# American Institute of Mathematical Sciences

March  2020, 10(1): 107-125. doi: 10.3934/naco.2019036

## Unified vector quasiequilibrium problems via improvement sets and nonlinear scalarization with stability analysis

 1 College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, China 2 Department of Basic, Yinchuan Energy College, Yinchuan, 750105, China

*Corresponding author: Chun-Rong Chen

Received  November 2018 Revised  April 2019 Published  May 2019

Fund Project: This research was supported by the National Natural Science Foundation of China (Grant numbers: 11301567 and 11571055) and the Fundamental Research Funds for the Central Universities (Grant number: 106112017CDJZRPY0020).

This paper has two objectives. The first one is to propose a new vector quasiequilibrium problem where the ordering relation is defined via an improvement set $D$, and its weak version, also their Minty-type dual problems and the corresponding set-valued cases. These models provide unified frameworks to deal with well-known exact and approximate vector quasiequilibrium problems with vector-valued or set-valued mappings. The second one is to study solution stability in the sense of Hölder continuity of the unique solution to parametric unified (resp. weak) vector quasiequilibrium problems, by employing the Gerstewitz scalarization techniques. In particular, we deduce a new stability result for the typical vector optimization problem related with (resp. weak) $D$-optimality, by considering perturbations of both the objective function and the feasible set.

