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An Extension of Yuan’s Lemma and Its Applications in Optimization

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  • Gabriel Haeser

    (University of São Paulo)

Abstract

We prove an extension of Yuan’s lemma to more than two matrices, as long as the set of matrices has rank at most 2. This is used to generalize the main result of Baccari and Trad (SIAM J Optim 15(2):394–408, 2005), where the classical necessary second-order optimality condition is proved, under the assumption that the set of Lagrange multipliers is a bounded line segment. We prove the result under the more general assumption that the Hessian of the Lagrangian, evaluated at the vertices of the Lagrange multiplier set, is a matrix set with at most rank 2. We apply the results to prove the classical second-order optimality condition to problems with quadratic constraints and without constant rank of the Jacobian matrix.

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  • 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.
    • Handle: RePEc:spr:joptap:v:174:y:2017:i:3:d:10.1007_s10957-017-1123-2
    DOI: 10.1007/s10957-017-1123-2
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    References listed on IDEAS

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    1. B. T. Polyak, 1998. "Convexity of Quadratic Transformations and Its Use in Control and Optimization," Journal of Optimization Theory and Applications, Springer, vol. 99(3), pages 553-583, December.
    2. 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.
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    Cited by:

    1. Roger Behling & Gabriel Haeser & Alberto Ramos & Daiana S. Viana, 2018. "On a Conjecture in Second-Order Optimality Conditions," Journal of Optimization Theory and Applications, Springer, vol. 176(3), pages 625-633, March.

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