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On the Classical Necessary Second-Order Optimality Conditions

Author

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  • A. Baccari

    (Ecole Supérieure des Sciences et Techniques de Tunis)

Abstract

In this paper, the property of a necessary second-order optimality condition to hold with the same Lagrange multiplier for all critical vectors is investigated. It is limited to nonconvex optimization Problems in ℝn with equality and inequality constraints; the Mangasarian-Fromovitz constraint qualification is assumed to be hold. A counterexample was given recently by Anitescu. We give some sufficient conditions and we prove that this property holds if n ≤ 2 or if the number of active inequality constraints is at most two. For three active inequality constraints and n =3, a counterexample is given.

Suggested Citation

  • A. Baccari, 2004. "On the Classical Necessary Second-Order Optimality Conditions," Journal of Optimization Theory and Applications, Springer, vol. 123(1), pages 213-221, October.
    • Handle: RePEc:spr:joptap:v:123:y:2004:i:1:d:10.1023_b:jota.0000043998.04008.e6
    DOI: 10.1023/B:JOTA.0000043998.04008.e6
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    Cited by:

    1. 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.
    2. Giorgio, 2019. "On Second-Order Optimality Conditions in Smooth Nonlinear Programming Problems," DEM Working Papers Series 171, University of Pavia, Department of Economics and Management.
    3. Gabriel Haeser, 2017. "An Extension of Yuan’s Lemma and Its Applications in Optimization," Journal of Optimization Theory and Applications, Springer, vol. 174(3), pages 641-649, September.
    4. María C. Maciel & Sandra A. Santos & Graciela N. Sottosanto, 2011. "On Second-Order Optimality Conditions for Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 149(2), pages 332-351, May.
    5. A. Izmailov & M. Solodov, 2009. "Examples of dual behaviour of Newton-type methods on optimization problems with degenerate constraints," Computational Optimization and Applications, Springer, vol. 42(2), pages 231-264, March.
    6. Mengmeng Song & Yong Xia, 2023. "Calabi-Polyak convexity theorem, Yuan’s lemma and S-lemma: extensions and applications," Journal of Global Optimization, Springer, vol. 85(3), pages 743-756, March.
    7. 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.
    8. 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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