Optimality for Multi-Objective Programming Involving Arcwise Connected d-Type-I Functions

ABSTRACT

This paper deals with the optimality conditions and dual theory of multi-objective programming problems involving generalized convexity. New classes of generalized type-I functions are introduced for arcwise connected functions, and examples are given to show the existence of these functions. By utilizing the new concepts, several sufficient optimality conditions and Mond-Weir type duality results are proposed for non-differentiable multi-objective programming problem.

This paper deals with the optimality conditions and dual theory of multi-objective programming problems involving generalized convexity. New classes of generalized type-I functions are introduced for arcwise connected functions, and examples are given to show the existence of these functions. By utilizing the new concepts, several sufficient optimality conditions and Mond-Weir type duality results are proposed for non-differentiable multi-objective programming problem.

KEYWORDS

Multi-Objective Programming, Pareto Efficient Solution, Arcwise Connected d-Type-I Functions, Optimality Conditions, Duality

Multi-Objective Programming, Pareto Efficient Solution, Arcwise Connected d-Type-I Functions, Optimality Conditions, Duality

Cite this paper

nullG. Yu and M. Wang, "Optimality for Multi-Objective Programming Involving Arcwise Connected d-Type-I Functions,"*American Journal of Operations Research*, Vol. 1 No. 4, 2011, pp. 243-248. doi: 10.4236/ajor.2011.14028.

nullG. Yu and M. Wang, "Optimality for Multi-Objective Programming Involving Arcwise Connected d-Type-I Functions,"

References

[1] M. A. Hanson and B. Mond, “Necessary and Sufficient Conditions in Constraint Optimization,” Mathematical Programming, Vol. 37, No. 1, 1987, pp. 51-58. doi:10.1007/BF02591683

[2] B. Aghezzaf and M. Hachimi, “Generalized Invexity and Duality in Multiobjective Programming Problems,” Journal of Global Optimization, Vol. 18, No. 1, 2000, pp. 91-101. doi:10.1023/A:1008321026317

[3] M. Hachimi and B. Aghezzaf, “Sufficiency and Duality in Differentiable Multiobjective Programming Involving Generalized Type I Functions,” Journal of Mathematical Analysis and Applications, Vol. 296, No. 2, 2004, pp. 382-392. doi:10.1016/j.jmaa.2003.12.042

[4] M. Hachimi and B. Aghezzaf, “Sufficiency and Duality in Nondifferentiable Multiobjective Programming Involving Generalized Type I Functions,” Journal of Mathematical Analysis and Applications, Vol. 319, No. 1, 2006, pp. 110-123. doi:10.1016/j.jmaa.2005.02.064

[5] R. N. Kaul, S. K. Suneja and M. K. Srivastava, “Optimality Criteria and Duality in Multiple Objective Optimization Involving Generalized Invexity,” Journal of Optimization Theory and Applications, Vol. 80, No. 3, 1994, pp. 465-482. doi:10.1007/BF02207775

[6] 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, No. 4, 2004, pp. 415-424. doi:10.1023/B:JOGO.0000047911.03061.88

[7] S. K. Mishra and M. A. Noor, “Some Nondifferentiable Multi-Objective Programming Problems,” Journal of Mathematic Analysis and Applications, Vol. 316, No. 2, 2006, pp. 472-482. doi:10.1016/j.jmaa.2005.04.067

[8] S. K. Mishra, S. Y. Wang and K. K. Lai, “Multiple Objective Fractional Programming Involving Semilocally Type I-Preinvex and Related Functions,” Journal of Mathematic Analysis and Applications, Vol. 310, No. 2, 2005, pp. 626-640.

[9] A. Mehra and D. Bhatia, “Optimality and Duality for Minmax Problems Involving Arcwise Connected and Generalized Arcwise Connected Functions,” Journal of Mathematical Analysis and Applications, Vol. 231, No. 2, 1999, pp. 425-445. doi:10.1006/jmaa.1998.6231

[10] N. G. Rueda, M. A. Hanson and C. Singh, “Optimality and Duality with Generalized Convexity,” Journal of Optimization Theory and Applications, Vol. 86, No. 2, 1995, pp. 491-500. doi:10.1007/BF02192091

[11] 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

[12] G. L. Yu and S. Y. Liu, “Optimality for (h, ?)-Multiobjective Programming Involving Generalized Type-I Functions,” Journal of Global Optimization, Vol. 41, No. 1, 2008, pp. 147-161. doi:10.1007/s10898-007-9196-3

[13] M. Avriel and I. Zang, “Generalized Arcwise-Connected Functions and Characterizations of Local-Global Minimum Properties,” Journal of Optimization Theory and Applications, Vol. 32, No. 4, 1980, pp. 407-425. doi:10.1007/BF00934030

[14] C. Singh, “Elementary Properties of Arcwise Connected Set and Functions,” Journal of Optimization Theory and Applications, Vol. 41, 1990, pp. 85-103.

