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Notes on Lipschitz properties of nonlinear scalarization functions with applications. (English) Zbl 1474.90413

Summary: Various kinds of nonlinear scalarization functions play important roles in vector optimization. Among them, the one commonly known as the Gerstewitz function is good at scalarizing. In linear normed spaces, the globally Lipschitz property of such function is deduced via primal and dual spaces approaches, respectively. The equivalence of both expressions for globally Lipschitz constants obtained by primal and dual spaces approaches is established. In particular, when the ordering cone is polyhedral, the expression for calculating Lipschitz constant is given. As direct applications of the Lipschitz property, several sufficient conditions for Hölder continuity of both single-valued and set-valued solution mappings to parametric vector equilibrium problems are obtained using the nonlinear scalarization approach.

MSC:

90C29 Multi-objective and goal programming
49K40 Sensitivity, stability, well-posedness
90C48 Programming in abstract spaces

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[1] Jahn, J., Scalarization in vector optimization, Mathematical Programming, 29, 2, 203-218 (1984) · Zbl 0539.90093 · doi:10.1007/BF02592221
[2] Jahn, J., Vector Optimization-Theory, Applications, and Extensions (2004), Berlin, Germany: Springer, Berlin, Germany · Zbl 1055.90065
[3] Zaffaroni, A., Degrees of efficiency and degrees of minimality, SIAM Journal on Control and Optimization, 42, 3, 1071-1086 (2003) · Zbl 1046.90084 · doi:10.1137/S0363012902411532
[4] Eichfelder, G., An adaptive scalarization method in multiobjective optimization, SIAM Journal on Optimization, 19, 4, 1694-1718 (2009) · Zbl 1187.90252 · doi:10.1137/060672029
[5] Flores-Bazán, F.; Hernández, E., A unified vector optimization problem: complete scalarizations and applications, Optimization, 60, 12, 1399-1419 (2011) · Zbl 1266.90162 · doi:10.1080/02331934.2011.641018
[6] Hiriart-Urruty, J.-B., Tangent cones, generalized gradients and mathematical programming in Banach spaces, Mathematics of Operations Research, 4, 1, 79-97 (1979) · Zbl 0409.90086 · doi:10.1287/moor.4.1.79
[7] Göpfert, A.; Riahi, H.; Tammer, C.; Zălinescu, C., Variational Methods in Partially Ordered Spaces (2003), New York, NY, USA: Springer, New York, NY, USA · Zbl 1140.90007
[8] Tammer, C.; Zălinescu, C., Lipschitz properties of the scalarization function and applications, Optimization, 59, 2, 305-319 (2010) · Zbl 1211.90220 · doi:10.1080/02331930801951033
[9] Chen, G.-y.; Huang, X.; Yang, X., Vector Optimization: Set-Valued and Variational Analysis (2005), Berlin, Germany: Springer, Berlin, Germany · Zbl 1104.90044
[10] Luc, D. T., Theory of Vector Optimization (1989), Berlin, Germany: Springer, Berlin, Germany
[11] Gerstewitz, C., Nichtkonvexe Dualität in der Vektoroptimierung, Leuna-Merseburg, 25, 3, 357-364 (1983) · Zbl 0548.90081
[12] Pascoletti, A.; Serafini, P., Scalarizing vector optimization problems, Journal of Optimization Theory and Applications, 42, 4, 499-524 (1984) · Zbl 0505.90072 · doi:10.1007/BF00934564
[13] Gerth, C.; Weidner, P., Nonconvex separation theorems and some applications in vector optimization, Journal of Optimization Theory and Applications, 67, 2, 297-320 (1990) · Zbl 0692.90063 · doi:10.1007/BF00940478
[14] Durea, M.; Tammer, C., Fuzzy necessary optimality conditions for vector optimization problems, Optimization, 58, 4, 449-467 (2009) · Zbl 1162.90597 · doi:10.1080/02331930701761615
[15] Chen, C. R.; Li, M. H., Hölder continuity of solutions to parametric vector equilibrium problems with nonlinear scalarization, Numerical Functional Analysis and Optimization, 35, 6, 685-707 (2014) · Zbl 1290.49073 · doi:10.1080/01630563.2013.818549
[16] Chen, C. R., Hölder continuity of the unique solution to parametric vector quasiequilibrium problems via nonlinear scalarization, Positivity, 17, 1, 133-150 (2013) · Zbl 1266.49043 · doi:10.1007/s11117-011-0153-5
[17] Li, M. H.; Li, S. J.; Chen, C. R., Hölder-likeness and contingent derivative of solutions to parametric weak vector equilibrium problems, Scientia Sinica. Series A, Mathematical, Physical, Astronomical and Technical Sciences, 43, 61-74 (2013) · Zbl 1488.90208
[18] Nam, N. M.; Zălinescu, C., Variational analysis of directional minimal time functions and applications to location problems, Set-Valued and Variational Analysis, 21, 2, 405-430 (2013) · Zbl 1321.49028 · doi:10.1007/s11228-013-0232-9
[19] Giannessi, F., Vector Variational Inequalities and Vector Equilibria: Mathematical Theories (2000), Dordrecht, The Netherlands: Kluwer Academic Publishers, Dordrecht, The Netherlands · Zbl 0952.00009
[20] Ansari, Q. H.; Yao, J. C., Recent Developments in Vector Optimization (2012), Berlin, Germany: Springer, Berlin, Germany · Zbl 1223.90006
