### Abstract

In this paper the notion of critical tangent cone CT (x|Q) to Q at x is introduced for the case when Q is a convex subset of a normed space X. If Q is closed with nonempty interior, and x ∈ Q, the nonemptiness of the Dubovitskii-Milyutin set of second-order admissible variations, V (x, d|Q), is then characterized by the condition d ∈ CT (x|Q). Furthermore, the support function of V (x, d|Q) is shown to be evaluated in terms of that support function of Q. For the cases when Q is the set of continuous or ℒ^{∞} selections of a certain set-valued map, the corresponding characterization of the cone CT (x|Q) and the formula for the support function of V (x, d|Q) are obtained in terms of more verifiable conditions.

Original language | English |
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Pages (from-to) | 241-258 |

Number of pages | 18 |

Journal | Set-Valued Analysis |

Volume | 12 |

Issue number | 1-2 |

DOIs | |

Publication status | Published - Jan 1 2004 |

### Keywords

- Critical cone
- Critical tangent cone
- First- and second-order optimality conditions
- Set-valued constraints

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

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## Cite this

*Set-Valued Analysis*,

*12*(1-2), 241-258. https://doi.org/10.1023/b:svan.0000023389.17834.95