The connectedness of the solutions set for set-valued vector equilibrium problems under improvement sets. (English) Zbl 1503.90127
Summary: In this paper, we provide the connectedness of the sets of weak efficient solutions, Henig efficient solutions and Benson proper efficient solutions for set-valued vector equilibrium problems under improvement sets.
MSC:
| 90C29 | Multi-objective and goal programming |
| 49J40 | Variational inequalities |
| 90C26 | Nonconvex programming, global optimization |
| 90C31 | Sensitivity, stability, parametric optimization |
| 49J53 | Set-valued and variational analysis |
References:
| [1] | Chicco, M.; Mignanego, F.; Pusillo, L.; Tijs, S., Vector optimization problems via improvement sets, J. Optim. Theory Appl., 150, 3, 516-529 (2011) · Zbl 1231.90329 · doi:10.1007/s10957-011-9851-1 |
| [2] | Gutierrez, C.; Jimenez, B.; Novo, V., Improvement sets and vector optimization, Eur. J. Oper. Res., 223, 2, 304-311 (2012) · Zbl 1292.90268 · doi:10.1016/j.ejor.2012.05.050 |
| [3] | Zhao, K. Q.; Yang, X. M.; Peng, J. W., Weak E-optimal solution in vector optimization, Taiwan. J. Math., 17, 4, 1287-1302 (2013) · Zbl 1304.90191 · doi:10.11650/tjm.17.2013.2721 |
| [4] | Zhao, K. Q.; Xia, Y. M.; Guo, H., A characterization of E-Benson proper efficiency via nonlinear scalarization in vector optimization, J. Appl. Math., 2014 (2014) · Zbl 1442.90176 |
| [5] | Zhao, K. Q.; Yang, X. M., E-Benson proper efficiency in vector optimization, Optimization, 64, 4, 739-752 (2015) · Zbl 1310.90094 · doi:10.1080/02331934.2013.798321 |
| [6] | Zhao, K. Q.; Xia, Y. M.; Yang, X. M., Nonlinear scalarization characterizations of E-efficiency in vector optimization, Taiwan. J. Math., 19, 2, 455-466 (2015) · Zbl 1357.90147 · doi:10.11650/tjm.19.2015.4360 |
| [7] | Zhao, K. Q.; Yang, X. M., Characterizations of the E-Benson proper efficiency in vector optimization problems, Numer. Algebra Control Optim., 3, 4, 643-653 (2013) · Zbl 1275.49061 · doi:10.3934/naco.2013.3.643 |
| [8] | Zhou, Z. A.; Yang, X. M.; Zhao, K. Q., E-super efficiency of set-valued optimization problems involving improvement sets, J. Ind. Manag. Optim., 12, 3, 1031-1039 (2016) · Zbl 1328.90118 · doi:10.3934/jimo.2016.12.1031 |
| [9] | Lalitha, C. S.; Chatterjee, P., Stability and scalarization in vector optimization using improvement sets, J. Optim. Theory Appl., 166, 3, 825-843 (2015) · Zbl 1342.49035 · doi:10.1007/s10957-014-0686-4 |
| [10] | Chicco, M.; Rossi, A., Existence of optimal points via improvement sets, J. Optim. Theory Appl., 167, 2, 487-501 (2015) · Zbl 1327.90230 · doi:10.1007/s10957-015-0744-6 |
| [11] | Zhou, Z. A.; Yang, X. M., Optimality conditions of generalized subconvexlike set-valued optimization problems based on the quasi-relativeinterior, J. Optim. Theory Appl., 150, 327-340 (2011) · Zbl 1229.90210 · doi:10.1007/s10957-011-9844-0 |
| [12] | Zhou, Z. A.; Chen, W.; Yang, X. M., Scalarizations and optimality of constrained set-valued optimization using improvement sets and image space analysis, J. Optim. Theory Appl., 183, 944-962 (2019) · Zbl 1472.90104 · doi:10.1007/s10957-019-01554-3 |
| [13] | Chen, C. R.; Zuo, X.; Lu, F.; Li, S. J., Vector equilibrium problems under improvement sets and linear scalarization with stability applications, Optim. Methods Softw., 31, 6, 1240-1257 (2016) · Zbl 1351.49009 · doi:10.1080/10556788.2016.1200043 |
| [14] | Giannessi, F., Vector Variational Inequalities and Vector Equilibria: Mathematical Theories (2000), Dordrecht: Kluwer Academic, Dordrecht · Zbl 0952.00009 · doi:10.1007/978-1-4613-0299-5 |
| [15] | Chen, G. Y.; Huang, X. X.; Yang, X. Q., Vector Optimization: Set-Valued and Variational Analysis (2005), Berlin: Springer, Berlin · Zbl 1104.90044 |