[15] R. N. Mukherjee and S. R. Yadav, “A Note on Arcwise Connected Sets and Functions,” Bulletin of the Australian Mathematical Society, Vol. 31, No. 3, 1985, pp. 369-375. doi:10.1017/S0004972700009333

[16] D. Bhatia and A. Mehra, “Optimality and Duality Involving Arcwise Connected Generalized Connected Functions,” Journal of Optimization Theory and Applications, Vol. 100, No. 1, 1999, pp. 181-194. doi:10.1023/A:1021725200423

[17] N. G. Rueda and M. A. Hanson, “Optimality Criteria in Mathematical Programming Involving Generalized Invexity,” Journal of Mathematical Analysis and Applications, Vol. 130, No. 2, 1998, pp. 375-385. doi:10.1016/0022-247X(88)90313-7

[18] S. Davar and A. Mehra, “Optimality and Duality for Fractional Programming Problems Involving Arcwise Connected Functions and Their Applications,” Journal of Mathematical Analysis and Applications, Vol. 263, No. 2, 2001, pp. 666-682. doi:10.1006/jmaa.2001.7651

[1] M. A. Hanson and B. Mond, “Necessary and Sufficient Conditions in Constraint Optimization,” Mathematical Programming, Vol. 37, No. 1, 1987, pp. 51-58. doi:10.1007/BF02591683

[2] B. Aghezzaf and M. Hachimi, “Generalized Invexity and Duality in Multiobjective Programming Problems,” Journal of Global Optimization, Vol. 18, No. 1, 2000, pp. 91-101. doi:10.1023/A:1008321026317

[3] M. Hachimi and B. Aghezzaf, “Sufficiency and Duality in Differentiable Multiobjective Programming Involving Generalized Type I Functions,” Journal of Mathematical Analysis and Applications, Vol. 296, No. 2, 2004, pp. 382-392. doi:10.1016/j.jmaa.2003.12.042

[4] M. Hachimi and B. Aghezzaf, “Sufficiency and Duality in Nondifferentiable Multiobjective Programming Involving Generalized Type I Functions,” Journal of Mathematical Analysis and Applications, Vol. 319, No. 1, 2006, pp. 110-123. doi:10.1016/j.jmaa.2005.02.064

[5] R. N. Kaul, S. K. Suneja and M. K. Srivastava, “Optimality Criteria and Duality in Multiple Objective Optimization Involving Generalized Invexity,” Journal of Optimization Theory and Applications, Vol. 80, No. 3, 1994, pp. 465-482. doi:10.1007/BF02207775

[6] 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, No. 4, 2004, pp. 415-424. doi:10.1023/B:JOGO.0000047911.03061.88

[7] S. K. Mishra and M. A. Noor, “Some Nondifferentiable Multi-Objective Programming Problems,” Journal of Mathematic Analysis and Applications, Vol. 316, No. 2, 2006, pp. 472-482. doi:10.1016/j.jmaa.2005.04.067

[8] S. K. Mishra, S. Y. Wang and K. K. Lai, “Multiple Objective Fractional Programming Involving Semilocally Type I-Preinvex and Related Functions,” Journal of Mathematic Analysis and Applications, Vol. 310, No. 2, 2005, pp. 626-640.

[9] A. Mehra and D. Bhatia, “Optimality and Duality for Minmax Problems Involving Arcwise Connected and Generalized Arcwise Connected Functions,” Journal of Mathematical Analysis and Applications, Vol. 231, No. 2, 1999, pp. 425-445. doi:10.1006/jmaa.1998.6231

[10] N. G. Rueda, M. A. Hanson and C. Singh, “Optimality and Duality with Generalized Convexity,” Journal of Optimization Theory and Applications, Vol. 86, No. 2, 1995, pp. 491-500. doi:10.1007/BF02192091

[11] 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

[12] G. L. Yu and S. Y. Liu, “Optimality for (h, ?)-Multiobjective Programming Involving Generalized Type-I Functions,” Journal of Global Optimization, Vol. 41, No. 1, 2008, pp. 147-161. doi:10.1007/s10898-007-9196-3

[13] M. Avriel and I. Zang, “Generalized Arcwise-Connected Functions and Characterizations of Local-Global Minimum Properties,” Journal of Optimization Theory and Applications, Vol. 32, No. 4, 1980, pp. 407-425. doi:10.1007/BF00934030

[14] C. Singh, “Elementary Properties of Arcwise Connected Set and Functions,” Journal of Optimization Theory and Applications, Vol. 41, 1990, pp. 85-103.

[15] R. N. Mukherjee and S. R. Yadav, “A Note on Arcwise Connected Sets and Functions,” Bulletin of the Australian Mathematical Society, Vol. 31, No. 3, 1985, pp. 369-375. doi:10.1017/S0004972700009333

[16] D. Bhatia and A. Mehra, “Optimality and Duality Involving Arcwise Connected Generalized Connected Functions,” Journal of Optimization Theory and Applications, Vol. 100, No. 1, 1999, pp. 181-194. doi:10.1023/A:1021725200423

[17] N. G. Rueda and M. A. Hanson, “Optimality Criteria in Mathematical Programming Involving Generalized Invexity,” Journal of Mathematical Analysis and Applications, Vol. 130, No. 2, 1998, pp. 375-385. doi:10.1016/0022-247X(88)90313-7

[18] S. Davar and A. Mehra, “Optimality and Duality for Fractional Programming Problems Involving Arcwise Connected Functions and Their Applications,” Journal of Mathematical Analysis and Applications, Vol. 263, No. 2, 2001, pp. 666-682. doi:10.1006/jmaa.2001.7651