Citation: Hong-Zhi Wei, Xin Zuo, Chun-Rong Chen. Unified vector quasiequilibrium problems via improvement sets and nonlinear scalarization with stability analysis. Numerical Algebra, Control & Optimization, 2020, 10 (1) : 107-125. doi: 10.3934/naco.2019036
##### References:
 [1] L. Q. Anh and P. Q. Khanh, On the Hölder continuity of solutions to parametric multivalued vector equilibrium problems, J. Math. Anal. Appl., 321 (2006), 308-315.   Google Scholar [2] L. Q. Anh and P. Q. Khanh, Uniqueness and Hölder continuity of the solution to multivalued equilibrium problems in metric spaces, J. Global Optim., 37 (2007), 449-465.   Google Scholar [3] L. Q. Anh and P. Q. Khanh, Sensitivity analysis for multivalued quasiequilibrium problems in metric spaces: Hölder continuity of solutions, J. Global Optim., 42 (2008), 515-531.   Google Scholar [4] L. Q. Anh and P. Q. Khanh, Semicontinuity of the approximate solution sets of multivalued quasiequilibrium problems, Numer. Funct. Anal. Optim., 29 (2008), 24-42.   Google Scholar [5] Q. H. Ansari and J.-C. Yao (eds.), Recent Developments in Vector Optimization, Springer, Berlin, 2012. doi: 10.1007/978-3-642-21114-0.  Google Scholar [6] M. Bianchi and R. Pini, Sensitivity for parametric vector equilibria, Optimization, 55 (2006), 221-230.  doi: 10.1080/02331930600662732.  Google Scholar [7] C. R. Chen, L. L. Li and M. H. Li, Hölder continuity results for nonconvex parametric generalized vector quasiequilibrium problems via nonlinear scalarizing functions, Optimization, 65 (2016), 35-51.  doi: 10.1080/02331934.2014.984707.  Google Scholar [8] C. R. Chen, Hölder continuity of the unique solution to parametric vector quasiequilibrium problems via nonlinear scalarization, Positivity, 17 (2013), 133-150.  doi: 10.1007/s11117-011-0153-5.  Google Scholar [9] C. R. Chen and M. H. Li, Hölder continuity of solutions to parametric vector equilibrium problems with nonlinear scalarization, Numer. Funct. Anal. Optim., 35 (2014), 685-707.  doi: 10.1080/01630563.2013.818549.  Google Scholar [10] C. R. Chen, X. Zuo, F. Lu and S. J. Li, Vector equilibrium problems under improvement sets and linear scalarization with stability applications, Optim. Methods Softw., 31 (2016), 1240-1257.  doi: 10.1080/10556788.2016.1200043.  Google Scholar [11] C. R. Chen, S. J. Li, J. Zeng and X. B. Li, Error analysis of approximate solutions to parametric vector quasiequilibrium problems, Optim. Lett., 5 (2011), 85-98.  doi: 10.1007/s11590-010-0192-z.  Google Scholar [12] C. R. Chen, S. J. Li and K. L. Teo, Solution semicontinuity of parametric generalized vector equilibrium problems, J. Global Optim., 45 (2009), 309-318.  doi: 10.1007/s10898-008-9376-9.  Google Scholar [13] G. Y. Chen, X. X. Huang and X. Q. Yang, Vector Optimization: Set-Valued and Variational Analysis, Springer, Berlin, 2005.   Google Scholar [14] M. Chicco, F. Mignanego, L. Pusillo and S. Tijs, Vector optimization problem via improvement sets, J. Optim. Theory Appl., 150 (2011), 516-529.  doi: 10.1007/s10957-011-9851-1.  Google Scholar [15] G. Debreu, Theory of Value: An Axiomatic Analysis of Economic Equilibrium, John Wiley, New York, 1959.   Google Scholar [16] M. Durea and R. Strugariu, Scalarization of constraints system in some vector optimization problems and applications, Optim. Lett., 8 (2014), 2021-2037.  doi: 10.1007/s11590-013-0690-x.  Google Scholar [17] M. Durea and C. Tammer, Fuzzy necessary optimality conditions for vector optimization problems, Optimization, 58 (2009), 449-467.  doi: 10.1080/02331930701761615.  Google Scholar [18] Chr. Gerstewitz (Tammer), Nichtkonvexe Dualität in der Vektoroptimierung, Wiss. Z. TH Leuna-Merseburg, 25 (1983), 357-364.  Google Scholar [19] C. Gerth (Tammer) and P. Weidner, Nonconvex separation theorems and some applications in vector optimization, J. Optim. Theory Appl., 67 (1990), 297-320. doi: 10.1007/BF00940478.  