# Levitin-Polyak well-posedness for lexicographic vector equilibrium problems

Volume 10, Issue 2, pp 354--367
Publication Date: February 20, 2017 Submission Date: August 13, 2016
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### Authors

Rabian Wangkeeree - Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand. - Research Center for Academic Excellence in Mathematics, Naresuan University, Phitsanulok 65000, Thailand. Thanatporn Bantaojai - Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand.

### Abstract

We introduce the notions of Levitin-Poljak (LP) well-posedness and LP well-posedness in the generalized sense for the lexicographic vector equilibrium problems. Then, we establish some sufficient conditions for lexicographic vector equilibrium problems to be LP well-posedness at the reference point. Numerous examples are provided to explain that all the assumptions we impose are very relaxed and cannot be dropped. The results in this paper unify, generalize and extend some known results in the literature.

### Keywords

• Levitin-polyak well-posedness
• lexicographic vector equilibrium problems
• metric spaces.

•  90C33
•  49K40

### References

• [1] L. Q. Anh, T. Q. Duy, A. Y. Kruger, N. H. Thao, Well-posedness for lexicographic vector equilibrium problems, Constructive nonsmooth analysis and related topics, Springer Optim. Appl., Springer, New York, 87 (2014), 159–174.

• [2] L. Q. Anh, P. Q. Khanh, D. T. M. Van, Well-posedness under relaxed semicontinuity for bilevel equilibrium and optimization problems with equilibrium constraints, J. Optim. Theory Appl., 153 (2012), 42–59.

• [3] L. Q. Anh, P. Q. Khanh, D. T. M Van, J.-C. Yao, Well-posedness for vector quasiequilibria, Taiwanese J. Math., 13 (2009), 713–737.

• [4] J. P. Aubin, H. Frankowska, Set-valued analysis, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA (1990)

• [5] J. Banaś, K. Goebel, Measures of noncompactness in Banach spaces, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, Inc., New York (1980)

• [6] M. Bianchi, I. V. Konnov, R. Pini, Lexicographic variational inequalities with applications, Optimization, 56 (2007), 355–367.

• [7] M. Bianchi, I. V. Konnov, R. Pini, Lexicographic and sequential equilibrium problems, J. Global Optim., 46 (2010), 551–560.

• [8] E. Blum, W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student, 63 (1994), 123–145.

• [9] E. Carlson, Generalized extensive measurement for lexicographic orders, J. Math. Psych., 54 (2010), 345–351.

• [10] L. C. Ceng, N. Hadjisavvas, S. Schaible, J.-C. Yao, Well-posedness for mixed quasivariational-like inequalities, J. Optim. Theory Appl., 139 (2008), 109–225.

• [11] J.-W. Chen, Z.-P. Wan, Y. J. Cho, Levitin-Polyak well-posedness by perturbations for systems of set-valued vector quasiequilibrium problems, Math. Methods Oper. Res., 77 (2013), 33–64.

• [12] G. P. Crespi, A. Guerraggio, M. Rocca, Well posedness in vector optimization problems and vector variational inequalities, J. Optim. Theory Appl., 132 (2007), 213–226.

• [13] J. Daneš, On the Istrăţescu’s measure of noncompactness, Bull. Math. Soc. Sci. Math. R. S. Roumanie (N.S.), 16 (1972), 403–406.

• [14] B. Djafari Rouhani, E. Tarafdar, P. J. Watson, Existence of solutions to some equilibrium problems, J. Optim. Theory Appl., 126 (2005), 97–107.

• [15] V. A. Emelichev, E. E. Gurevsky, K. G. Kuzmin , On stability of some lexicographic integer optimization problem, Control Cybernet., 39 (2010), 811–826.

• [16] Y.-P. Fang, R. Hu, N.-J. Huang, Well-posedness for equilibrium problems and for optimization problems with equilibrium constraints, Comput. Math. Appl., 55 (2008), 89–100.

• [17] Y.-P. Fang, N.-J. Huang, J.-C. Yao, Well-posedness of mixed variational inequalities, inclusion problems and fixed point problems, J. Global Optim., 41 (2008), 117–133.

• [18] F. Flores-Bazán, Existence theorems for generalized noncoercive equilibrium problems: the quasi-convex case, SIAM J. Optim., 11 (2001), 675–690.

• [19] E. C. Freuder, R. Heffernan, R. J. Wallace, N. Wilson, Lexicographically-ordered constraint satisfaction problems, Constraints, 15 (2010), 1–28.

