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**\((p,r)\)-invex sets and functions.**
*(English)*
Zbl 1051.90018

Summary: Notions of invexity of a function and of a set are generalized. The notion of an invex function with respect to \(\eta\) can be further extended with the aid of \(p\)-invex sets. Slight generalization of the notion of \(p\)-invex sets with respect to \(\eta\) leads to a new class of functions. A family of real functions called, in general, \((p, r)\)-pre-invex functions with respect to \(\eta\) (without differentiability) or \((p,r)\)-invex functions with respect to \(\eta\) (in the differentiable case) is introduced. Some (geometric) properties of these classes of functions are derived. Sufficient optimality conditions are obtained for a nonlinear programming problem involving \((p, r)\)-invex functions with respect to \(\eta\) .

### MSC:

90C26 | Nonconvex programming, global optimization |

26B25 | Convexity of real functions of several variables, generalizations |

90C29 | Multi-objective and goal programming |

### Keywords:

\((p,r)\)-invex set with respect to \(\eta\); \(r\)-invex set with respect to \(\eta\); \((p,r)\)-pre-invex function with respect to \(\eta\); \((p,r)\)-invex function with respect to \(\eta\)
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\textit{T. Antczak}, J. Math. Anal. Appl. 263, No. 2, 355--379 (2001; Zbl 1051.90018)

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### References:

[1] | Antczak, T., Invexity in optimization theory, Proceedings of the V mathematical conference, (1998), Lesko |

[2] | T. Antczak, r-Invexity in Mathematical Programming, to be published. · Zbl 1097.90042 |

[3] | Antczak, T., (p,r)-invex sets and functions, (1998), University of Lódz |

[4] | Avriel, M., r-convex functions, Math. programming, 2, 309-323, (1972) · Zbl 0249.90063 |

[5] | Bazaraa, M.S.; Sherali, H.D.; Shetty, C.M., Nonlinear programming: theory and algorithms, (1991), Wiley New York |

[6] | Ben-Israel, A.; Mond, B., What is invexity?, J. austral. math. soc. ser. B, 28, 1-9, (1986) · Zbl 0603.90119 |

[7] | Hanson, M.A., On sufficiency of the kuhn – tucker conditions, J. math. anal. appl., 80, 545-550, (1981) · Zbl 0463.90080 |

[8] | Hanson, M.A.; Mond, B., Necessary and sufficient conditions in constrained optimization, Math. programming, 37, 51-58, (1987) · Zbl 0622.49005 |

[9] | Mangasarian, O.L., Nonlinear programming, (1969), McGraw-Hill New York · Zbl 0194.20201 |

[10] | Martin, D.H., The essence of invexity, J. optim. theory appl., 42, 65-76, (1985) · Zbl 0552.90077 |

[11] | Mirtinovic, D.S., Elementarne nierówności, (1972), PWN Warsaw |

[12] | Rockafellar, R.T., Convex analysis, (1970), Princeton Univ. Press Princeton · Zbl 0202.14303 |

[13] | Rueda, N.G.; Hanson, M.A., Optimality criteria in mathematical programming involving generalized invexity, J. math. anal. appl., 130, 375-385, (1988) · Zbl 0647.90076 |

[14] | Weir, T.; Jeyakumar, V., A class of nonconvex functions and mathematical programming, Bull. austral. math. soc., 38, 177-189, (1988) · Zbl 0639.90082 |

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