Google Scholar [20] F. Giannessi (ed.), Vector Variational Inequalities and Vector Equilibria: Mathematical Theories, Kluwer Academic Publishers, Dordrecht, 2000. doi: 10.1007/978-1-4613-0299-5.  Google Scholar [21] X. H. Gong and J. C. Yao, Lower semicontinuity of the set of efficient solutions for generalized systems, J. Optim. Theory Appl., 138 (2008), 197-205.  doi: 10.1007/s10957-008-9379-1.  Google Scholar [22] A. Göpfert, H. Riahi, C. Tammer and C. Zălinescu, Variational Methods in Partially Ordered Spaces, Springer, New York, 2003.   Google Scholar [23] C. Gutiérrez, B. Jiménez and V. Novo, Improvement sets and vector optimization, Eur. J. Oper. Res., 223 (2012), 304-311.  doi: 10.1016/j.ejor.2012.05.050.  Google Scholar [24] J.-B. Hiriart-Urruty, Tangent cones, generalized gradients and mathematical programming in Banach spaces, Math. Oper. Res., 4 (1979), 79-97.  doi: 10.1287/moor.4.1.79.  Google Scholar [25] J. Jahn, Vector Optimization-Theory, Applications, and Extensions, 2nd ed., Springer, Berlin, 2011. doi: 10.1007/978-3-540-24828-6.  Google Scholar [26] A. A. Khan, C. Tammer and C. Zălinescu, Set-Valued Optimization, Springer, Berlin, 2015.  doi: 10.1007/978-3-642-54265-7.  Google Scholar [27] K. Kimura and J. C. Yao, Semicontinuity of solution mappings of parametric generalized vector equilibrium problems, J. Optim. Theory Appl., 138 (2008), 429-443.  doi: 10.1007/s10957-008-9386-2.  Google Scholar [28] C. S. Lalitha and P. Chatterjee, Stability and scalarization in vector optimization using improvement sets, J. Optim. Theory Appl., 166 (2015), 825-843.  doi: 10.1007/s10957-014-0686-4.  Google Scholar [29] S. J. Li, C. R. Chen, X. B. Li and K. L. Teo, Hölder continuity and upper estimates of solutions to vector quasiequilibrium problems, Eur. J. Oper. Res., 210 (2011), 148-157.  doi: 10.1016/j.ejor.2010.10.005.  Google Scholar [30] S. J. Li and X. B. Li, Hölder continuity of solutions to parametric weak generalized Ky Fan inequality, J. Optim. Theory Appl., 149 (2011), 540-553.  doi: 10.1007/s10957-011-9803-9.  Google Scholar [31] D. T. Luc, Theory of Vector Optimization, Springer, Berlin, 1989.   Google Scholar [32] N. M. Nam and C. Zălinescu, Variational analysis of directional minimal time functions and applications to location problems, Set-Valued Var. Anal., 21 (2013), 405-430.  doi: 10.1007/s11228-013-0232-9.  Google Scholar [33] P. Oppezzi and A. Rossi, Improvement sets and convergence of optimal points, J. Optim. Theory Appl., 165 (2015), 405-419.  doi: 10.1007/s10957-014-0669-5.  Google Scholar [34] Z. Y. Peng, X. M. Yang and J. W. Peng, On the lower semicontinuity of the solution mappings to parametric weak generalized Ky Fan inequality, J. Optim. Theory Appl., 152 (2012), 256-264.  doi: 10.1007/s10957-011-9883-6.  Google Scholar [35] R. T. Rockafellar and R. J. -B. Wets, Variational Analysis, Springer, Berlin, 1998.   Google Scholar [36] P. H. Sach and L. A. Tuan, New scalarizing approach to the stability analysis in parametric generalized Ky Fan inequality problems, J. Optim. Theory Appl., 157 (2013), 347-364.  doi: 10.1007/s10957-012-0105-7.  Google Scholar [37] C. Tammer and C. Zălinescu, Lipschitz properties of the scalarization function and applications, Optimization, 59 (2010), 305-319.  doi: 10.1080/02331930801951033.  Google Scholar [38] A. Zaffaroni, Degrees of efficiency and degrees of minimality, SIAM J. Control Optim., 42 (2003), 1071-1086.  doi: 10.1137/S0363012902411532.  Google Scholar [39] K. Q. Zhao and X. M. Yang, A unified stability result with perturbations in vector optimization, Optim. Lett., 7 (2013), 1913-1919.  doi: 10.1007/s11590-012-0533-1.  Google Scholar [40] K. Q. Zhao and X. M. Yang, $E$-Benson proper efficiency in vector optimization, Optimization, 64 (2015), 739-752.  doi: 10.1080/02331934.2013.798321.  Google Scholar [41] K. Q. Zhao, G. Y. Chen and X. M. Yang, Approximate proper efficiency in vector optimization, Optimization, 64 (2015), 1777-1793.  doi: 10.1080/02331934.2014.979818.  Google Scholar

