### 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 |
---|---|

Pages (from-to) | 241-258 |

Number of pages | 18 |

Journal | Set-Valued Analysis |

Volume | 12 |

Issue number | 1-2 |

Publication status | Published - Mar 2004 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

### Cite this

*Set-Valued Analysis*,

*12*(1-2), 241-258.

**Critical and critical tangent cones in optimization problems.** / Páles, Z.; Zeidan, Vera.

Research output: Contribution to journal › Article

*Set-Valued Analysis*, vol. 12, no. 1-2, pp. 241-258.

}

TY - JOUR

T1 - Critical and critical tangent cones in optimization problems

AU - Páles, Z.

AU - Zeidan, Vera

PY - 2004/3

Y1 - 2004/3

N2 - 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.

AB - 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.

KW - Critical cone

KW - Critical tangent cone

KW - First- and second-order optimality conditions

KW - Set-valued constraints

UR - http://www.scopus.com/inward/record.url?scp=3943058201&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=3943058201&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:3943058201

VL - 12

SP - 241

EP - 258

JO - Set-Valued and Variational Analysis

JF - Set-Valued and Variational Analysis

SN - 1877-0533

IS - 1-2

ER -