Skip to main content
SpringerLink
Log in

Robust Duality in Parametric Convex Optimization

  • Published:
Set-Valued and Variational Analysis Aims and scope Submit manuscript

Abstract

Modelling of convex optimization in the face of data uncertainty often gives rise to families of parametric convex optimization problems. This motivates us to present, in this paper, a duality framework for a family of parametric convex optimization problems. By employing conjugate analysis, we present robust duality for the family of parametric problems by establishing strong duality between associated dual pair. We first show that robust duality holds whenever a constraint qualification holds. We then show that this constraint qualification is also necessary for robust duality in the sense that the constraint qualification holds if and only if robust duality holds for every linear perturbation of the objective function. As an application, we obtain a robust duality theorem for the best approximation problems with constraint data uncertainty under a strict feasibility condition.

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. Bank, B., Guddat, J., Klatte, D., Kummer, B., Tammer, K.: Nonlinear Parametric Optimization. Birkhäuser Verlag, Basel-Boston (1983)

    Google Scholar 

  2. Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilber Spaces. CMS Books in Mathematics. Springer, New York Dordrecht Heidelberg London (2011)

    Book  Google Scholar 

  3. Boţ, R.I.: Conjugate Duality in Convex Optimization. Lecture Notes in Economics and Mathematical Systems, vol. 637. Springer, Berlin Heidelberg (2010)

    Google Scholar 

  4. Boţ, R.I., Grad, S.-M.: Lower semicontinuous type regularity conditions for subdifferential calculus. Optim. Methods Softw. 25(1), 37–48 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Boţ, R.I., Grad, S.-M., Wanka, G.: New regularity conditions for strong and total Fenchel-Lagrange duality in infinite dimensional spaces. Nonlinear Anal. 69(1), 323–336 (2007)

    Google Scholar 

  6. Boţ, R.I., Grad, S.-M., Wanka, G.: On strong and total Lagrange duality for convex optimization problems. J. Math. Anal. Appl. 337(2), 1315–1325 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Boţ, R.I., Wanka, G.: A weaker regularity condition for subdifferential calculus and Fenchel duality in infinite dimensional spaces. Nonlinear Anal. 64(12), 2787–2804 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  8. Burachik, R.S., Jeyakumar, V.: A new geometric condition for Fenchel’s duality in infinite dimensional spaces. Math. Program. 104(2–3), 229–233 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  9. Charnes, A., Cooper, W.W., Kortanek, K.O.: Duality in semi-infinite programs and some works of Haar and Carathéodory. Manage. Sci. 9, 209–228 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ekeland, I., Temam, R.: Convex Analysis and Variational Problems. North-Holland Publishing Company, Amsterdam (1976)

  11. Goberna, M.A., López, M.A.: Linear Semi-Infinite Optimization. Wiley, Chichester (1998)

    MATH  Google Scholar 

  12. Guddat, J., Guerra, V.F., Jongen, H.: Parametric Optimization: Singularities, Pathfollowing and Jumps. B. G. Teubner, Stuttgart; Wiley, Chichester (1990)

  13. Haar, A.: Über lineare Ungleichungen. Acta Sci. Math. 2, 1–14 (1924)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  15. 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 

  16. Jeyakumar, V., Wang, J.H., Li, G.: Lagrange multiplier characterizations of robust best approximations under constraint data uncertainty. J. Math. Anal. Appl. 393, 285–297 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Levy, A.B., Mordukhovich, B.S.: Coderivatives in parametric optimization. Math. Program. 99(A), 311–327 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  18. Li, G.Y., Jeyakumar, V., Lee, G.M.: Robust conjugate duality for convex optimization under uncertainty with application to data classification. Nonlinear Anal. 74(6), 2327–2341 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  19. Li, G., Ng, K.F.: On extension of Fenchel duality and its application. SIAM J. Optim. 19, 1489–1509 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  20. Qi, H.D.: Some theoretical aspects of Newton’s method for constrained best interpolation. In: Jeyakumar, V., Rubinov, A. (eds.) Continuous Optimization, Current Trends and Modern Applications. Applied Optimization, vol. 99, pp. 23–49. Springer, New York (2005)

  21. Zălinescu, C.: Convex Analysis in General Vector Spaces. World Scientific, River Edge (2002)

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Chemnitz University of Technology, 09107, Chemnitz, Germany

    R. I. Boţ

  2. Department of Applied Mathematics, University of New South Wales, Sydney, 2052, Australia

    V. Jeyakumar & G. Y. Li

Authors
  1. R. I. Boţ
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. Jeyakumar
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. G. Y. Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. Jeyakumar.

Additional information

The authors are grateful to the referees for their constructive comments and helpful suggestions which have contributed to the final preparation of the paper.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Boţ, R.I., Jeyakumar, V. & Li, G.Y. Robust Duality in Parametric Convex Optimization. Set-Valued Var. Anal 21, 177–189 (2013). https://doi.org/10.1007/s11228-012-0219-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11228-012-0219-y

Keywords

Mathematics Subject Classifications (2010)

Search

Navigation