Critical sets in parametric optimization

H.Th. Jongen, P. Jonker, F. Twilt

Research output: Contribution to journalArticleAcademic

78 Citations (Scopus)
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Abstract

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.
Original languageUndefined
Pages (from-to)333-353
JournalMathematical programming
Volume34
Issue number3
DOIs
Publication statusPublished - 1986

Keywords

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

Cite this

Jongen, H. T., Jonker, P., & Twilt, F. (1986). Critical sets in parametric optimization. Mathematical programming, 34(3), 333-353. https://doi.org/10.1007/BF01582234
Jongen, H.Th. ; Jonker, P. ; Twilt, F. / Critical sets in parametric optimization. In: Mathematical programming. 1986 ; Vol. 34, No. 3. pp. 333-353.
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Jongen, HT, Jonker, P & Twilt, F 1986, 'Critical sets in parametric optimization' 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.

In: Mathematical programming, Vol. 34, No. 3, 1986, p. 333-353.

Research output: Contribution to journalArticleAcademic

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.

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KW - Kuhn-Tucker Set

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