Skip to main content
SpringerLink
Log in

On \({\epsilon}\) -solutions for convex optimization problems with uncertainty data

  • Published:
Positivity Aims and scope Submit manuscript

Abstract

We consider \({\epsilon}\) -solutions (approximate solutions) for a robust convex optimization problem in the face of data uncertainty. Using robust optimization approach (worst-case approach), we establish an optimality theorem and duality theorems for \({\epsilon}\) -solutions for the robust convex optimization problem. Moreover, we give an example illustrating the duality theorems.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

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

    Article  MathSciNet  MATH  Google Scholar 

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

  3. Hamel A.: An \({\epsilon}\) -Lagrange multiplier rule for a mathematical programming problem on Banach spaces. Optimization 49, 137–149 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  4. Gutiérrez C., Jiménez B., Novo V.: Multiplier rules and saddle-point theorems for Helbig’s approximate solutions in convex Pareto problems. J. Global. Optim. 32, 367–383 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  5. Liu J.C.: \({\epsilon}\) -Duality theorem of nondifferentiable nonconvex multiobjective programming. J. Optim. Th. Appl. 69, 153–167 (1991)

    Article  MATH  Google Scholar 

  6. Liu J.C.: \({\epsilon}\) -Pareto optimality for nondifferentiable multiobjective programming via penalty function. J. Math. Anal. Appl. 198, 248–261 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  7. Strodiot J.J., Nguyen V.H., Heukemes N.: \({\epsilon}\) -Optimal solutions in nondifferentiable convex programming and some related questions. Math. Program. 25, 307–328 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  8. Yokoyama K.: Epsilon approximate solutions for muliobjective programming problems. J. Math. Anal. Appl. 203, 142–149 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  9. Govil M.G., Mehra A.: \({\epsilon}\) -Optimality for multiobjective programming on a Banach space. Eur. J. Oper. Res. 157, 106–112 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  10. Lee G.M., Lee J.H.: \({\epsilon}\) -Duality for convex semidefinite optimization problem with conic constraints. J. Ineq. Appl. 2010, 106–112 (2010)

    Google Scholar 

  11. Beck A., Ben-Tal A.: Duality in robust optimization: primal worst equals dual best. Oper. Res. Lett. 37, 1–6 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ben-Tal A., Nemirovski A.: Robust optimization—methodology and applications. Math. Program. Ser. B 92, 453–480 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  13. Ben-Tal, A., Ghaoui, L.E., Nemirovski, A.: Robust optimization. In: Princeton Series in Applied Mathematics (2009)

  14. Bersimas D., Brown D.: Constructing uncertainty sets for robust linear optimization. Oper. Res. 57, 1483–1495 (2009)

    Article  MathSciNet  Google Scholar 

  15. Bersimas D., Pachamanova D., Sim M.: Robust linear optimization under general norms. Oper. Res. Lett. 32, 510–516 (2004)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  18. Rockafellar R.T.: Convex Analysis. Princeton Univ. Press, Princeton (1970)

    MATH  Google Scholar 

  19. Boyd, S., Vandemberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)

  20. Hiriart-Urruty J.B., Lemarechal C.: Convex Analysis and Minimization Algorithms. Springer, Berlin (1993)

    Google Scholar 

  21. 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  MathSciNet  MATH  Google Scholar 

  22. 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  MathSciNet  MATH  Google Scholar 

  23. Li C., Ng K.F., Pong T.K.: Constraint qualifications for convex inequality systems with applications in constrainted optimization. SIAM. J. Optim. 19, 163–187 (2008)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, Pukyong National University, Busan, 608-737, Korea

    Jae Hyoung Lee & Gue Myung Lee

Authors
  1. Jae Hyoung Lee
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gue Myung Lee
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gue Myung Lee.

Additional information

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

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (No. 2011-0018619).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lee, J.H., Lee, G.M. On \({\epsilon}\) -solutions for convex optimization problems with uncertainty data. Positivity 16, 509–526 (2012). https://doi.org/10.1007/s11117-012-0186-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11117-012-0186-4

Keywords

Mathematics Subject Classification

Search

Navigation