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##### References:
 [1] L. Q. Anh and P. Q. Khanh, On the Hölder continuity of solutions to parametric multivalued vector equilibrium problems, J. Math. Anal. Appl., 321 (2006), 308-315.   Google Scholar [2] L. Q. Anh and P. Q. Khanh, Uniqueness and Hölder continuity of the solution to multivalued equilibrium problems in metric spaces, J. Global Optim., 37 (2007), 449-465.   Google Scholar [3] L. Q. Anh and P. Q. Khanh, Sensitivity analysis for multivalued quasiequilibrium problems in metric spaces: Hölder continuity of solutions, J. Global Optim., 42 (2008), 515-531.   Google Scholar [4] L. Q. Anh and P. Q. Khanh, Semicontinuity of the approximate solution sets of multivalued quasiequilibrium problems, Numer. Funct. Anal. Optim., 29 (2008), 24-42.   Google Scholar [5] Q. H. Ansari and J.-C. Yao (eds.), Recent Developments in Vector Optimization, Springer, Berlin, 2012. doi: 10.1007/978-3-642-21114-0.  Google Scholar [6] M. Bianchi and R. Pini, Sensitivity for parametric vector equilibria, Optimization, 55 (2006), 221-230.  doi: 10.1080/02331930600662732.  Google Scholar [7] C. R. Chen, L. L. Li and M. H. Li, Hölder continuity results for nonconvex parametric generalized vector quasiequilibrium problems via nonlinear scalarizing functions, Optimization, 65 (2016), 35-51.  doi: 10.1080/02331934.2014.984707.  Google Scholar [8] C. R. Chen, Hölder continuity of the unique solution to parametric vector quasiequilibrium problems via nonlinear scalarization, Positivity, 17 (2013), 133-150.  doi: 10.1007/s11117-011-0153-5.  Google Scholar [9] C. R. Chen and M. H. Li, Hölder continuity of solutions to parametric vector equilibrium problems with nonlinear scalarization, Numer. Funct. Anal. Optim., 35 (2014), 685-707.  doi: 10.1080/01630563.2013.818549.  Google Scholar [10] C. R. Chen, X. Zuo, F. Lu and S. J. Li, Vector equilibrium problems under improvement sets and linear scalarization with stability applications, Optim. Methods Softw., 31 (2016), 1240-1257.  doi: 10.1080/10556788.2016.1200043.  Google Scholar [11] C. R. Chen, S. J. Li, J. Zeng and X. B. Li, Error analysis of approximate solutions to parametric vector quasiequilibrium problems, Optim. Lett., 5 (2011), 85-98.  doi: 10.1007/s11590-010-0192-z.  Google Scholar [12] C. R. Chen, S. J. Li and K. L. Teo, Solution semicontinuity of parametric generalized vector equilibrium problems, J. Global Optim., 45 (2009), 309-318.  doi: 10.1007/s10898-008-9376-9.  Google Scholar [13] G. Y. Chen, X. X. Huang and X. Q. Yang, Vector Optimization: Set-Valued and Variational Analysis, Springer, Berlin, 2005.   Google Scholar [14] M. Chicco, F. Mignanego, L. Pusillo and S. Tijs, Vector optimization problem via improvement sets, J. Optim. Theory Appl., 150 (2011), 516-529.  doi: 10.1007/s10957-011-9851-1.  Google Scholar [15] G. Debreu, Theory of Value: An Axiomatic Analysis of Economic Equilibrium, John Wiley, New York, 1959.   Google Scholar [16] M. Durea and R. Strugariu, Scalarization of constraints system in some vector optimization problems and applications, Optim. Lett., 8 (2014), 2021-2037.  doi: 10.1007/s11590-013-0690-x.  Google Scholar [17] M. Durea and C. Tammer, Fuzzy necessary optimality conditions for vector optimization problems, Optimization, 58 (2009), 449-467.  doi: 10.1080/02331930701761615.  Google Scholar [18] Chr. Gerstewitz (Tammer), Nichtkonvexe Dualität in der Vektoroptimierung, Wiss. Z. TH Leuna-Merseburg, 25 (1983), 357-364.  Google Scholar [19] C. Gerth (Tammer) and P. Weidner, Nonconvex separation theorems and some applications in vector optimization, J. Optim. Theory Appl., 67 (1990), 297-320. doi: 10.1007/BF00940478.  Google Scholar [20] F. Giannessi (ed.), Vector Variational Inequalities and Vector Equilibria: Mathematical Theories, Kluwer Academic Publishers, Dordrecht, 2000. doi: 10.1007/978-1-4613-0299-5.  