• [20] J. Hadamard, Sur le problémes aux dérivées partielles et leur signification physique [On the problems about partial derivatives and their physical signicance] , Bull. Univ. Princeton, 13 (1902), 49–52.

• [21] N. X. Hai, P. Q. Khanh, Existence of solutions to general quasiequilibrium problems and applications, J. Optim. Theory Appl., 133 (2007), 317–327.

• [22] A. D. Ioffe, R. E. Lucchetti, J. P. Revalski, Almost every convex or quadratic programming problem is well posed, Math. Oper. Res., 29 (2004), 369–382.

• [23] K. Kimura, Y.-C. Liou, S.-Y. Wu, J.-C. Yao , Well-posedness for parametric vector equilibrium problems with applications, J. Ind. Manag. Optim., 4 (2008), 313–327.

• [24] I. V. Konnov, M. S. S. Ali, Descent methods for monotone equilibrium problems in Banach spaces, J. Comput. Appl. Math., 188 (2006), 165–179.

• [25] A. S. Konsulova, J. P. Revalski, Constrained convex optimization problemswell-posedness and stability, Numer. Funct. Anal. Optim., 15 (1994), 889–907.

• [26] M. Küçük, M. Soyertem, Y. Küçük, On constructing total orders and solving vector optimization problems with total orders, J. Global Optim., 50 (2011), 235–247.

• [27] C. S. Lalitha, G. Bhatia, Levitin-Polyak well-posedness for parametric quasivariational inequality problem of the Minty type, Positivity, 16 (2012), 527–541.

• [28] E. S. Levitin, B. T. Poljak, Convergence of minimizing sequences in problems on the relative extremum, (Russian) Dokl. Akad. Nauk SSSR, 168 (1966), 997–1000.

• [29] S. J. Li, M. H. Li, Levitin-Polyak well-posedness of vector equilibrium problems, Math. Methods Oper. Res., 69 (2009), 125–140.

• [30] M. B. Lignola, Well-posedness and L-well-posedness for quasivariational inequalities, J. Optim. Theory Appl., 128 (2006), 119–138.

• [31] X. J. Long, N.-J. Huang, K. L. Teo, Levitin-Polyak well-posedness for equilibrium problems with functional constraints, J. Inequal. Appl., 2008 (2008), 14 pages.

• [32] P. Loridan, $\epsilon$-solutions in vector minimization problems, J. Optim. Theory Appl., 43 (1984), 265–276.

• [33] R. Lucchetti, F. Patrone, A characterization of Tyhonov well-posedness for minimum problems, with applications to variational inequalities, Numer. Funct. Anal. Optim., 3 (1981), 461–476.

• [34] M. Margiocco, F. Patrone, L. Pusillo Chicco, A new approach to Tikhonov well-posedness for Nash equilibria, Optimization, 40 (1997), 385–400.

• [35] J. Morgan, V. Scalzo, Pseudocontinuity in optimization and nonzero-sum games, J. Optim. Theory Appl., 120 (2004), 181–197.

• [36] J.-W. Peng, Y. Wang, L.-J. Zhao, Generalized Levitin-Polyak well-posedness of vector equilibrium problems, Fixed Point Theory Appl., 2009 (2009 ), 14 pages.

• [37] J.-W. Peng, S.-Y. Wu, Y. Wang, Levitin-Polyak well-posedness of generalized vector quasi-equilibrium problems with functional constraints, J. Global Optim., 52 (2012), 779–795.

• [38] V. Rakočević, Measures of noncompactness and some applications, Filomat, 12 (1998), 87–120.

• [39] J. P. Revalski, Hadamard and strong well-posedness for convex programs, SIAM J. Optim., 7 (1997), 519–526.

• [40] I. Sadeqi, C. G. Alizadeh, Existence of solutions of generalized vector equilibrium problems in reflexive Banach spaces, Nonlinear Anal., 74 (2011), 2226–2234.

• [41] A. N. Tikhonov, On the stability of the functional optimization problem, USSR Comput. Math. Math. Phys., 6 (1966), 28–33.

• [42] J. Yu, H. Yang, C. Yu, Well-posed Ky Fan’s point, quasi-variational inequality and Nash equilibrium problems, Nonlinear Anal., 66 (2007), 777–790.

• [43] T. Zolezzi, Well-posedness criteria in optimization with application to the calculus of variations, Nonlinear Anal., 25 (1995), 437–453.

• [44] T. Zolezzi, Well-posedness and optimization under perturbations, Optimization with data perturbations, II, Ann. Oper. Res., 101 (2001), 351–361.