| [16] | Ansari, Q. H.; Yao, J. C., Recent Developments in Vector Optimization (2012), Berlin: Springer, Berlin · Zbl 1223.90006 · doi:10.1007/978-3-642-21114-0 |
| [17] | Lee, G. M.; Kim, D. S.; Lee, B. S.; Yun, N. D., Vector variational inequality as a tool for studying vector optimization problems, Nonlinear Anal., 34, 5, 745-765 (1998) · Zbl 0956.49007 · doi:10.1016/S0362-546X(97)00578-6 |
| [18] | Cheng, Y. H., On the connectedness of the solution set for the weak vector variational inequality, J. Math. Anal. Appl., 260, 1, 1-5 (2001) · Zbl 0990.49010 · doi:10.1006/jmaa.2000.7389 |
| [19] | Gong, X. H., Efficiency and Henig efficiency for vector equilibrium problems, J. Optim. Theory Appl., 108, 1, 139-154 (2001) · Zbl 1033.90119 · doi:10.1023/A:1026418122905 |
| [20] | Gong, X. H., Connectedness of the solution sets and scalarization for vector equilibrium problems, J. Optim. Theory Appl., 133, 2, 151-161 (2007) · Zbl 1155.90018 · doi:10.1007/s10957-007-9196-y |
| [21] | Gong, X. H.; Yao, J. C., Connectedness of the set of efficient solutions for generalized systems, J. Optim. Theory Appl., 138, 2, 189-196 (2008) · Zbl 1302.49011 · doi:10.1007/s10957-008-9378-2 |
| [22] | Chen, B.; Gong, X. H.; Yuan, S. M., Connectedness and compactness of weak efficient solutions set for set-valued vector equilibrium problems, J. Inequal. Appl., 2008 (2008) · Zbl 1157.49009 · doi:10.1155/2008/581849 |
| [23] | Chen, B.; Liu, Q. Y.; Liu, Z. B.; Huang, N. J., Connectedness of approximate solutions set for vector equilibrium problems in Hausdorff topological vector spaces, Fixed Point Theory Appl., 2011 (2011) · Zbl 1315.47055 · doi:10.1186/1687-1812-2011-36 |
| [24] | Han, Y.; Huang, N. J., The connectedness of the solutions set for generalized vector equilibrium problems, Optimization, 65, 2, 357-367 (2016) · Zbl 1334.49029 · doi:10.1080/02331934.2015.1044899 |
| [25] | Han, Y.; Huang, N. J., Some characterizations of the approximate solutions to generalized vector equilibrium problems, J. Ind. Manag. Optim., 12, 3, 1135-1151 (2016) · Zbl 1328.90147 · doi:10.3934/jimo.2016.12.1135 |
| [26] | Qiu, Q. S., Henig efficiency in vector optimization with nearly cone-subconvexlike set-valued functions, Acta Math. Appl. Sin., 23, 2, 319-328 (2007) · Zbl 1198.90329 · doi:10.1007/s10255-007-0374-3 |
| [27] | Jahn, J., Mathematical Vector Optimization in Partially-Ordered Linear Spaces (1986), Frankfurt am Main: Peter Lang, Frankfurt am Main · Zbl 0578.90048 |
| [28] | Cheng, Y. H.; Fu, W. T., Strong efficiency in a locally convex space, Math. Methods Oper. Res., 50, 3, 373-384 (1999) · Zbl 0947.90104 · doi:10.1007/s001860050076 |
| [29] | Song, W., Lagrangian duality for minimization of nonconvex multifunctions, J. Optim. Theory Appl., 93, 1, 167-182 (1997) · Zbl 0901.90161 · doi:10.1023/A:1022658019642 |
| [30] | Breckner, W. W.; Kassay, G., A systematization of convexity concepts for sets and functions, J. Convex Anal., 4, 109-127 (1997) · Zbl 0885.52003 |
| [31] | Jahn, J., Vector Optimization. Theory, Applications, and Extensions (2011), Berlin: Springer, Berlin · Zbl 1401.90206 |
| [32] | Gong, X. H.; Fu, W. T.; Liu, W.; Giannessi, F., Super efficiency for a vector equilibrium in locally convex topological vector spaces, Vector Variational Inequalities and Vector Equilibria: Mathematical Theories, 233-252 (2000), Dordrecht: Kluwer Academic, Dordrecht · Zbl 1019.49016 · doi:10.1007/978-1-4613-0299-5_13 |
| [33] | Brown, R., Topology (1988), New York: Ellis Horwood, New York · Zbl 0655.55001 |
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