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**Invexity and generalized convexity.**
*(English)*
Zbl 0731.26009

A differentiable function \(f:{\mathbb{R}}^ n\to {\mathbb{R}}\) is said to be invex if there exists a function \(\eta (y,x)\in {\mathbb{R}}^ n\) such that, for all \(y,x\in {\mathbb{R}}^ n\)
\[
f(y)-f(x)\geq \eta (y,x)^ t\nabla f(x).
\]
Various extensions of such functions including pseudo- and quasi-invex have been defined and their relationship to each other and other generalizations of convexity have been studied. For the non- differentiable case, f is said to be pre-invex if
\[
f(x+t\eta (y,x))\leq tf(y)+(1-t)f(x),\quad 0\leq t\leq 1.
\]
This comprehensive paper brings together many of these scattered results which are then studied, compared, and extended. Some definitions that are introduced include \(\eta\)-invex subsets, pseudo and quasi-pre-invex functions. One very minor correction. Reference 2 should be to the Journal, not the Bulletin, of the Australian Mathematical Society.

Reviewer: B.Mond (Bundoora)

### MSC:

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

90C30 | Nonlinear programming |

Full Text:
DOI

### References:

[1] | Avriel Diewert, Generalized concavity (1988) |

[2] | DOI: 10.1017/S0334270000005142 · Zbl 0603.90119 |

[3] | Bector C.R., Congressus Numerantium 52 pp 53– (1986) |

[4] | DOI: 10.1007/BF00932539 · Zbl 0325.26007 |

[5] | DOI: 10.1017/S0004972700004895 · Zbl 0452.90066 |

[6] | DOI: 10.1017/S1446788700022126 |

[7] | DOI: 10.1016/0022-247X(81)90123-2 · Zbl 0463.90080 |

[8] | DOI: 10.1016/0377-2217(82)90181-3 · Zbl 0501.90090 |

[9] | Rapcsak T., Publicationes Matematicae 34 pp 35– (1987) |

[10] | DOI: 10.1016/0022-247X(88)90113-8 · Zbl 0663.90087 |

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