Advertisement

A note on approximate Karush–Kuhn–Tucker conditions in locally Lipschitz multiobjective optimization

  • Nguyen Van Tuyen
  • Jen-Chih Yao
  • Ching-Feng Wen
Original Paper

Abstract

In the recent paper of Giorgi et al. (J Optim Theory Appl 171:70–89, 2016), the authors introduced the so-called approximate Karush–Kuhn–Tucker (AKKT) condition for smooth multiobjective optimization problems and obtained some AKKT-type necessary optimality conditions and sufficient optimality conditions for weak efficient solutions of such a problem. In this note, we extend these optimality conditions to locally Lipschitz multiobjective optimization problems using Mordukhovich subdifferentials. Furthermore, we prove that, under some suitable additional conditions, an AKKT condition is also a KKT one.

Keywords

Approximate optimality conditions Multiobjective optimization problems Locally Lipschitz functions Mordukhovich subdifferential 

Notes

Acknowledgements

The authors would like to thank the referees for their constructive comments which significantly improve the presentation of the paper. J.-C. Yao and C.-F. Wen are supported by the Taiwan MOST (Grant Nos. 106-2923-E-039-001-MY3, 106-2115-M-037-001), respectively, as well as the Grant from Research Center for Nonlinear Analysis and Optimization, Kaohsiung Medical University, Taiwan.

References

  1. 1.
    Andreani, R., Birgin, E.G., Martínez, J.M., Schuverdt, M.L.: On augmented Lagrangian methods with general lower-level constraints. SIAM J. Optim. 18, 1286–1309 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Andreani, R., Martínez, J.M., Svaiter, B.F.: A new sequential optimality condition for constrained optimization and algorithmic consequences. SIAM J. Optim. 20, 3533–3554 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Andreani, R., Haeser, G., Martínez, J.M.: On sequential optimality conditions for smooth constrained optimization. Optimization 60, 627–641 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Andreani, R., Martínez, J.M., Ramos, A., Silva, P.J.S.: A cone-continuity constraint qualification and algorithmic consequences. SIAM J. Optim. 26, 96–110 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Birgin, E.G., Martínez, J.M.: Practical augmented Lagrangian methods for constrained optimization. In: Nicholas, J.H. (ed.) Fundamental of Algorithms, pp. 1–220. SIAM Publications, Philadelphia (2014)Google Scholar
  6. 6.
    Borwein, J.M., Zhu, Q.J.: Techniques of Variational Analysis. Springer, New York (2005)zbMATHGoogle Scholar
  7. 7.
    Chuong, T.D., Kim, D.S.: Optimality conditions and duality in nonsmooth multiobjective optimization problems. Ann. Oper. Res. 217, 117–136 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dutta, J., Deb, K., Tulshyan, R., Arora, R.: Approximate KKT points and a proximity measure for termination. J. Glob. Optim. 56, 1463–1499 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Giorgi, G., Jiménez, B., Novo, V.: Approximate Karush–Kuhn–Tucker condition in multiobjective optimization. J. Optim. Theory Appl. 171, 70–89 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Haeser, G., Schuverdt, M.L.: On approximate KKT condition and its extension to continuous variational inequalities. J. Optim. Theory Appl. 149, 528–539 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Haeser, G., Melo, V.V.: Convergence detection for optimization algorithms: approximate-KKT stopping criterion when Lagrange multipliers are not available. Oper. Res. Lett. 43, 484–488 (2015)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Mangasarian, O.L.: Nonlinear Programming. McGraw-Hill, New York (1969)zbMATHGoogle Scholar
  13. 13.
    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation I: Basis Theory. Springer, Berlin (2006)Google Scholar
  14. 14.
    Penot, J.P.: Calculus without derivatives. In: Sheldon, A., Kenneth, R. (eds.) Graduate Texts in Mathematics, vol. 266, pp. 1–524. Springer, New York (2013)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematical SciencesUniversity of Electronic Science and Technology of ChinaChengduPeople’s Republic of China
  2. 2.Department of MathematicsHanoi Pedagogical University 2Xuan Hoa, Phuc YenVietnam
  3. 3.Center for General EducationChina Medical UniversityTaichungTaiwan
  4. 4.Center for Fundamental Science; and Research Center for Nonlinear Analysis and OptimizationKaohsiung Medical UniversityKaohsiungTaiwan
  5. 5.Department of Medical ResearchKaohsiung Medical University HospitalKaohsiungTaiwan

Personalised recommendations