[21] Anh, L. Q.; Khanh, P. Q., On the Hölder continuity of solutions to parametric multivalued vector equilibrium problems, Journal of Mathematical Analysis and Applications, 321, 1, 308-315 (2006) · Zbl 1104.90041 · doi:10.1016/j.jmaa.2005.08.018
[22] Anh, L. Q.; Khanh, P. Q., Uniqueness and Hölder continuity of the solution to multivalued equilibrium problems in metric spaces, Journal of Global Optimization, 37, 3, 449-465 (2007) · Zbl 1156.90025 · doi:10.1007/s10898-006-9062-8
[23] Anh, L. Q.; Khanh, P. Q., Sensitivity analysis for multivalued quasiequilibrium problems in metric spaces: Hölder continuity of solutions, Journal of Global Optimization, 42, 4, 515-531 (2008) · Zbl 1188.90274 · doi:10.1007/s10898-007-9268-4
[24] Bianchi, M.; Pini, R., Sensitivity for parametric vector equilibria, Optimization, 55, 3, 221-230 (2006) · Zbl 1149.90156 · doi:10.1080/02331930600662732
[25] Li, X. B.; Long, X. J.; Zeng, J., Hölder continuity of the solution set of the Ky Fan inequality, Journal of Optimization Theory and Applications, 158, 2, 397-409 (2013) · Zbl 1272.90113 · doi:10.1007/s10957-012-0249-5
[26] Li, S. J.; Li, X. B.; Teo, K. L., The Hölder continuity of solutions to generalized vector equilibrium problems, European Journal of Operational Research, 199, 2, 334-338 (2009) · Zbl 1176.90643 · doi:10.1016/j.ejor.2008.12.024
[27] Li, S. J.; Li, X. B., Hölder continuity of solutions to parametric weak generalized Ky Fan inequality, Journal of Optimization Theory and Applications, 149, 3, 540-553 (2011) · Zbl 1229.90214 · doi:10.1007/s10957-011-9803-9
[28] Li, S. J.; Chen, C. R.; Li, X. B.; Teo, K. L., Hölder continuity and upper estimates of solutions to vector quasiequilibrium problems, European Journal of Operational Research, 210, 2, 148-157 (2011) · Zbl 1236.90113 · doi:10.1016/j.ejor.2010.10.005
[29] Chen, C. R.; Li, S. J.; Zeng, J.; Li, X. B., Error analysis of approximate solutions to parametric vector quasiequilibrium problems, Optimization Letters, 5, 1, 85-98 (2011) · Zbl 1213.90260 · doi:10.1007/s11590-010-0192-z
[30] Chen, C.; Li, S., Upper Hölder estimates of solutions to parametric primal and dual vector quasi-equilibria, Journal of Industrial and Management Optimization, 8, 3, 691-703 (2012) · Zbl 1287.49040 · doi:10.3934/jimo.2012.8.691
[31] Anh, L. Q.; Khanh, P. Q.; Tam, T. N., On Hölder continuity of approximate solutions to parametric equilibrium problems, Nonlinear Analysis: Theory, Methods & Applications A: Theory and Methods, 75, 4, 2293-2303 (2012) · Zbl 1237.49032 · doi:10.1016/j.na.2011.10.029
[32] Li, X. B.; Li, S. J.; Chen, C. R., Lipschitz continuity of an approximate solution mapping to equilibrium problems, Taiwanese Journal of Mathematics, 16, 3, 1027-1040 (2012) · Zbl 1245.26004
[33] Chieu, N. H.; Yao, J.-C.; Yen, N. D., Relationships between Robinson metric regularity and Lipschitz-like behavior of implicit multifunctions, Nonlinear Analysis: Theory, Methods & Applications A: Theory and Methods, 72, 9-10, 3594-3601 (2010) · Zbl 1186.49015 · doi:10.1016/j.na.2009.12.039
[34] Yen, N. D.; Yao, J.-C.; Kien, B. T., Covering properties at positive-order rates of multifunctions and some related topics, Journal of Mathematical Analysis and Applications, 338, 1, 467-478 (2008) · Zbl 1137.47038 · doi:10.1016/j.jmaa.2007.05.041
[35] Chen, C. R.; Li, S. J., Semicontinuity results on parametric vector variational inequalities with polyhedral constraint sets, Journal of Optimization Theory and Applications, 158, 1, 97-108 (2013) · Zbl 1272.90096 · doi:10.1007/s10957-012-0199-y
[36] Sach, P. H.; Tuan, L. A., New scalarizing approach to the stability analysis in parametric generalized Ky Fan inequality problems, Journal of Optimization Theory and Applications, 157, 2, 347-364 (2013) · Zbl 1283.90041 · doi:10.1007/s10957-012-0105-7
[37] Sach, P. H., New nonlinear scalarization functions and applications, Nonlinear Analysis: Theory, Methods & Applications A: Theory and Methods, 75, 4, 2281-2292 (2012) · Zbl 1267.90156 · doi:10.1016/j.na.2011.10.028
[38] Lalitha, C. S.; Bhatia, G., Stability of parametric quasivariational inequality of the Minty type, Journal of Optimization Theory and Applications, 148, 2, 281-300 (2011) · Zbl 1233.90263 · doi:10.1007/s10957-010-9755-5
[39] Gutiérrez, C.; Jiménez, B.; Novo, V., New second-order directional derivative and optimality conditions in scalar and vector optimization, Journal of Optimization Theory and Applications, 142, 1, 85-106 (2009) · Zbl 1206.90210 · doi:10.1007/s10957-009-9525-4
[40] Sawaragi, Y.; Nakayama, H.; Tanino, T., Theory of Multiobjective Optimization (1985), Orlando, Fla, USA: Academic Press, Orlando, Fla, USA · Zbl 0566.90053
[41] Li, X. B.; Li, S. J., Continuity of approximate solution mappings for parametric equilibrium problems, Journal of Global Optimization, 51, 3, 541-548 (2011) · Zbl 1229.90235 · doi:10.1007/s10898-010-9641-6
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