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The Importance of Being Normal, Regular and Proper in the Calculus of Variations

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  • Jorge A. Becerril

    (Universidad Nacional Autónoma de México)

  • Javier F. Rosenblueth

    (Universidad Nacional Autónoma de México)

Abstract

Some classical references in the literature include second-order necessary conditions for the isoperimetric problem of Lagrange in the calculus of variations involving inequality and equality constraints. Those conditions take into account, in the sets of critical directions, the sign of the Lagrange multipliers of the extremal under consideration. However, they hold under a strong assumption of normality relative to a set defined only by equality constraints for active indices. In this paper, we show how the same conditions can be preserved under a weaker assumption, thus providing a wider range of applicability.

Suggested Citation

  • Jorge A. Becerril & Javier F. Rosenblueth, 2017. "The Importance of Being Normal, Regular and Proper in the Calculus of Variations," Journal of Optimization Theory and Applications, Springer, vol. 172(3), pages 759-773, March.
    • Handle: RePEc:spr:joptap:v:172:y:2017:i:3:d:10.1007_s10957-017-1070-y
    DOI: 10.1007/s10957-017-1070-y
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    References listed on IDEAS

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    1. Aram Arutyunov & Fernando Lobo Pereira, 2006. "Second-Order Necessary Optimality Conditions for Problems Without A Priori Normality Assumptions," Mathematics of Operations Research, INFORMS, vol. 31(1), pages 1-12, February.
    2. G. Cullingford & J. D. C. A. Prideaux, 1973. "A Variational Study of Optimal Resource Profiles," Management Science, INFORMS, vol. 19(9), pages 1067-1081, May.
    3. Hadley, G. & Kemp, M. C., 1971. "Variational Methods in Economics," Elsevier Monographs, Elsevier, edition 1, number 9780720436013 edited by Bliss, C. J..
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    Cited by:

    1. Jorge Becerril & Karla Cortez, 2019. "Normality and Uniqueness of Multipliers in Isoperimetric Control Problems," Journal of Optimization Theory and Applications, Springer, vol. 182(3), pages 947-964, September.

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