Taiwanese Journal of Mathematics

OPTIMALITY CONDITIONS FOR EFFICIENT SOLUTION TO THE VECTOR EQUILIBRIUM PROBLEMS WITH CONSTRAINTS

Xun-Hua Gong

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Abstract

In this paper, by using the generalization of Ljusternik theorem, the open mapping theorem of convex process, and the convex sets separation theorem, we give the necessary conditions for the efficient solution to the constrained vector equilibrium problems without requiring that the ordering cone in the objective space has a nonempty interior and without requiring that the the convexity conditions. By the assumption of the convexity, we give the sufficient conditions for the efficient solution. As applications, we give the necessary conditions and the sufficient conditions for efficient solution to the constrained vector variational inequalities and constrained vector optimization problems.

Article information

Source
Taiwanese J. Math., Volume 16, Number 4 (2012), 1453-1473.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500406743

Digital Object Identifier
doi:10.11650/twjm/1500406743

Mathematical Reviews number (MathSciNet)
MR2951147

Zentralblatt MATH identifier
1258.49023

Subjects
Primary: 49J52: Nonsmooth analysis [See also 46G05, 58C50, 90C56] 49J50: Fréchet and Gateaux differentiability [See also 46G05, 58C20] 90C29: Multi-objective and goal programming 90C46: Optimality conditions, duality [See also 49N15]

Keywords
vector equilibrium problems efficient solution optimality conditions

Citation

Gong, Xun-Hua. OPTIMALITY CONDITIONS FOR EFFICIENT SOLUTION TO THE VECTOR EQUILIBRIUM PROBLEMS WITH CONSTRAINTS. Taiwanese J. Math. 16 (2012), no. 4, 1453--1473. doi:10.11650/twjm/1500406743. https://projecteuclid.org/euclid.twjm/1500406743


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References

  • N. J. Huang, J. Li and H. B. Thompson, Stability for parametric implicit vector equilibrium problems, Mathematical and Computer Modelling, 43 (2006), 1267-1274.
  • F. Giannessi, G. Mastroeni and L. Pellegrini, On the theory of vector optimization and variational inequalities, image space analysis and separation, in: Vector Variational Inequalities and Vector Equilibria: Mathematical Theories, F. Giannessi, (ed.), Kluwer, Dordrecht, 2000, pp. 153-215.
  • J. Morgan and M. Romaniello, Scalarization and Kuhn-Tucker-Like conditions for weak vector generalized quasivarational Inequalities, Journal of Optimization Theory and Applications, 130 (2006), 309-316.
  • X. Q. Yang and X. Y. Zheng, Approximate solutions and optimality conditions of vector variatinal inequalities in Banach spaces, Journal of Global Optimization, 40 (2008), 455-462.
  • X. H. Gong, Optimality conditions for vector equilibrium problems, Journal of Mathematical Analysis and Applications, 342 (2008), 1455-1466.
  • Q. S. Qiu, Optimality conditions of globally efficient solution for vector equilibrium problems with generalized convexity, Hindawi Publishing Corporation Journal of Inequalities and Applications Vol. 2009, Article ID 898213, 13 pages, doi:10.1155/2009/898213.
  • X. H. Gong, Scalarization and optimality conditions for vector equilibrium problems, Nonlinear Analysis, 73 (2010) 3598-3612.
  • X. H. Gong and Y. J. Yao, Connectedness of the set of efficient solutions for generalized systems, Journal of Optimization Theory and Applications, 138 (2008), 189-196.
  • X. H. Gong and Y. J. Yao, Lower semicontinuity of the set of efficient solutions for generalized systems, Journal of Optimization Theory and Applications, 138 (2008), 197-205.
  • J. Jahn, Mathematical Vector Optimization in Partially-Ordered Linear Spaces, Germany: Peter Lang, Frankfurt an Main, 1986.
  • J. P. Aubin and H. F. Frankowska, Set-Valued Analysis, Birkh$\ddot{a}$user, Boston, 1990.
  • W. Y. Chen, Nonlinear Functional Analysis, Gansu people Press, 1981, (in Chinese).