On a Conjecture in Second-Order Optimality Conditions

  • Roger Behling
  • Gabriel Haeser
  • Alberto Ramos
  • Daiana S. Viana
Article
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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.

Keywords

Nonlinear optimization Constraint qualifications Second-order optimality conditions Singular value decomposition 

Mathematics Subject Classification

90C46 90C30 15B10 

Notes

Acknowledgements

This research was partially conducted while the second author held a Visiting Scholar position at Department of Management Science and Engineering, Stanford University, Stanford CA, USA. The fourth author was a Ph.D. student at Department of Applied Mathematics, University of São Paulo-SP, Brazil. We thank the referees and the handling editor for insightful remarks. This research was funded by FAPESP Grants 2013/05475-7 and 2016/02092-8, by CNPq Grants 454798/2015-6, 303264/2015-2 and 481992/2013-8, and CAPES.

References

  1. 1.
    Behling, R., Haeser, G., Ramos, A., Viana, D.: On a conjecture in second-order optimality conditions. arXiv:1706.07833 (2017). Extended Technical Report
  2. 2.
    Baccari, A., Trad, A.: On the classical necessary second-order optimality conditions in the presence of equality and inequality constraints. SIAM J. Optim. 15(2), 394–408 (2005)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Andreani, R., Martínez, J.M., Schuverdt, M.L.: On second-order optimality conditions for nonlinear programming. Optimization 56, 529–542 (2007)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Shen, C., Xue, W., An, Y.: A new result on second-order necessary conditions for nonlinear programming. Oper. Res. Lett. 43, 117–122 (2015)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Minchenko, L., Leschov, A.: On strong and weak second-order necessary optimality conditions for nonlinear programming. Optimization 65(9), 1693–1702 (2016)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Haeser, G.: An extension of Yuan’s Lemma and its applications in optimization. J. Optim. Theory Appl. 174(3), 641–649 (2017)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Haeser, G., Ramos, A.: A note on the smoothness of multi-parametric singular value decomposition with applications in optimization. Optimization Online (2017)Google Scholar
  8. 8.
    Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)CrossRefMATHGoogle Scholar
  9. 9.
    Ben-Tal, A.: Second-order and related extremality conditions in nonlinear programming. J. Optim. Theory Appl. 31(2), 143–165 (1980)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Yuan, Y.: On a subproblem of trust region algorithms for constrained optimization. Math. Program. 47, 53–63 (1990)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Andreani, R., Behling, R., Haeser, G., Silva, P.J.S.: On second order optimality conditions for nonlinear optimization. Optim. Methods Softw. 32(1), 22–38 (2017)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Minchenko, L., Stakhovski, S.: On relaxed constant rank regularity condition in mathematical programming. Optimization 60(4), 429–440 (2011)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Andreani, R., Echagüe, C.E., Schuverdt, M.L.: Constant-rank condition and second-order constraint qualification. J. Optim. Theory Appl. 146(2), 255–266 (2010)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Minchenko, L., Stakhovski, S.: Parametric nonlinear programming problems under the relaxed constant rank condition. SIAM J. Optim. 21, 314–332 (2011)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Bomze, I.: Copositivity for second-order optimality conditions in general smooth optimization problems. Optimization 65(4), 779–795 (2016)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Federal University of Santa CatarinaBlumenauBrazil
  2. 2.Department of Applied MathematicsUniversity of São PauloSão PauloBrazil
  3. 3.Department of MathematicsFederal University of ParanáCuritibaBrazil
  4. 4.Center of Exact and Technological SciencesFederal University of AcreRio BrancoBrazil

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