##
**New optimality conditions for a nondifferentiable fractional semipreinvex programming problem.**
*(English)*
Zbl 1266.90146

Summary: We study a nondifferentiable fractional programming problem as follows: \((P)\min_{x \in K}f(x)/g(x)\) subject to \(x \in K \subseteq X\), \(h_i(x) \leq 0\), \(i = 1, 2, \dots, m\), where \(K\) is a semiconnected subset in a locally convex topological vector space \(X\), \(f : K \to \mathbb R\), \(g : K \to \mathbb R_+\) and \(h_i : K \to \mathbb R\), \(i = 1, 2, \dots, m\). If \(f\), \(-g\), and \(h_i\), \(i = 1, 2, \dots, m\), are arc-directionally differentiable, semipreinvex maps with respect to a continuous map \(\gamma : [0, 1] \to K \subseteq X\) satisfying \(\gamma(0) = 0\) and \(\gamma'(O^+) \in K\), then the necessary and sufficient conditions for optimality of \((P)\) are established.

### MSC:

90C26 | Nonconvex programming, global optimization |

PDF
BibTeX
XML
Cite

\textit{Y.-C. Chen} and \textit{W.-S. Du}, J. Appl. Math. 2013, Article ID 527183, 5 p. (2013; Zbl 1266.90146)

Full Text:
DOI

### References:

[1] | B. D. Craven, “Invex functions and constrained local minima,” Bulletin of the Australian Mathematical Society, vol. 24, no. 3, pp. 357-366, 1981. · Zbl 0452.90066 |

[2] | B. D. Craven and B. M. Glover, “Invex functions and duality,” Australian Mathematical Society, vol. 39, no. 1, pp. 1-20, 1985. · Zbl 0565.90064 |

[3] | F. H. Clarke, Optimization and Nonsmooth Analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, New York, NY, USA, 1983. · Zbl 0582.49001 |

[4] | F. H. Clarke, Optimization and Nonsmooth Analysis, John Wiley & Sons, New York, NY, USA, 2nd edition, 1983. · Zbl 0582.49001 |

[5] | H. Dietrich, “On the convexification procedure for nonconvex and nonsmooth infinite-dimensional optimization problems,” Journal of Mathematical Analysis and Applications, vol. 161, no. 1, pp. 28-34, 1991. · Zbl 0755.49013 |

[6] | H.-C. Lai, “Optimality conditions for semi-preinvex programming,” Taiwanese Journal of Mathematics, vol. 1, no. 4, pp. 389-404, 1997. · Zbl 0894.90164 |

[7] | H. C. Lai and C. P. Ho, “Duality theorem of nondifferentiable convex multiobjective programming,” Journal of Optimization Theory and Applications, vol. 50, no. 3, pp. 407-420, 1986. · Zbl 0577.90077 |

[8] | H.-C. Lai and L.-J. Lin, “Moreau-Rockafellar type theorem for convex set functions,” Journal of Mathematical Analysis and Applications, vol. 132, no. 2, pp. 558-571, 1988. · Zbl 0656.90097 |

[9] | H. C. Lai and P. Szilágyi, “Alternative theorems and saddlepoint results for convex programming problems of set functions with values in ordered vector spaces,” Acta Mathematica Hungarica, vol. 63, no. 3, pp. 231-241, 1994. · Zbl 0808.90108 |

[10] | H. C. Lai and J. C. Liu, “Minimax fractional programming involving generalised invex functions,” The ANZIAM Journal, vol. 44, no. 3, pp. 339-354, 2003. · Zbl 1042.90046 |

[11] | J.-C. Chen and H.-C. Lai, “Fractional programming for variational problem with F,\rho ,\theta -invexity,” Journal of Nonlinear and Convex Analysis, vol. 4, no. 1, pp. 25-41, 2003. · Zbl 1038.90083 |

[12] | J. C. Liu, “Optimality and duality for generalized fractional programming involving nonsmooth (F,\rho )-convex functions,” Computers and Mathematics with Applications, vol. 42, pp. 437-446, 1990. |

[13] | M. A. Hanson, “On sufficiency of the Kuhn-Tucker conditions,” Journal of Mathematical Analysis and Applications, vol. 80, no. 2, pp. 545-550, 1981. · Zbl 0463.90080 |

[14] | M. A. Hanson and B. Mond, “Necessary and sufficient conditions in constrained optimization,” Mathematical Programming, vol. 37, no. 1, pp. 51-58, 1987. · Zbl 0622.49005 |

[15] | S. K. Mishra and G. Giorgi, Invexity and Optimization, vol. 88 of Nonconvex Optimization and Its Applications, Springer, Berlin, Germnay, 2008. · Zbl 1155.90016 |

[16] | S. K. Mishra, S. Wang, and K. K. Lai, V-Invex Functions and Vector Optimization, vol. 14 of Springer Optimization and Its Applications, Springer, New York, NY, USA, 2008. · Zbl 1142.90002 |

[17] | S. K. Mishra, S.-Y. Wang, and K. K. Lai, Generalized Convexity and Vector Optimization, vol. 90 of Nonconvex Optimization and Its Applications, Springer, Berlin, Germnay, 2009. · Zbl 1162.49001 |

[18] | S. K. Mishra and V. Laha, “On approximately star-shaped functions and approximate vector variational inequalities,” Journal of Optimization Theory and Applications. In press. · Zbl 1366.90191 |

[19] | S. K. Mishra and V. Laha, “On V-r-invexity and vector variational Inequalities,” Filomat, vol. 26, no. 5, pp. 1065-1073, 2012. · Zbl 1289.49016 |

[20] | S. K. Mishra and B. B. Upadhyay, “Nonsmooth minimax fractional programming involving \eta -pseudolinear functions,” Optimization. In press. · Zbl 1291.90174 |

[21] | S. K. Mishra and M. Jaiswal, “Optimality conditions and duality for nondifferentiable multiobjective semi-infinite programming,” Vietnam Journal of Mathematics, vol. 40, pp. 331-343, 2012. · Zbl 1302.90200 |

[22] | 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, pp. 375-385, 1988. · Zbl 0647.90076 |

[23] | T. W. Reiland, “Nonsmooth invexity,” Bulletin of the Australian Mathematical Society, vol. 12, pp. 437-446, 1990. · Zbl 0711.90072 |

[24] | T. Weir, “Programming with semilocally convex functions,” Journal of Mathematical Analysis and Applications, vol. 168, no. 1, pp. 1-12, 1992. · Zbl 0762.90064 |

[25] | T. Weir and V. Jeyakumar, “A class of nonconvex functions and mathematical programming,” Bulletin of the Australian Mathematical Society, vol. 38, no. 2, pp. 177-189, 1988. · Zbl 0639.90082 |

[26] | T. Antczak, “(p, r)-invex sets and functions,” Journal of Mathematical Analysis and Applications, vol. 263, no. 2, pp. 355-379, 2001. · Zbl 1051.90018 |

[27] | Y. L. Ye, “D-invexity and optimality conditions,” Journal of Mathematical Analysis and Applications, vol. 162, no. 1, pp. 242-249, 1991. · Zbl 0755.90074 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.