## Abstract

For a nonempty closed set C in a real normed vector space X and an inequality solution set, we present several sufficient conditions for the tangent and contingent cones to their intersection to contain the intersections of the corresponding cones. We not only express the contingent cone to a solution set of inequalities and equalities by the directional (or Frchet) derivatives of the active inequality constraint functions and the Frchet derivatives of the equality constraint functions but also the tangent cone by the Clarke (or lower Dini, or upper Dini) derivatives of the active inequality constraint functions and the directional derivatives of the equality constraint functions. By using a simple property of the function d_{C} - d _{C}^{c}, we characterize these cones by the hypertangent and hypercontingent vectors to the set C. Furthermore, these results allow us to present new constraint qualifications for the Karush-Kuhn-Tucker conditions.

Original language | English |
---|---|

Pages (from-to) | 1203-1220 |

Number of pages | 18 |

Journal | Nonlinear Analysis, Theory, Methods and Applications |

Volume | 73 |

Issue number | 5 |

DOIs | |

Publication status | Published - 1 Sept 2010 |

## Keywords

- Contingent cone
- Hypercontingent vector
- Hypertangent vector
- Karush-Kuhn-Tucker condition
- Mangasarian-Fromovitz constraint qualification
- Tangent cone