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On a Conjecture in Second-Order Optimality Conditions

Author

Listed:
  • Roger Behling

    (Federal University of Santa Catarina)

  • Gabriel Haeser

    (University of São Paulo)

  • Alberto Ramos

    (Federal University of Paraná)

  • Daiana S. Viana

    (Federal University of Acre)

Abstract

In this paper, we deal with a conjecture formulated in Andreani et al. (Optimization 56:529–542, 2007), which states that whenever a local minimizer of a nonlinear optimization problem fulfills the Mangasarian–Fromovitz constraint qualification and the rank of the set of gradients of active constraints increases at most by one in a neighborhood of the minimizer, a second-order optimality condition that depends on one single Lagrange multiplier is satisfied. This conjecture generalizes previous results under a constant rank assumption or under a rank deficiency of at most one. We prove the conjecture under the additional assumption that the Jacobian matrix has a smooth singular value decomposition. Our proof also extends to the case of the strong second-order condition, defined in terms of the critical cone instead of the critical subspace.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:176:y:2018:i:3:d:10.1007_s10957-018-1229-1
    DOI: 10.1007/s10957-018-1229-1
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    References listed on IDEAS

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    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. 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.
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