Skip to main content
Springer Nature Link
Log in

An exact explicit dual for the linear copositive programming problem

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

Abstract

Recently, for a linear copositive programming problem, we formulated an exact explicit dual problem in the form of the extended Lagrange-Slater dual. This dual problem is formulated using only the data of the primal copositive problem, satisfies the strong duality relation, and is obtained without any regularity assumptions due to the use of a concept of the normalized immobile index set. The constraints of the exact explicit dual problem are formulated in terms of completely positive matrices and their number is presented in terms of a finite integer parameter \(m_0\). In this paper, we prove that \(m_0\le 2n\), where n is the dimension of the primal variable’s space.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /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.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ahmed, F., Dür, M., Still, G.: Copositive programming via semi-infinite optimization. J. Optim. Theory Appl. 159, 322–340 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bomze, I.M.: Copositive optimization - recent developments and applications. EJOR. 216(3), 509–520 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bonnans, J.F., Shapiro, A.: Perturbation analysis of optimization problems. Springer-Verlag, New-York (2000)

    Book  MATH  Google Scholar 

  4. Borwein, J.M., Wolkowicz, H.: Regularizing the abstract convex program. J. Math. Anal. Appl. 83, 495–530 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  5. Dür, M.: Copositive programming: a survey. In: Diehl, M., Glineur, F., Jarlebring, E., Michielis, W. (eds.) Recent advances in optimization and its applications in engineering, pp. 3–20. Springer, Berlin (2010)

    Chapter  Google Scholar 

  6. Gowda, M. S., Sznajder, R.: On the non-homogeneity of completely positive cones, Technical Report trGOW11-04, Dep. Mathematics and Statistics University of Maryland, USA. http://www.optimization-online.org/DB (2011) Accessed 7 Dec 2020

  7. Jeyakumar, V.: A note on strong duality in convex semidefinite optimization: necessary and sufficient conditions. Optim. Lett. 2, 15–25 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Kostyukova, O.I., Tchemisova, T.V.: On equivalent representations and properties of faces of the cone of copositive matrices. arXiv:2012.03610v1 [math.OC] (2020) Accessed Dec 7 (2020)

  9. Kostyukova, O.I., Tchemisova, T.V.: Optimality conditions for convex semi-infinite programming problems with finitely representable compact index sets. J. Optim. Theory Appl. 175(1), 76–103 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  10. Kostyukova, O.I., Tchemisova, T.V.: Optimality criteria without constraint qualification for linear semidefinite problems. J. Math. Sci. 182(2), 126–143 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kostyukova, O.I., Tchemisova, T.V.: On strong duality in linear copositive programming. J. Glob. Optim. (2021). https://doi.org/10.1007/s10898-021-00995-3

  12. Kostyukova, O.I., Tchemisova, T.V., Dudina, O.S.: Immobile indices and CQ-free optimality criteria for linear copositive programming problems. Set-Valued Var. Anal. 28, 89–107 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  13. Levin, V.L.: Application of E. Helly’s theorem to convex programming, problems of best approximation and related questions. Math. USSR Sbornik 8(2), 235–247 (1969)

    Article  Google Scholar 

  14. Pólik, I., Terlaky, T.: Exact duality for optimization over symmetric cones. AdvOL-Report No. 2007/10 McMaster University, Advanced Optimization Lab., Hamilton, Canada. http://www.optimization-online.org/DB_FILE/2007/08/1754.pdf (2007). Accessed 7 Dec 2020

  15. Ramana, M.V.: An exact duality theory for semidefinite programming and its complexity implications. Math. Prog. 77, 129–162 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ramana, M.V., Tuncel, L., Wolkowicz, H.: Strong duality for semidefinite programming. SIAM J. Optim. 7(3), 641–662 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  17. Tunçel, L., Wolkowicz, H.: Strong duality and minimal representations for cone optimization. Comput. Optim. Appl. 53, 619–648 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. Wolkowicz, H., Saigal, R., Vandenberghe, L.: Handbook of semidefinite programming. Theory, algorithms, and applications. Springer, New York (2000)

    Book  MATH  Google Scholar 

  19. Zǎlinescu, C.: On duality gap in linear conic problems. Optim. Lett. 6, 393–402 (2012)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work was supported by the state research program "Convergence" (Republic of Belarus), Task 1.3.01, and the Portuguese funds through CIDMA - Centre for Research and Development in Mathematics and Applications and FCT - Portuguese Foundation for Science and Technology, within the project UIDB/04106/2020. The authors thank the anonymous reviewers whose important comments and suggestions helped to improve the presentation of this paper.

Author information

Authors and Affiliations

  1. Institute of Mathematics, National Academy of Sciences of Belarus, Surganov Str. 11, 220072, Minsk, Belarus

    O. I. Kostyukova

  2. Mathematical Department, University of Aveiro, Campus Universitario Santiago, 3810-193, Aveiro, Portugal

    T. V. Tchemisova

Authors
  1. O. I. Kostyukova
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. T. V. Tchemisova
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to T. V. Tchemisova.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kostyukova, O.I., Tchemisova, T.V. An exact explicit dual for the linear copositive programming problem. Optim Lett 17, 107–120 (2023). https://doi.org/10.1007/s11590-022-01870-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-022-01870-0

Keywords