Skip to main content
Log in

Characterizations of robust solution set of convex programs with uncertain data

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

Abstract

In this paper, we study convex programming problems with data uncertainty in both the objective function and the constraints. Under the framework of robust optimization, we employ a robust regularity condition, which is much weaker than the ones in the open literature, to establish various properties and characterizations of the set of all robust optimal solutions of the problems. These are expressed in term of subgradients, Lagrange multipliers and epigraphs of conjugate functions. We also present illustrative examples to show the significances of our theoretical 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. Mangasarian, O.L.: A simple characterization of solution sets of convex programs. Oper. Res. Lett. 7, 21–26 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  2. Burke, J.V., Ferris, M.: Characterization of solution sets of convex programs. Oper. Res. Lett. 10, 57–60 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  3. Jeyakumar, V., Yang, X.Q.: On characterizing the solution sets of pseudolinear programs. J. Optim. Theory Appl. 87(3), 747–755 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  4. Penot, J.P.: Characterization of solution sets of quasiconvex programs. J. Optim. Theory Appl. 117, 627–636 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  5. 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 

  6. Wu, Z.L., Wu, S.Y.: Characterizations of the solution sets of convex programs and variational inequality problems. J. Optim. Theory Appl. 130, 339–358 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  8. Son, T.Q., Dinh, N.: Characterizations of optimal solution sets of convex infinite programs. TOP. 16, 147–163 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Yang, X.M.: On characterizing the colution cets of pseudoinvex extremum problems. J. Optim. Theory Appl. 140, 537–542 (2009)

    Article  MathSciNet  Google Scholar 

  10. Lalitha, C.S., Mehta, M.: Characterizations of solution sets of mathematical programs in terms of Lagrange multipliers. Optimization 58, 995–1007 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Castellani, M., Giuli, M.: A characterization of the solution set of pseudoconvex extremum problems. J. Convex Anal. 19, 113–123 (2012)

    MathSciNet  MATH  Google Scholar 

  12. Zhao, K.Q., Tang, L.P.: On characterizing solution set of non-differentiable \(\eta \)-pseudolinear extremum problem. Optimization 61, 239–249 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. Zhao, K.Q., Yang, X.M.: Characterizations of the solution set for a class of nonsmooth optimization problems. Optim. Lett. 7, 685–694 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. Son, T.Q., Kim, D.S.: A new approach to characterize the solution set of a pseudoconvex programming problem. J. Comput. Appl. Math. 261, 333–340 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ben-Tal, A., Ghaoui, L.E., Nemirovski, A.: Robust Optimization. Princeton University Press, Princeton (2009)

    Book  MATH  Google Scholar 

  16. 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 

  17. Goberna, M.A., Jeyakumar, V., Li, G., Lopez, M.: Robust linear semi-infinite programming duality. Math. Program, Series B 139, 185–203 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. Jeyakumar, V., Li, G.: Characterizing robust set containments and solutions of uncertain linear programs without qualifications. Oper. Res. Lett. 38, 188–194 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  19. Jeyakumar, V., Li, G.: Strong duality in robust convex programming: complete characterizations. SIAM J. Optim. 20(6), 3384–3407 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  20. Jeyakumar, V., Li, G., Wang, J.H.: Some robust convex programs without a duality gap. J. Convex Anal. 20(2), 377–394 (2013)

    MathSciNet  MATH  Google Scholar 

  21. Jeyakumar, V., Lee, G.M., Li, G.: Characterizing robust solution sets of convex programs under data uncertainty. J. Optim. Theory Appl. 164, 407–435 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. Sun, X.K., Peng, Z.Y., Guo, X.L.: Some characterizations of robust optimal solutions for uncertain convex optimization problems. Optim. Lett. 10, 1463–1478 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  23. Sun, X.K., Long, X.J., Fu, H.Y., Li, X.B.: Some characterizations of robust optimal solutions for uncertain fractional optimization and applications. J. Ind. Manag. Optim. 13, 803–824 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  24. Bot, R.I., Wanka, G.: A weaker regularity condition for subdifferential calculus and Fenchel duality in infinite dimensional spaces. Nonlinear Anal. 64, 2787–2804 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  25. Fang, D.H., Li, C., Yang, X.Q.: Stable and total Fenchel duality for DC optimization problems in locally convex spaces. SIAM J. Optim. 21, 730–760 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors express their deep gratitude to the anonymous referees and the associate editor for their valuable comments and suggestions that helped to improve this article. Xiao-Bing Li would like to thank Prof. Song Wang for his hospitality during his stay from February 2015 to February 2016 at the School of Mathematics and Statistics of the Curtin University in Perth. This research was partially supported by the Basic and Advanced Research Project of Chongqing (cstc2015jcyjBX0131, cstc2015jcyjA30009), the Program of Chongqing Innovation Team Project in University under Grant (CXTDX201601022) and the Program for Core Young Teacher of the Municipal Higher Education of Chongqing ([2014]47).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Xiao-Bing Li.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, XB., Wang, S. Characterizations of robust solution set of convex programs with uncertain data. Optim Lett 12, 1387–1402 (2018). https://doi.org/10.1007/s11590-017-1187-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-017-1187-9

Keywords

Navigation