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**Generalized second-order mixed symmetric duality in nondifferentiable mathematical programming.**
*(English)*
Zbl 1229.49033

Summary: This paper is concerned with a pair of second-order mixed symmetric dual programs involving nondifferentiable functions. Weak, strong, and converse duality theorems are proved for the aforementioned pair using the notion of second-order \(F\)-convexity/pseudoconvexity assumptions.

### MSC:

49N15 | Duality theory (optimization) |

90C46 | Optimality conditions and duality in mathematical programming |

### Keywords:

mixed symmetric dual programs; nondifferentiable functions; weak duality theorems; strong duality theorems; converse duality theorems; second-order \(F\)-convexity; second-order \(F\)-pseudoconvexity
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\textit{R. P. Agarwal} et al., Abstr. Appl. Anal. 2011, Article ID 103597, 14 p. (2011; Zbl 1229.49033)

### References:

[1] | G. B. Dantzig, E. Eisenberg, and R. W. Cottle, “Symmetric dual nonlinear programs,” Pacific Journal of Mathematics, vol. 15, pp. 809-812, 1965. · Zbl 0136.14001 |

[2] | B. Mond, “A symmetric dual theorem for non-linear programs,” Quarterly of Applied Mathematics, vol. 23, pp. 265-269, 1965. · Zbl 0136.13907 |

[3] | M. S. Bazaraa and J. J. Goode, “On symmetric duality in nonlinear programming,” Operations Research, vol. 21, pp. 1-9, 1973. · Zbl 0259.90034 |

[4] | S. Chandra and I. Husain, “Symmetric dual nondifferentiable programs,” Bulletin of the Australian Mathematical Society, vol. 24, no. 2, pp. 295-307, 1981. · Zbl 0478.90061 |

[5] | S. Chandra, B. D. Craven, and B. Mond, “Generalized concavity and duality with a square root term,” Optimization, vol. 16, no. 5, pp. 653-662, 1985. · Zbl 0575.49008 |

[6] | B. Mond and M. Schechter, “Nondifferentiable symmetric duality,” Bulletin of the Australian Mathematical Society, vol. 53, no. 2, pp. 177-188, 1996. · Zbl 0846.90100 |

[7] | P. Kumar and D. Bhatia, “A note on symmetric duality for multiobjective non-linear program,” Opsearch, vol. 32, no. 3, pp. 172-183, 1995. |

[8] | O. L. Mangasarian, “Second- and higher-order duality in nonlinear programming,” Journal of Mathematical Analysis and Applications, vol. 51, no. 3, pp. 607-620, 1975. · Zbl 0313.90052 |

[9] | C. R. Bector and S. Chandra, “Second order symmetric and self-dual programs,” Opsearch, vol. 23, no. 2, pp. 89-95, 1986. · Zbl 0604.90117 |

[10] | S. H. Hou and X. M. Yang, “On second-order symmetric duality in nondifferentiable programming,” Journal of Mathematical Analysis and Applications, vol. 255, no. 2, pp. 491-498, 2001. · Zbl 0986.90054 |

[11] | S. Chandra, I. Husain, and Abha, “On mixed symmetric duality in mathematical programming,” Opsearch, vol. 36, no. 2, pp. 165-171, 1999. · Zbl 1141.90539 |

[12] | X. M. Yang, K. L. Teo, and X. Q. Yang, “Mixed symmetric duality in nondifferentiable mathematical programming,” Indian Journal of Pure and Applied Mathematics, vol. 34, no. 5, pp. 805-815, 2003. · Zbl 1053.90133 |

[13] | I. Ahmad, “Multiobjective mixed symmetric duality with invexity,” New Zealand Journal of Mathematics, vol. 34, no. 1, pp. 1-9, 2005. · Zbl 1255.90094 |

[14] | C. R. Bector, S. Chandra, and Abha, “On mixed symmetric duality in multiobjective programming,” Opsearch, vol. 36, no. 4, pp. 399-407, 1999. · Zbl 1141.90517 |

[15] | S. Chandra, A. Goyal, and I. Husain, “On symmetric duality in mathematical programming with F-convexity,” Optimization, vol. 43, no. 1, pp. 1-18, 1998. · Zbl 0905.90167 |

[16] | T. R. Gulati, I. Ahmad, and I. Husain, “Second order symmetric duality with generalized convexity,” Opsearch, vol. 38, no. 2, pp. 210-222, 2001. · Zbl 1278.90431 |

[17] | T. R. Gulati and S. K. Gupta, “Wolfe type second-order symmetric duality in nondifferentiable programming,” Journal of Mathematical Analysis and Applications, vol. 310, no. 1, pp. 247-253, 2005. · Zbl 1079.90148 |

[18] | M. Schechter, “More on subgradient duality,” Journal of Mathematical Analysis and Applications, vol. 71, no. 1, pp. 251-262, 1979. · Zbl 0421.90062 |

[19] | R. P. Agarwal, I. Ahmad, and A. Jayswal, “Higher order symmetric duality in nondifferentiable multi-objective programming problems involving generalized cone convex functions,” Mathematical and Computer Modelling, vol. 52, no. 9-10, pp. 1644-1650, 2010. · Zbl 1205.90254 |

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