ABSTRACT

In** **this paper** **we apply the directional derivative technique to characterize D-multifunction, quasi D-multifunction and use them to obtain** ***ε*-optimality for set valued vector optimization problem with multivalued maps. We introduce the notions of local and partial-*ε*-minimum (weak) point and study *ε*-optimality, *ε*-Lagrangian multiplier theorem and *ε*-duality results.

Cite this paper

S. Suneja and M. Sharma, "*ε*-Optimality in Multivalued Optimization," *American Journal of Operations Research*, Vol. 3 No. 4, 2013, pp. 413-420. doi: 10.4236/ajor.2013.34039.

S. Suneja and M. Sharma, "

References

[1] H. W. Kuhn and A. W. Tucker, “Nonlinear Programming Proceedings of the 2nd Berkeley Symposium on Mathematical Statistics and Probability,” University of California Press, Berkeley, 1951, pp. 481-492.

[2] A. M. Geoffrion, “Proper Efficiency and Theory of Vector Maximization,” Journal of Mathematical Analysis and Applications, Vol. 35, 1991, pp. 175-184.

[3] J. M. Borwein, “Proper Efficient Points for Maximizations with Respect to Cones,” SIAM Journal on Control and Optimization, Vol. 15, No. 1, 1977, pp. 57-63. doi:10.1137/0315004

[4] H. P. Benson, “An Improved Definition of Proper Efficiency for Vector Maximization with Respect to Cones,” Journal of Mathematical Analysis and Applications, Vol. 71, No. 1, 1979, pp. 232-241. doi:10.1016/0022-247X(79)90226-9

[5] J. M. Borwein and D. M. Zhauang, “Super Efficiency in Convex Vector Optimization,” Zeitschrift für Operations Research, Vol. 35, No. 3, 1991, pp. 175-184. doi:10.1007/BF01415905

[6] M. Chinaie and J. Zafarani, “Image Space Analysis and Scalarization of Multivalued Optimization,” Journal of Optimization Theory and Applications, Vol. 142, No. 3, 2009, pp. 451-467.

[7] A. Hamel, “An ε-Lagrange Multiplier Rule for a Mathematical Programming Problem on Banach Spaces,” Optimization, Vol. 49, No. 1-2, 2001, pp. 137-149. doi:10.1080/02331930108844524

[8] W. D. Rong and Y. N. Wu, “ε-Weak Minimal Solutions of Vector Optimization Problems with Set-Valued Maps,” Journal of Optimization Theory and Applications, Vol. 106, No. 3, 2000, pp. 569-579. doi:10.1023/A:1004657412928

[9] M. Chinaie and J. Zafarani, “Image Space Analysis and Scalarization of Multivalued Optimization,” Journal of Optimization Theory and Applications, Vol. 106, No. 3, 2010, pp. 1-11.

[10] X. Q. Yang, “Directional Derivatives for Set-Valued Mappings and Applications,” Mathematical Methods of Operations Research, Vol. 48, No. 2, 1998, pp. 273-285. doi:10.1007/s001860050028

[11] J. Benoist, J. M. Borwein and N. A. Popovici, “Characterization of Quasiconvex Vector Valued Functions,” Proceedings of the American Mathematical Socitty, Vol. 131, 2003, pp. 1109-1113.

[12] J. Benoist and N. Popovici, “Characterizations of Convex and Quasiconvex Set-Valued Maps,” Mathematical Methods of Operations Research, Vol. 57, No. 3, 2003, pp. 427-435. doi:10.1007/978-3-540-24828-6

[13] J. Jahn, “Vector Optimization Theory, Applications and Extensions,” Springer, Berlin, 2004.

[14] T. Illes and G. Kassay, “Theorem of Alternative and Optimality Conditions for Convexlike and General Convexlike Programming,” Journal of Optimization Theory and Applications, Vol. 101, No. 2, 1999, pp. 243-257. doi:10.1023/A:1021781308794

[1] H. W. Kuhn and A. W. Tucker, “Nonlinear Programming Proceedings of the 2nd Berkeley Symposium on Mathematical Statistics and Probability,” University of California Press, Berkeley, 1951, pp. 481-492.

[2] A. M. Geoffrion, “Proper Efficiency and Theory of Vector Maximization,” Journal of Mathematical Analysis and Applications, Vol. 35, 1991, pp. 175-184.

[3] J. M. Borwein, “Proper Efficient Points for Maximizations with Respect to Cones,” SIAM Journal on Control and Optimization, Vol. 15, No. 1, 1977, pp. 57-63. doi:10.1137/0315004

[4] H. P. Benson, “An Improved Definition of Proper Efficiency for Vector Maximization with Respect to Cones,” Journal of Mathematical Analysis and Applications, Vol. 71, No. 1, 1979, pp. 232-241. doi:10.1016/0022-247X(79)90226-9

[5] J. M. Borwein and D. M. Zhauang, “Super Efficiency in Convex Vector Optimization,” Zeitschrift für Operations Research, Vol. 35, No. 3, 1991, pp. 175-184. doi:10.1007/BF01415905

[6] M. Chinaie and J. Zafarani, “Image Space Analysis and Scalarization of Multivalued Optimization,” Journal of Optimization Theory and Applications, Vol. 142, No. 3, 2009, pp. 451-467.

[7] A. Hamel, “An ε-Lagrange Multiplier Rule for a Mathematical Programming Problem on Banach Spaces,” Optimization, Vol. 49, No. 1-2, 2001, pp. 137-149. doi:10.1080/02331930108844524

[8] W. D. Rong and Y. N. Wu, “ε-Weak Minimal Solutions of Vector Optimization Problems with Set-Valued Maps,” Journal of Optimization Theory and Applications, Vol. 106, No. 3, 2000, pp. 569-579. doi:10.1023/A:1004657412928

[9] M. Chinaie and J. Zafarani, “Image Space Analysis and Scalarization of Multivalued Optimization,” Journal of Optimization Theory and Applications, Vol. 106, No. 3, 2010, pp. 1-11.

[10] X. Q. Yang, “Directional Derivatives for Set-Valued Mappings and Applications,” Mathematical Methods of Operations Research, Vol. 48, No. 2, 1998, pp. 273-285. doi:10.1007/s001860050028

[11] J. Benoist, J. M. Borwein and N. A. Popovici, “Characterization of Quasiconvex Vector Valued Functions,” Proceedings of the American Mathematical Socitty, Vol. 131, 2003, pp. 1109-1113.

[12] J. Benoist and N. Popovici, “Characterizations of Convex and Quasiconvex Set-Valued Maps,” Mathematical Methods of Operations Research, Vol. 57, No. 3, 2003, pp. 427-435. doi:10.1007/978-3-540-24828-6

[13] J. Jahn, “Vector Optimization Theory, Applications and Extensions,” Springer, Berlin, 2004.

[14] T. Illes and G. Kassay, “Theorem of Alternative and Optimality Conditions for Convexlike and General Convexlike Programming,” Journal of Optimization Theory and Applications, Vol. 101, No. 2, 1999, pp. 243-257. doi:10.1023/A:1021781308794