Optimality for Henig Proper Efficiency in Vector Optimization Involving Dini Set-Valued Directional Derivatives

ABSTRACT

This note studies the optimality conditions of vector optimization problems involving generalized convexity in locally convex spaces. Based upon the concept of Dini set-valued directional derivatives, the necessary and sufficient optimality conditions are established for Henig proper and strong minimal solutions respectively in generalized preinvex vector optimization problems.

This note studies the optimality conditions of vector optimization problems involving generalized convexity in locally convex spaces. Based upon the concept of Dini set-valued directional derivatives, the necessary and sufficient optimality conditions are established for Henig proper and strong minimal solutions respectively in generalized preinvex vector optimization problems.

KEYWORDS

Vector Optimization, Dini Set-Valued Directional Derivative, Generalized Preinvex Function, Henig Proper Efficiency

Vector Optimization, Dini Set-Valued Directional Derivative, Generalized Preinvex Function, Henig Proper Efficiency

Cite this paper

nullG. Yu and H. Bai, "Optimality for Henig Proper Efficiency in Vector Optimization Involving Dini Set-Valued Directional Derivatives,"*Applied Mathematics*, Vol. 2 No. 7, 2011, pp. 922-925. doi: 10.4236/am.2011.27126.

nullG. Yu and H. Bai, "Optimality for Henig Proper Efficiency in Vector Optimization Involving Dini Set-Valued Directional Derivatives,"

References

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[5] L. Batista Dos Santos, R. Osuna-ómez, M. A.Rojas-Medar and A. Rufián-Lizana, “Preinvex Functions and Weak Efficient Solutions for Some Vectorial Optimization Problem in Banach Spaces,” Computers and Mathematics with Applications, Vol. 48, No. 5-6, 2004, pp. 885-895. doi:10.1016/j.camwa.2003.05.013

[6] A. J. V. Brandāo, M. A. Rojas-Medar and G. N. Silva, “Optimality Conditions for Pareto Nonsmooth Nonconvex Programming in Banach Spaces,” Journal of Optimization Theory and Applications, Vol. 103, No. 1, 1999, pp. 65-73. doi:10.1023/A:1021769232224

[7] S. K. Mishra, G. Giorgi and S. Y. Wang, “Duality in Vector Optimization in Banach Spaces with Generalized Convexity,” Journal of Global Optimization, Vol. 29, pp. 2004, pp. 415-424. doi:10.1016/j.camwa.2007.05.002

[8] G. L. Yu and S. Y. Liu, “Some Vector Optimization Problems in Banach Spaces with Generalized Convexity,” Computers and Mathematics with Applications, Vol. 54, No. 11-12, 2007, pp. 1403-1410. doi:10.1016/j.camwa.2007.05.002

[9] J. H. Qiu, “Cone-Directed Contingent Derivatives and Generalized Preinvex Set-Valued Optimization,” Acta Mathematica Scientia, Vol. 27, No. 1, 2007, pp. 211-218. doi:10.1016/S0252-9602(07)60019-8

[10] M. I. Henig, “Proper Efficiency with Respect to Cones,” Journal of Optimization Theory and Applications, Vol. 36, No. 3, 1982, pp. 387-407. doi:10.1007/BF00934353

[11] Borwein and D. M. Zhuang, “Super Efficiency in Vector Optimization,” Transactions of the American Mathematical Society, Vol. 338, No. 1, 1993, pp. 105-122. doi:10.2307/2154446

[12] X. H. Gong, “Optimality Conditions for Henig and Globally Proper Efficient Solutions with Ordering Cone Has Empty Interior,” Journal of Mathematical Analysis and Applications, Vol. 307, No. 1, 2005, pp. 12-31. doi:10.1016/j.jmaa.2004.10.001

[1] X. Q. Yang, “A Generalized Upper Dini-Directional Derivative in Vector Optimization,” Optimization, Vol. 43, No. 4, 1997, pp. 339-351. doi:10.1080/02331939808844392

[2] I. Ginchev, A. Guerraggio and M. Rocca, “Dini Set-Valued Derivative in Locally Lipschitz Vector Optimization,” Journal of Optimization Theory and Applications, Vol. 142, No. 1, 2009, pp. 87-105. doi:10.1007/s10957-009-9551-2

[3] T. Weir and V. Jeyakumar, “A Class of Nonconvex Functions and Mathematical Programming,” Bulletin of Australian Mathematical Society, Vol. 38, No. 2, 1988, pp. 177-189. doi:10.1017/S0004972700027441

[4] T. Weir and B. Mond, “Preinvex Functions in Multiple- Objective Optimization,” Journal of Mathematical Analysis and Applications, Vol. 136, No. 1, 1988, pp. 29-38. doi:10.1016/0022-247X(88)90113-8

[5] L. Batista Dos Santos, R. Osuna-ómez, M. A.Rojas-Medar and A. Rufián-Lizana, “Preinvex Functions and Weak Efficient Solutions for Some Vectorial Optimization Problem in Banach Spaces,” Computers and Mathematics with Applications, Vol. 48, No. 5-6, 2004, pp. 885-895. doi:10.1016/j.camwa.2003.05.013

[6] A. J. V. Brandāo, M. A. Rojas-Medar and G. N. Silva, “Optimality Conditions for Pareto Nonsmooth Nonconvex Programming in Banach Spaces,” Journal of Optimization Theory and Applications, Vol. 103, No. 1, 1999, pp. 65-73. doi:10.1023/A:1021769232224

[7] S. K. Mishra, G. Giorgi and S. Y. Wang, “Duality in Vector Optimization in Banach Spaces with Generalized Convexity,” Journal of Global Optimization, Vol. 29, pp. 2004, pp. 415-424. doi:10.1016/j.camwa.2007.05.002

[8] G. L. Yu and S. Y. Liu, “Some Vector Optimization Problems in Banach Spaces with Generalized Convexity,” Computers and Mathematics with Applications, Vol. 54, No. 11-12, 2007, pp. 1403-1410. doi:10.1016/j.camwa.2007.05.002

[9] J. H. Qiu, “Cone-Directed Contingent Derivatives and Generalized Preinvex Set-Valued Optimization,” Acta Mathematica Scientia, Vol. 27, No. 1, 2007, pp. 211-218. doi:10.1016/S0252-9602(07)60019-8

[10] M. I. Henig, “Proper Efficiency with Respect to Cones,” Journal of Optimization Theory and Applications, Vol. 36, No. 3, 1982, pp. 387-407. doi:10.1007/BF00934353

[11] Borwein and D. M. Zhuang, “Super Efficiency in Vector Optimization,” Transactions of the American Mathematical Society, Vol. 338, No. 1, 1993, pp. 105-122. doi:10.2307/2154446

[12] X. H. Gong, “Optimality Conditions for Henig and Globally Proper Efficient Solutions with Ordering Cone Has Empty Interior,” Journal of Mathematical Analysis and Applications, Vol. 307, No. 1, 2005, pp. 12-31. doi:10.1016/j.jmaa.2004.10.001