Google Scholar [21] X. H. Gong and J. C. Yao, Lower semicontinuity of the set of efficient solutions for generalized systems, J. Optim. Theory Appl., 138 (2008), 197-205.  doi: 10.1007/s10957-008-9379-1.  Google Scholar [22] A. Göpfert, H. Riahi, C. Tammer and C. Zălinescu, Variational Methods in Partially Ordered Spaces, Springer, New York, 2003.   Google Scholar [23] C. Gutiérrez, B. Jiménez and V. Novo, Improvement sets and vector optimization, Eur. J. Oper. Res., 223 (2012), 304-311.  doi: 10.1016/j.ejor.2012.05.050.  Google Scholar [24] J.-B. Hiriart-Urruty, Tangent cones, generalized gradients and mathematical programming in Banach spaces, Math. Oper. Res., 4 (1979), 79-97.  doi: 10.1287/moor.4.1.79.  Google Scholar [25] J. Jahn, Vector Optimization-Theory, Applications, and Extensions, 2nd ed., Springer, Berlin, 2011. doi: 10.1007/978-3-540-24828-6.  Google Scholar [26] A. A. Khan, C. Tammer and C. Zălinescu, Set-Valued Optimization, Springer, Berlin, 2015.  doi: 10.1007/978-3-642-54265-7.  Google Scholar [27] K. Kimura and J. C. Yao, Semicontinuity of solution mappings of parametric generalized vector equilibrium problems, J. Optim. Theory Appl., 138 (2008), 429-443.  doi: 10.1007/s10957-008-9386-2.  Google Scholar [28] C. S. Lalitha and P. Chatterjee, Stability and scalarization in vector optimization using improvement sets, J. Optim. Theory Appl., 166 (2015), 825-843.  doi: 10.1007/s10957-014-0686-4.  Google Scholar [29] S. J. Li, C. R. Chen, X. B. Li and K. L. Teo, Hölder continuity and upper estimates of solutions to vector quasiequilibrium problems, Eur. J. Oper. Res., 210 (2011), 148-157.  doi: 10.1016/j.ejor.2010.10.005.  Google Scholar [30] S. J. Li and X. B. Li, Hölder continuity of solutions to parametric weak generalized Ky Fan inequality, J. Optim. Theory Appl., 149 (2011), 540-553.  doi: 10.1007/s10957-011-9803-9.  Google Scholar [31] D. T. Luc, Theory of Vector Optimization, Springer, Berlin, 1989.   Google Scholar [32] N. M. Nam and C. Zălinescu, Variational analysis of directional minimal time functions and applications to location problems, Set-Valued Var. Anal., 21 (2013), 405-430.  doi: 10.1007/s11228-013-0232-9.  Google Scholar [33] P. Oppezzi and A. Rossi, Improvement sets and convergence of optimal points, J. Optim. Theory Appl., 165 (2015), 405-419.  doi: 10.1007/s10957-014-0669-5.  Google Scholar [34] Z. Y. Peng, X. M. Yang and J. W. Peng, On the lower semicontinuity of the solution mappings to parametric weak generalized Ky Fan inequality, J. Optim. Theory Appl., 152 (2012), 256-264.  doi: 10.1007/s10957-011-9883-6.  Google Scholar [35] R. T. Rockafellar and R. J. -B. Wets, Variational Analysis, Springer, Berlin, 1998.   Google Scholar [36] P. H. Sach and L. A. Tuan, New scalarizing approach to the stability analysis in parametric generalized Ky Fan inequality problems, J. Optim. Theory Appl., 157 (2013), 347-364.  doi: 10.1007/s10957-012-0105-7.  Google Scholar [37] C. Tammer and C. Zălinescu, Lipschitz properties of the scalarization function and applications, Optimization, 59 (2010), 305-319.  doi: 10.1080/02331930801951033.  Google Scholar [38] A. Zaffaroni, Degrees of efficiency and degrees of minimality, SIAM J. Control Optim., 42 (2003), 1071-1086.  doi: 10.1137/S0363012902411532.  Google Scholar [39] K. Q. Zhao and X. M. Yang, A unified stability result with perturbations in vector optimization, Optim. Lett., 7 (2013), 1913-1919.  doi: 10.1007/s11590-012-0533-1.  Google Scholar [40] K. Q. Zhao and X. M. Yang, $E$-Benson proper efficiency in vector optimization, Optimization, 64 (2015), 739-752.  doi: 10.1080/02331934.2013.798321.  Google Scholar [41] K. Q. Zhao, G. Y. Chen and X. M. Yang, Approximate proper efficiency in vector optimization, Optimization, 64 (2015), 1777-1793.  doi: 10.1080/02331934.2014.979818.  Google Scholar
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