**Volume 10, Issue 1, pp 84--91**

**Publication Date**: 2017-01-26

**Yang Yanlong**
- School of computer science and technology, Guizhou University, Guiyang 550025, China.

**Deng Xicai**
- Department of Mathematics and Computer, Guizhou Normal College, Guiyang 550018, China.

**Xiang Shuwen**
- School of computer science and technology, Guizhou University, Guiyang 550025, China.

**Jia Wensheng**
- School of computer science and technology, Guizhou University, Guiyang 550025, China.

In this paper, we first construct a complete metric space \(\Lambda\) consisting of a class of strong vector equilibrium problems (for short, (SVEP)) satisfying some conditions. Under the abstract framework, we introduce a notion of well-posedness for the (SVEP), which unifies its Hadamard and Tikhonov well-posedness. Furthermore, we prove that there exists a dense \(G_{\delta}\) set Q of \(\Lambda\) such that each (SVEP) in Q is well-posed, that is, the majority (in Baire category sense) of (SVEP) in \(\Lambda\) is well-posed. Finally, metric characterizations on the well-posedness for the (SVEP) are given.

Strong vector equilibrium problems, well-posedness, dense set, metric characterizations.

[1] C. D. Aliprantis, K. C. Border,/ Infinite-dimensional analysis,/ A hitchhiker’s guide, Second edition, Springer-Verlag, Berlin,/ (1999).

[2] L. Anderlini, D. Canning,/ Structural stability implies robustness to bounded rationality,/ J. Econom. Theory,/ 101 (2001), 395–422.

[3] G. Beer,/ On a generic optimization theorem of Petar Kenderov,/ Nonlinear Anal.,/ 12 (1988), 647–655.

[4] M. Bianchi, G. Kassay, R. Pini,/ Well-posedness for vector equilibrium problems,/ Math. Methods Oper. Res.,/ 70 (2009), 171–182.

[5] G.-Y. Chen, X.-X. Huang, X.-Q. Yang,/ Vector optimization, Set-valued and variational analysis,/ Lecture Notes in Economics and Mathematical Systems, Springer-Verlag, Berlin,/ (2005).

[6] X.-C. Deng, S.-W. Xiang,/ Well-posed generalized vector equilibrium problems,/ J. Inequal. Appl.,/ 2014 (2014), 12 pages.

[7] D. Dentcheva, S. Helbig,/ On variational principles, level sets, well-posedness, and \(\varepsilon\)-solutions in vector optimization,/ J. Optim. Theory Appl.,/ 89 (1996), 329–349.

[8] A. L. Dontchev, T. Zolezzi,/ Well-posed optimization problems,/ Lecture Notes in Mathematics, Springer-Verlag, Berlin,/ (1993).

[9] M. K. Fort,/ Points of continuity of semicontinuous functions,/ Publ. Math. Debrecen.,/ 2 (1951), 100–102.

[10] C. Gerth, P. Weidner,/ Nonconvex separation theorems and some applications in vector optimization,/ J. Optim. Theory Appl.,/ 67 (1990), 297–320.

[11] F. Giannessi (ed.),/ Vector variational inequalities and vector equilibria,/ Mathematical theories, Nonconvex Optimization and its Applications, 38. Kluwer Academic Publishers, Dordrecht,/ (2000).

[12] X.-X. Huang,/ Extended well-posedness properties of vector optimization problems,/ J. Optim. Theory Appl.,/ 106 (2000), 165–182.

[13] X.-X. Huang,/ Extended and strongly extended well-posedness of set-valued optimization problems,/ Math. Methods Oper. Res.,/ 53 (2001), 101–116.

[14] X.-X. Huang,/ Pointwise well-posedness of perturbed vector optimization problems in a vector-valued variational principle,/ J. Optim. Theory Appl.,/ 108 (2001), 671–686.

[15] P. S. Kenderov,/ Most of the optimization problems have unique solution,/ in: B Brosowski, F. Deutsch (Eds.), Proceedings, Oberwolhfach on Parametric Optimization, in: Birkhauser International Series of Numerical Mathematics, Birkhauser, Basel,/ 72 (1984), 203–216.

[16] P. S. Kenderov, N. K. Ribarska,/ Most of the two-person zero-sum games have unique solution,/ Workshop/ Miniconference on Functional Analysis and Optimization, Canberra, (1988), 73–82, Proc. Centre Math. Anal. Austral. Nat. Univ., Austral. Nat. Univ., Canberra,/ 20 (1988).

[17] E. S. Levitin, B. T. Polyak,/ Convergence of minimizing sequences in conditional extremum problems,/ Dokl. Akad. Nauk SSSR,/ 168 (1966), 764–767.

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

[19] S. J. Li, W. Y. Zhang,/ Hadamard well-posed vector optimization problems,/ J. Global Optim.,/ 46 (2010), 383–393.

[20] D. T. Luc,/ Theory of vector optimization,/ Lecture Notes in Economics and Mathematical Systems, Springer-Verlag, Berlin,/ (1989).

[21] R. Lucchetti,/ Well-posedness towards vector optimizationn,/ Lecture Notes in Economics and Mathematical Systems, Springer-Verlag, Berlin,/ 294 (1987), 194–207.

[22] E. Miglierina, E. Molho,/ Well-posedness and convexity in vector optimization,/ Math. Methods Oper. Res.,/ 58 (2003), 375–385.

[23] E. Miglierina, E. Molho, M. Rocca,/ ell-posedness and scalarization in vector optimization,/ J. Optim. Theory Appl.,/ 126 (2005), 391–409.

[24] D. T. Peng, J. Yu, N. H. Xiu,/ Generic uniqueness of solutions for a class of vector Ky Fan inequalities,/ J. Optim. Theory Appl.,/ 155 (2012), 165–179.

[25] K.-K. Tan, J. Yu, X.-Z. Yuan,/ The uniqueness of saddle points,/ Bull. Polish Acad. Sci. Math.,/ 43 (1995), 119–129.

[26] A. N. Tyhonov,/ On the stability of the functional optimization problem,/ U.S.S.R. Comput. Math. Math. Phys.,/ 6 (1966), 28–33.

[27] J. Yu,/ Essential equilibria of n-person noncooperative games,/ J. Math. Econom.,/ 31 (1999), 361–372.

[28] J. Yu, D.-T. Peng, S.-W. Xiang, Generic uniqueness of equilibrium points,/ Nonlinear Anal.,/ 74 (2011), 6326–6332.

[29] J. Yu, H. Yang, C. Yu,/ Structural stability and robustness to bounded rationality for non-compact cases,/ J. Global Optim.,/ 44 (2009), 149–157.

[30] C. Yu, J. Yu,/ On structural stability and robustness to bounded rationality,/ Nonlinear Anal.,/ 65 (2006), 583–592.

[31] A. J. Zaslavski,/ Generic well-posedness of minimization problems with mixed continuous constraints,/ Nonlinear Anal.,/ 64 (2006), 2381–2399.

[32] A. J. Zaslavski,/ Generic existence of Lipschitzian solutions of optimal control problems without convexity assumptions,/ J. Math. Anal. Appl.,/ 335 (2007), 962–973.

[33] W.-B. Zhang, N.-J. Huang, D. O’Regan,/ Generalized well-posedness for symmetric vector quasi-equilibrium problems,/ J. Appl. Math.,/ 2015 (2015), 10 pages.