well-posedness for set optimization problems, J. Global Optim., 62, 763-773 (2015) · Zbl 1353.90139
[29] Luc, D. T., (Theory of Vector Optimization. Theory of Vector Optimization, Lecture Notes in Economics and Mathematical Systems, vol. 319 (1989), Springer: Springer Berlin)
[30] Mao, J. Y.; Wang, S. H.; Han, Y., The stability of the solution sets for set optimization problems via improvement sets, Optimization, 68, 2171-2193 (2019)
[31] Peng, Z. Y.; Wang, Z. Y.; Yang, X. M., Connectedness of solution sets for weak generalized symmetric Ky Fan inequality problems via addition-invariant sets, J. Optim. Theory Appl., 185, 188-206 (2020) · Zbl 1436.49021
[32] Qiu, Q. S.; Yang, X. M., Connectedness of Henig weakly efficient solution set for set-valued optimization problems, J. Optim. Theory Appl., 152, 439-449 (2012) · Zbl 1250.90079
[33] Sun, E. J., On the connectedness of the efficient set for strictly quasiconvex vector minimization problems, J. Optim. Theory Appl., 89, 475-481 (1996) · Zbl 0851.90108
[34] Xu, Y. D.; Li, S. J., Continuity of the solution set mappings to a parametric set optimization problem, Optim. Lett., 8, 2315-2327 (2014) · Zbl 1332.90290
[35] Zhang, C. L.; Huang, N. J., On the stability of minimal solutions for parametric set optimization problems, Appl. Anal. (2019)
[36] Zhang, W. Y.; Li, S. J.; Teo, K. L., Well-posedness for set optimization problems, Nonlinear Anal., 71, 3769-3778 (2009) · Zbl 1165.90680
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.