Semicontinuity of the minimal solution set mappings for parametric set-valued vector optimization problems. (English) Zbl 1488.90180
Summary: With the help of a level mapping, this paper mainly investigates the semicontinuity of minimal solution set mappings for set-valued vector optimization problems. First, we introduce a kind of level mapping which generalizes one given in [Y. Han and X. Gong, Optimization 65, No. 7, 1337–1347 (2016; Zbl 1346.90680)]. Then, we give a sufficient condition for the upper semicontinuity and the lower semicontinuity of the level mapping. Finally, in terms of the semicontinuity of the level mapping, we establish the upper semicontinuity and the lower semicontinuity of the minimal solution set mapping to parametric set-valued vector optimization problems under the C-Hausdorff continuity instead of the continuity in the sense of Berge.
MSC:
| 90C29 | Multi-objective and goal programming |
| 90C31 | Sensitivity, stability, parametric optimization |
| 49J40 | Variational inequalities |
| 49K40 | Sensitivity, stability, well-posedness |
Keywords:
set-valued vector optimization problems; level mapping; solution set mapping; upper semicontinuity; lower semicontinuityCitations:
Zbl 1346.90680References:
| [1] | Luc, DT, Lecture Notes in Economics and Mathematical System. Theory of Vector Optimization (1989), Berlin: Springer, Berlin · Zbl 0695.90002 · doi:10.1007/978-3-642-50280-4 |
| [2] | Sawaragi, Y.; Makayama, H.; Tanino, T., Theory of Multiobjective Optimization (1985), New York: Academic Press, New York · Zbl 0566.90053 |
| [3] | Lucchetti, RE; Miglierina, E., Stability for convex vector optimization problems, Optimization, 53, 517-528 (2004) · Zbl 1153.90536 · doi:10.1080/02331930412331327166 |
| [4] | Huang, XX, Stability in vector-valued and set-valued optimization, Math. Methods Oper. Res., 52, 185-193 (2000) · Zbl 1026.49011 · doi:10.1007/s001860000085 |
| [5] | Huang, XX; Yang, XQ, On characterizations of proper efficiency for nonconvex multiobjective optimization, J. Global Optim., 23, 213-231 (2002) · Zbl 1026.90079 · doi:10.1023/A:1016522528364 |
| [6] | Lalitha, C.; Chatterjee, P., Stability and scalarization of weak efficient, efficient and henig proper efficient sets using generalized quasiconvexities, J. Optim. Theory Appl., 155, 941-961 (2012) · Zbl 1272.90079 · doi:10.1007/s10957-012-0106-6 |
| [7] | Anh, LQ; Hung, NV, On the stability of solution mappings parametric generalized vector quasivariational inequality problems of the Minty type, Filomat, 31, 747-757 (2017) · Zbl 1488.90196 · doi:10.2298/FIL1703747A |
| [8] | Anh, LQ; Hung, NV, Stability of solution mappings for parametric bilevel vector equilibrium problems, Comput. Appl. Math., 37, 1537-1549 (2018) · Zbl 1433.49008 · doi:10.1007/s40314-016-0411-z |
| [9] | Hung, NV; Hai, NM, Stability of approximating solutions to parametric bilevel vector equilibrium problems and applications, Comput. Appl. Math., 38, 57 (2019) · Zbl 1438.90346 · doi:10.1007/s40314-019-0823-7 |
| [10] | Hung, NV, On the stability of the solution mapping for parametric traffic network problems, Indagat. Math., 29, 885-894 (2018) · Zbl 1394.90179 · doi:10.1016/j.indag.2018.01.007 |
| [11] | Cheng, YH; Zhu, DL, Global stability for the weak vector variational inequality, J. Global Optim., 32, 543-550 (2005) · Zbl 1097.49006 · doi:10.1007/s10898-004-2692-9 |
| [12] | Gong, XH, Continuity of the solution set to parametric weak vector equilibrium problems, J. Optim. Theory Appl., 139, 35-46 (2008) · Zbl 1189.90195 · doi:10.1007/s10957-008-9429-8 |
