Higher-Order Duality for Minimax Fractional Type Programming Involving Symmetric Matrices

ABSTRACT

Convexity and generalized convexity play important roles in optimization theory. With the development of programming problem, there has been a growing interest in the higher-order dual problem and a lot of related generalized convexities are given. In this paper, we give the convexity of (F, α ,p ,d ,b , φ )_{β} vector-pseudo- quasi-Type I and formulate a higher-order duality for minimax fractional type programming involving symmetric matrices, and give the weak, strong and strict converse duality theorems under the condition of higher-order (F, α ,p ,d ,b , φ )_{β} vector-pseudoquasi-Type I.

Convexity and generalized convexity play important roles in optimization theory. With the development of programming problem, there has been a growing interest in the higher-order dual problem and a lot of related generalized convexities are given. In this paper, we give the convexity of (F, α ,p ,d ,b , φ )

KEYWORDS

Higher-Order (F, α, p, d, b, φ )_{β} Vector-Pseudoquasi-Type I,
Higher-Order Duality,
Minimax Fractional Type Programming,
Positive Semidefinite Symmetric Matrix

Higher-Order (F, α, p, d, b, φ )

Cite this paper

nullC. Jin and C. Cheng, "Higher-Order Duality for Minimax Fractional Type Programming Involving Symmetric Matrices,"*Applied Mathematics*, Vol. 2 No. 11, 2011, pp. 1387-1392. doi: 10.4236/am.2011.211196.

nullC. Jin and C. Cheng, "Higher-Order Duality for Minimax Fractional Type Programming Involving Symmetric Matrices,"

References

[1] W. E. Schmitendorf, “Necessary Conditions and Sufficient Conditions for Static Minimax Problems,” Journal of Mathematical Analysis and Applications, Vol. 57, No. 3-4, 1977, pp. 683-693. doi:10.1016/0022-247X(77)90255-4

[2] S. Tanimoto, “Duality for a Class of Nondifferentiable Ma thematical Programming Problems,” Journal of Mathematical Analysis and Applications, Vol. 79, No. 2, 1981, pp. 286-294. doi:10.1016/0022-247X(81)90025-1

[3] T. Weir, “Pseudoconvex Minimax Programming,” Utilitas Mathematica, Vol. 42, 1992, pp. 234-240.

[4] S. K. Mishra and N. G. Rueda, “Second-Order Duality for Nondifferentiable Minimax Programming Involving Generalized Type I Functions,” Journal of Optimization Theory and Applications, Vol. 130, No. 3, 2006, pp. 477-486. doi:10.1007/s10957-006-9113-9

[5] Z. Husain, A. Jayswal and I. Ahmad, “Second-Order Duality for Nondifferentiable Minimax Programming Problems with Generalized Convexity,” Journal of Glo- bal Optimization, Vol. 44, No. 4, 2009, pp. 593-608. doi:10.1007/s10898-008-9360-4

[6] O. L. Mangasarian, “Second- and Higher-Order Duality in Nonlinear Programming,” Journal of Mathematical Ana- lysis and Applications, Vol. 51, No. 3, 1975, pp. 607-620. doi:10.1016/0022-247X(75)90111-0

[7] J. Zhang, “Generalized Convexity and Higher Order Duality for Mathematical Programming Problem,” Ph.D. Dissertation, La Trobe University, Melbourne, 1998.

[8] S. K. Mishra and N. G. Rueda, “Higher Order Generalized Invexity and Duality in Nondifferentiable Mathematical Programming,” Journal of Mathematical Analysis and Applications, Vol. 272, No. 2, 2002, pp. 496-506. doi:10.1016/S0022-247X(02)00170-1

[9] I. Ahmad, Z. Huasin and S. Sharma, “Higher-Order Duality in Nondifferentiable Minimax Programming with Generalized Type I Functions,” Journal of Optimization Theory and Applications, Vol. 141, No. 1, 2009, pp. 1-12. doi:10.1007/s10957-008-9474-3

[10] A. Jayswal and I. Stancu-Minasian, “Higher-Order Duality for Nondifferentiable Minimax Programming Problem with Generalized Convexity,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 74, No. 2, 2011, pp. 616-625. doi:10.1016/j.na.2010.09.016

[11] H. C. Lai and J. C. Liu, “Necessary and Sufficient Conditions for Minimax Fractional Programming,” Journal of Mathematical Analysis and Applications, Vol. 230, No. 2, 1999, pp. 311-328. doi:10.1006/jmaa.1998.6204

[1] W. E. Schmitendorf, “Necessary Conditions and Sufficient Conditions for Static Minimax Problems,” Journal of Mathematical Analysis and Applications, Vol. 57, No. 3-4, 1977, pp. 683-693. doi:10.1016/0022-247X(77)90255-4

[2] S. Tanimoto, “Duality for a Class of Nondifferentiable Ma thematical Programming Problems,” Journal of Mathematical Analysis and Applications, Vol. 79, No. 2, 1981, pp. 286-294. doi:10.1016/0022-247X(81)90025-1

[3] T. Weir, “Pseudoconvex Minimax Programming,” Utilitas Mathematica, Vol. 42, 1992, pp. 234-240.

[4] S. K. Mishra and N. G. Rueda, “Second-Order Duality for Nondifferentiable Minimax Programming Involving Generalized Type I Functions,” Journal of Optimization Theory and Applications, Vol. 130, No. 3, 2006, pp. 477-486. doi:10.1007/s10957-006-9113-9

[5] Z. Husain, A. Jayswal and I. Ahmad, “Second-Order Duality for Nondifferentiable Minimax Programming Problems with Generalized Convexity,” Journal of Glo- bal Optimization, Vol. 44, No. 4, 2009, pp. 593-608. doi:10.1007/s10898-008-9360-4

[6] O. L. Mangasarian, “Second- and Higher-Order Duality in Nonlinear Programming,” Journal of Mathematical Ana- lysis and Applications, Vol. 51, No. 3, 1975, pp. 607-620. doi:10.1016/0022-247X(75)90111-0

[7] J. Zhang, “Generalized Convexity and Higher Order Duality for Mathematical Programming Problem,” Ph.D. Dissertation, La Trobe University, Melbourne, 1998.

[8] S. K. Mishra and N. G. Rueda, “Higher Order Generalized Invexity and Duality in Nondifferentiable Mathematical Programming,” Journal of Mathematical Analysis and Applications, Vol. 272, No. 2, 2002, pp. 496-506. doi:10.1016/S0022-247X(02)00170-1

[9] I. Ahmad, Z. Huasin and S. Sharma, “Higher-Order Duality in Nondifferentiable Minimax Programming with Generalized Type I Functions,” Journal of Optimization Theory and Applications, Vol. 141, No. 1, 2009, pp. 1-12. doi:10.1007/s10957-008-9474-3

[10] A. Jayswal and I. Stancu-Minasian, “Higher-Order Duality for Nondifferentiable Minimax Programming Problem with Generalized Convexity,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 74, No. 2, 2011, pp. 616-625. doi:10.1016/j.na.2010.09.016

[11] H. C. Lai and J. C. Liu, “Necessary and Sufficient Conditions for Minimax Fractional Programming,” Journal of Mathematical Analysis and Applications, Vol. 230, No. 2, 1999, pp. 311-328. doi:10.1006/jmaa.1998.6204