### Abstract

Original language | Undefined |
---|---|

Pages (from-to) | 333-353 |

Journal | Mathematical programming |

Volume | 34 |

Issue number | 3 |

DOIs | |

Publication status | Published - 1986 |

### Keywords

- Linear Index
- Generalized Critical PointCritical Point
- Mangasarian-Fromowitz Constraint
- Quadratic Index
- Parametric Optimization
- IR-85874
- Qualification
- Kuhn-Tucker Set

### Cite this

*Mathematical programming*,

*34*(3), 333-353. https://doi.org/10.1007/BF01582234

}

*Mathematical programming*, vol. 34, no. 3, pp. 333-353. https://doi.org/10.1007/BF01582234

**Critical sets in parametric optimization.** / Jongen, H.Th.; Jonker, P.; Twilt, F.

Research output: Contribution to journal › Article › Academic

TY - JOUR

T1 - Critical sets in parametric optimization

AU - Jongen, H.Th.

AU - Jonker, P.

AU - Twilt, F.

PY - 1986

Y1 - 1986

N2 - We deal with one-parameter families of optimization problems in finite dimensions. The constraints are both of equality and inequality type. The concept of a ‘generalized critical point’ (g.c. point) is introduced. In particular, every local minimum, Kuhn-Tucker point, and point of Fritz John type is a g.c. point. Under fairly weak (even generic) conditions we study the set∑ consisting of all g.c. points. Due to the parameter, the set∑ is pieced together from one-dimensional manifolds. The points of∑ can be divided into five (characteristic) types. The subset of ‘nondegenerate critical points’ (first type) is open and dense in∑ (nondegenerate means: strict complementarity, nondegeneracy of the corresponding quadratic form and linear independence of the gradients of binding constraints). A nondegenerate critical point is completely characterized by means of four indices. The change of these indices along∑ is presented. Finally, the Kuhn-Tucker subset of∑ is studied in more detail, in particular in connection with the (failure of the) Mangasarian-Fromowitz constraint qualification.

AB - We deal with one-parameter families of optimization problems in finite dimensions. The constraints are both of equality and inequality type. The concept of a ‘generalized critical point’ (g.c. point) is introduced. In particular, every local minimum, Kuhn-Tucker point, and point of Fritz John type is a g.c. point. Under fairly weak (even generic) conditions we study the set∑ consisting of all g.c. points. Due to the parameter, the set∑ is pieced together from one-dimensional manifolds. The points of∑ can be divided into five (characteristic) types. The subset of ‘nondegenerate critical points’ (first type) is open and dense in∑ (nondegenerate means: strict complementarity, nondegeneracy of the corresponding quadratic form and linear independence of the gradients of binding constraints). A nondegenerate critical point is completely characterized by means of four indices. The change of these indices along∑ is presented. Finally, the Kuhn-Tucker subset of∑ is studied in more detail, in particular in connection with the (failure of the) Mangasarian-Fromowitz constraint qualification.

KW - Linear Index

KW - Generalized Critical PointCritical Point

KW - Mangasarian-Fromowitz Constraint

KW - Quadratic Index

KW - Parametric Optimization

KW - IR-85874

KW - Qualification

KW - Kuhn-Tucker Set

U2 - 10.1007/BF01582234

DO - 10.1007/BF01582234

M3 - Article

VL - 34

SP - 333

EP - 353

JO - Mathematical programming

JF - Mathematical programming

SN - 0025-5610

IS - 3

ER -