| [13] | Gong, XH; Yao, JC, Lower semicontinuity of the set of efficient solutions for generalized systems, J. Optim. Theory Appl., 138, 197-205 (2008) · Zbl 1302.49018 · doi:10.1007/s10957-008-9379-1 |
| [14] | Chen, CR; Li, SJ, On the solution continuity of parametric generalized systems, Pac. J. Optim., 6, 141-151 (2010) · Zbl 1190.49032 |
| [15] | Han, Y.; Gong, XH, Semicontinuity of solution mappings to parametric generalized vector equilibrium problems, Numer. Func. Anal. Opt., 37, 1420-1437 (2016) · Zbl 1356.49038 · doi:10.1080/01630563.2016.1216446 |
| [16] | Han, Y.; Huang, NJ, Stability of efficient solutions to parametric generalized vector equilibrium problems, Sci. Sin. Math., 47, 397-408 (2017) · Zbl 1499.90245 · doi:10.1360/012016-12 |
| [17] | Han, Y.; Huang, NJ, Some characterizations of the approximate solutions to generalized vector equilibrium problems, J. Ind. Manag. Optim., 12, 1135-1151 (2016) · Zbl 1328.90147 · doi:10.3934/jimo.2016.12.1135 |
| [18] | Wang, QL; Li, XB; Zeng, J., Semicontinuity of approximate solution mappings for parametric generalized weak vector equilibrium problems, J. Nonlinear Sci. Appl., 10, 2678-2688 (2017) · Zbl 1412.49056 · doi:10.22436/jnsa.010.05.34 |
| [19] | Sach, PH; Tuan, LA, New scalarizing approach to the stability analysis in parametric generalized Ky Fan inequality problems, J. Optim. Theory Appl., 157, 347-364 (2013) · Zbl 1283.90041 · doi:10.1007/s10957-012-0105-7 |
| [20] | Anh, LQ; Bantaojai, T.; Hung, NV; Tam, VM; Wangkeeree, R., Painlevé-Kuratowski convergences of the solution sets for generalized vector quasi-equilibrium problems, Comput. Appl. Math., 37, 3832-3845 (2018) · Zbl 1406.49013 · doi:10.1007/s40314-017-0548-4 |
| [21] | Anh, LQ; Hung, NV, Gap functions and Hausdorff continuity of solution mappings to parametric strong vector quasiequilibrium problems, J. Ind. Manag. Optim., 14, 65-79 (2018) · Zbl 1412.90145 |
| [22] | Hung, NV, On the lower semicontinuity of the solution sets for parametric generalized vector mixed quasivariational inequality problems, Bull. Korean Math. Soc., 52, 1777-1795 (2015) · Zbl 1357.90155 · doi:10.4134/BKMS.2015.52.6.1777 |
| [23] | Hung, NV, Stability of a solution set for parametric generalized vector mixed quasivariational inequality problem, J. Inequal. Appl., 276, 1-14 (2013) · Zbl 1282.90186 |
| [24] | Han, Y.; Gong, XH, Continuity of the efficient solution mapping for vector optimization problem, Optimization, 65, 1337-1347 (2016) · Zbl 1346.90680 · doi:10.1080/02331934.2016.1154554 |
| [25] | Xu, YD; Li, SJ, On the solution continuity of parametric set optimization problems, Math. Methods Oper. Res., 84, 223-237 (2016) · Zbl 1356.90144 · doi:10.1007/s00186-016-0541-5 |
| [26] | Khoshkhabar-amiranloo, S., Stability of minimal solutions to parametric set optimization problems, Appl. Anal., 97, 1-13 (2018) · Zbl 1416.49022 · doi:10.1080/00036811.2017.1376320 |
| [27] | Guu, SM; Huang, NJ; Li, J., Scalarization approaches for set-valued vector optimization problem and vector variational inequalities, J. Math. Anal. Appl., 356, 564-576 (2009) · Zbl 1176.90538 · doi:10.1016/j.jmaa.2009.03.040 |
| [28] | Benoist, J.; Popovici, N., Characterizations of convex and quasiconvex set-valued maps, Math. Methods Oper. Res., 57, 427-435 (2003) · Zbl 1047.54012 · doi:10.1007/s001860200260 |
| [29] | Aubin, JP; Ekeland, I., Applied Nonlinear Analysis (1984), New York: Wiley, New York · Zbl 0641.47066 |
| [30] | Karaman, S.; Soyertem, M.; Güvenç, IA; Tozkan, D.; Küçük, M.; Küçük, Y., Partail order relations on family of sets and scalarizations for set optimization, Positivity, 22, 783-802 (2018) · Zbl 1426.90207 · doi:10.1007/s11117-017-0544-3 |
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