Skip to main content
Log in

On quasi \(\epsilon \)-solution for robust convex optimization problems

  • Original Paper
  • Published:
Optimization Letters Aims and scope Submit manuscript

Abstract

This paper devotes to the quasi \(\epsilon \)-solution (one sort of approximate solutions) for a robust convex optimization problem in the face of data uncertainty. Using robust optimization approach (worst-case approach), we establish approximate optimality theorem and approximate duality theorems in term of Wolfe type on quasi \(\epsilon \)-solution for the robust convex optimization problem. Moreover, some examples are given to illustrate the obtained results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Ben-Tal, A., Nemirovski, A.: A selected topics in robust convex optimization. Math. Progr. Ser. B. 112, 125–158 (2008)

  2. Boyd, S., Vandemberghe, L.: Convex optimization. Cambridge Univ. Press, Cambridge (2004)

    Book  Google Scholar 

  3. Chuong, T.D., Kim, D.S.: Approximate solutions of multiobjective optimization problems. Positivity 20, 187–207 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  4. Dutta, J.: Necessary optimality conditions and saddle points for approximate optimization in Banach spaces. Top 13, 127–143 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  5. Dutta, J., Deb, K., Tulshyan, R., Arora, R.: Approximate KKT points and a proximity measure for termination. J. Glob. Optim. 56, 1463–1499 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  6. Ekeland, I.: On the variational principle. J. Math. Anal. Appl. 47, 324–353 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  7. Houda, M.: Comparison of approximations in stochastic and robust optimization programs. In: Hušková, M., Janžura, M. (eds.) Prague stochastics 2006, pp. 418–425. Matfyzpress, Prague (2006)

    Google Scholar 

  8. Jeyakumar, V.: Asymptotic dual conditions characterizing optimality for convex programs. J. Optim. Theory Appl. 93, 153–165 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  9. Jeyakumar, V., Lee, G.M., Dinh, N.: Characterization of solution sets of convex vector minimization problems. Eur. J. Oper. Res. 174, 1380–1395 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  10. Jeyakumar, V., Lee, G.M., Dinh, N.: New sequential Lagrange multiplier conditions characterizing optimality without constraint qualification for convex programs. SIAM J. Optim. 14, 534–547 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  11. Jeyakumar, V., Li, G.Y.: Strong duality in robust convex programming: complete characterizations. SIAM J. Optim. 20, 3384–3407 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  12. Lee, J.H., Lee, G.M.: On \(\epsilon \)-solutions for convex optimization problems with uncertainty data. Positivity 16, 509–526 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  13. Loridan, P.: Necessary conditions for \(\epsilon \)-optimality. Math. Progr. Stud. 19, 140–152 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  14. Rockafellar, R.T.: Convex analysis. Princeton Univ. Press, Princeton (1970)

    Book  MATH  Google Scholar 

  15. Rockafellar, R.T., Wets, R.J.B.: Variational analysis. Springer, Berlin (2001)

    MATH  Google Scholar 

  16. Son, T.Q., Strodiot, J.J., Nguyen, V.H.: \(\epsilon \)-optimality and \(\epsilon \)-Lagrangian duality for a nonconvex programming problem with an infinite number of constraints. J. Optim. Theory Appl. 141, 389–409 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  17. Sun, X.K., Peng, Z.Y., Guo, X.L.: Some characterizations of robust optimal solutions for uncertain convex optimization problems. Optim. Lett. (2015). doi:10.1007/s11590-015-0946-8

    Google Scholar 

Download references

Acknowledgments

The authors would like to express their sincere thanks to anonymous referees for variable suggestions and comments for the paper.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Liguo Jiao.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lee, J.H., Jiao, L. On quasi \(\epsilon \)-solution for robust convex optimization problems. Optim Lett 11, 1609–1622 (2017). https://doi.org/10.1007/s11590-016-1067-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-016-1067-8

Keywords

Mathematics Subject Classification

Navigation