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http://dx.doi.org/10.4134/BKMS.2013.50.3.777

PROPER EFFICIENCY FOR SET-VALUED OPTIMIZATION PROBLEMS AND VECTOR VARIATIONAL-LIKE INEQUALITIES  

Long, Xian Jun (College of Mathematics and Statistics Chongqing Technology and Business University)
Quan, Jing (Department of Mathematics Yibin University)
Wen, Dao-Jun (College of Mathematics and Statistics Chongqing Technology and Business University)
Publication Information
Bulletin of the Korean Mathematical Society / v.50, no.3, 2013 , pp. 777-786 More about this Journal
Abstract
The purpose of this paper is to establish some relationships between proper efficiency of set-valued optimization problems and proper efficiency of vector variational-like inequalities under the assumptions of generalized cone-preinvexity. Our results extend and improve the corresponding results in the literature.
Keywords
set-valued optimization problem; vector variational-like inequality; proper efficiency; contingent epiderivative; generalized cone-preinvexity;
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1 F. Giannessi, Theorem of alternative, quadratic programs, and complementarity problem, In: R. W. Cottle, F. Giannessi, J. L. Lions(Eds.), Variational Inequality and Complementarity Problem, pp. 151-186. John Wiley and Sons, Chichester, UK, 1980.
2 F. Giannessi, Vector Variational Inequilities and Vector Equilibria: Mathematical Theories, Kluwer Academic, Dordrechet, 2000.
3 F. Giannessi, On Minty variational principle, In: F. Giannessi, S. Komlosi, T. Tapcsack(eds.), New Trends in Mathematical Programming, pp. 93-99. Kluwer Academic, Dordrechet, 1998.
4 X. H. Gong, Optimality conditions for Henig and globally proper efficient solutions with ordering cone has empty interior, J. Math. Anal. Appl. 307 (2005), no. 1, 12-31.   DOI   ScienceOn
5 J. Zeng and S. J. Li, On vector variational-like inequalities and set-valued optimization problems, Optim. Lett. 5 (2011), no. 1, 55-69.   DOI   ScienceOn
6 M. I. Henig, Proper efficiency with respect to cones, J. Optim. Theory Appl. 36 (1982), no. 3, 387-407.   DOI
7 J. Jahn and R. Rauh, Contingent epiderivatives and set-valued optimization, Math. Methods Oper. Res. 46 (1997), no. 2, 193-211.   DOI   ScienceOn
8 A. Al-Homidan and Q. H. Ansari, Generalized Minty vector variational-like inequalities and vector optimization problems, J. Optim. Theory Appl. 144 (2010), no. 1, 1-11.   DOI
9 Q. H. Ansari and G. M. Lee, Nonsmooth vector optimization problems and Minty vector variational inequalities, J. Optim. Theory Appl. 145 (2000), no. 1, 1-16.
10 Q. H. Ansari, M. Rezaie, and J. Zafarani, Generalized vector variational-like inequalities and vector optimization, J. Global Optim. 53 (2012), no. 2, 271-284.   DOI
11 Q. H. Ansari and J. C. Yao, On nondifferentiable and nonconvex vector optimization problems, J. Optim. Theory Appl. 106 (2000), no. 3, 475-488.   DOI   ScienceOn
12 D. Bhatia and A. Mehra, Lagrangian duality for preinvex set-valued functions, J. Math. Anal. Appl. 214 (1997), no. 2, 599-612.   DOI   ScienceOn
13 J. M. Borwein and D. Zhang, Super efficiency in vector optimization, Trans. Amer. Math. Soc. 338 (1993), no. 1, 105-122.   DOI
14 W. Liu and X. H. Gong, Proper efficiency for set-valued vector optimization problems and vector variational inequalities, Math. Methods Oper. Res. 51 (2000), no. 3, 443-457.   DOI
15 G. Bouligand, Sur l'existence des demi-tangentes a une courbe de Jordan, Fundamenta Math. 15 (1930), 215-215.   DOI
16 H. W. Corley, Optimality conditions for maximizations of set-valued functions, J. Optim. Theory Appl. 58 (1988), no. 1, 1-10.   DOI
17 G. Y. Chen, X. X. Huang, and X. Q. Yang, Vector Optimization: Set-Valued and Variational Analysis, Springer-Verlag, Berlin, Heidelberg, 2005.
18 D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and their Applications, Academic Press, New York, 1980.
19 G. M. Lee, D. S. Kim, B. S. Lee, and N. D. Yen, Vector variational inequalities as a tool for studying vector optimization problems, Nonlinear Anal. 34 (1998), no. 5, 745-765.   DOI   ScienceOn
20 X. J. Long, J. W. Peng, and S. Y. Wu, Generalized vector variational-like inequalities and nonsmooth vector optimization problems, Optimization 61 (2012), 1075-1086.   DOI
21 S. K. Mishra and S. Y. Wang, Vector variational-like inequalities and non-smooth vector optimization problems, Nonlinear Anal. 64 (2006), no. 9, 1939-1945.   DOI   ScienceOn
22 X. M. Yang and X. Q. Yang, Vector variational-like inequality with pseudoinvexity, Optimization 55 (2006), no. 1-2, 157-170.   DOI   ScienceOn
23 J. H. Qiu, Cone-directed contingent derivatives and generalized preinvex set-valued optimization, Acta Math. Sci. Ser. B Engl. Ed. 27 (2007), no. 1, 211-218.
24 M. Rezaie and J. Zafarani, Vector optimization and variational-like inequalities, J. Global Optim. 43 (2009), no. 1, 47-66.   DOI
25 G. Ruiz-Garzon, R. Osuna-Gomez, and A. Rufian-Lizana, Relationships between vector variational-like inequality and optimization problems, European J. Oper. Res. 157 (2004), no. 1, 113-119.   DOI   ScienceOn
26 X. M. Yang, X. Q. Yang, and K. L. Teo, Some remarks on the Minty vector variational inequality, J. Optim. Theory Appl. 121 (2004), no. 1, 193-201.   DOI   ScienceOn