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A Guided Tour in Constraint Qualifications for Nonlinear Programming under Differentiability Assumptions

Author

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  • Giorgio Giorgi

    (Department of Economics and Management, University of Pavia)

Abstract

It is well-known that the celebrated Kuhn-Tucker or Karush-Kuhn-Tucker necessary optimality conditions hold at a local solution point of a nonlinear programming problem if some regularity conditions, usually called "constraint qualifcations", are satisfied. In the present paper we give an up-to-date overview of several constraint quali?cations proposed in the literature for a nonlinear programming problem, under differentiability assumptions. In particular, we point out the various implications existing among the constraint qualifications considered. For the reader's convenience we shall consider separately the case of inequality constraints only and the case of mixed equality and inequality constraints. Some remarks on second-order constraint qualifications are made and some historical notes on this subject are given.

Suggested Citation

  • Giorgio Giorgi, 2018. "A Guided Tour in Constraint Qualifications for Nonlinear Programming under Differentiability Assumptions," DEM Working Papers Series 160, University of Pavia, Department of Economics and Management.
    • Handle: RePEc:pav:demwpp:demwp0160
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    File URL: http://dem-web.unipv.it/web/docs/dipeco/quad/ps/RePEc/pav/demwpp/DEMWP0160.pdf
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    References listed on IDEAS

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    1. B. Bernholtz, 1964. "A New Derivation of the Kuhn-Tucker Conditions," Operations Research, INFORMS, vol. 12(2), pages 295-299, April.
    2. R. Andreani & J. M. Martinez & M. L. Schuverdt, 2005. "On the Relation between Constant Positive Linear Dependence Condition and Quasinormality Constraint Qualification," Journal of Optimization Theory and Applications, Springer, vol. 125(2), pages 473-483, May.
    3. R. Andreani & C. E. Echagüe & M. L. Schuverdt, 2010. "Constant-Rank Condition and Second-Order Constraint Qualification," Journal of Optimization Theory and Applications, Springer, vol. 146(2), pages 255-266, August.
    4. David G. Luenberger & Yinyu Ye, 2008. "Linear and Nonlinear Programming," International Series in Operations Research and Management Science, Springer, edition 0, number 978-0-387-74503-9, September.
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    6. Giorgio Giorgi & Bienvenido Jiménez & Vicente Novo, 2014. "Some Notes on Approximate Optimality Conditions in Scalar and Vector Optimization Problems," DEM Working Papers Series 095, University of Pavia, Department of Economics and Management.
    7. Alberto Cambini & Laura Martein, 2009. "Generalized Convexity and Optimization," Lecture Notes in Economics and Mathematical Systems, Springer, number 978-3-540-70876-6, December.
    8. Giorgio Giorgi & Bienvenido Jiménez & Vicente Novo, 2016. "Approximate Karush–Kuhn–Tucker Condition in Multiobjective Optimization," Journal of Optimization Theory and Applications, Springer, vol. 171(1), pages 70-89, October.
    9. Lei Guo & Jin Zhang & Gui-Hua Lin, 2014. "New Results on Constraint Qualifications for Nonlinear Extremum Problems and Extensions," Journal of Optimization Theory and Applications, Springer, vol. 163(3), pages 737-754, December.
    10. D.P. Bertsekas & A.E. Ozdaglar, 2002. "Pseudonormality and a Lagrange Multiplier Theory for Constrained Optimization," Journal of Optimization Theory and Applications, Springer, vol. 114(2), pages 287-343, August.
    11. Giorgio Giorgi, 2017. "Minimum Principle-Type Necessary Optimality Conditions in Scalar and Vector Optimization. An Account," Journal of Mathematics Research, Canadian Center of Science and Education, vol. 9(4), pages 168-184, August.
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    Cited by:

    1. G. Haeser & A. Ramos, 2020. "New Constraint Qualifications with Second-Order Properties in Nonlinear Optimization," Journal of Optimization Theory and Applications, Springer, vol. 184(2), pages 494-506, February.
    2. Giorgio Giorgi, 2019. "Notes on Constraint Qualifications for Second-Order Optimality Conditions," Journal of Mathematics Research, Canadian Center of Science and Education, vol. 11(5), pages 16-32, October.

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