Skip to main content
SpringerLink
Log in

Understanding Badly and Well-Behaved Linear Matrix Inequalities Via Semi-infinite Optimization

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

In this paper, we use a linear semi-infinite optimization approach to study badly and well-behaved linear matrix inequalities. We utilize a result on uniform LP duality of linear semi-infinite optimization problems to prove recent results obtained by Pataki. Such an approach not only provides alternative proofs of known results, but also gives new insights about badly and well-behaved linear matrix inequalities in terms of a cone and a linear subspace associated with the corresponding linear semi-infinite systems. Furthermore, when the linear matrix inequality constraint of the primal semidefinite optimization problem is badly behaved, we give a characterization of objective functions for the primal linear semidefinite optimization problem with which strong duality holds.

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

Data Availability

Data sharing was not applicable to this article as no datasets were generated or analyzed during the current study.

References

  1. Duffin, R.J.: Infinite programs. In: Kuhn, H.W., Tucker, A.W. (eds.) Linear Inequalities and Related Systems, pp. 157–170. Princeton University Press, Princeton (1956)

    Google Scholar 

  2. Duffin, R.J., Jeroslow, R., Karlovitz, L.A.: Duality in semi-infinite linear programming. In: Semi-infinite Programming and Applications (Austin, Tex., 1981), vol. 215 of Lecture Notes in Economics and Mathematical Systems, pp. 50–62. Springer, Berlin (1983)

  3. Hettich, R., Kortanek, K.O.: Semi-infinite programming: theory, methods, and applications. SIAM Rev. 35, 380–429 (1983)

    Article  MathSciNet  Google Scholar 

  4. Kortanek, K.O., Zhang, Q.: Perfect duality in semi-infinite and semidefinite programming. Math. Program. Ser. A 91, 127–144 (2001)

    Article  MathSciNet  Google Scholar 

  5. Krishnan, K., Mitchell, J.E.: Semi-infinite linear programming approaches to semidefinite programming problems. In: Pardalos, P., Wolkowicz, H. (eds.) Novel Approaches to Hard Discrete Optimization Problems, Fields Institute Communications Series. vol. 37, pp. 123–142 (2003)

  6. Luo, Z.Q., Sturm, J.F., Zhang, S.: Duality Results for Conic Convex Programming. Econometric Institute, Erasmus University Rotterdam, Technical report (1997)

  7. Pataki, G.: Bad semidefinite programs: they all look the same. SIAM J. Optim. 27, 146–172 (2017)

    Article  MathSciNet  Google Scholar 

  8. Pataki, G.: Characterizing bad semidefinite programs: normal forms and short proofs. SIAM Rev. 61, 839–859 (2019)

    Article  MathSciNet  Google Scholar 

  9. Pataki, G.: On Positive Duality Gaps in Semidefinite Programming, arXiv:1812.11796

  10. Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)

    Book  Google Scholar 

  11. Todd, M.J.: Semidefinite optimization. Acta Numer. 10, 515–560 (2001)

    Article  MathSciNet  Google Scholar 

  12. Vandenberghe, L., Boyd, S.: Semidefinite programming. SIAM Rev. 38, 49–95 (1996)

    Article  MathSciNet  Google Scholar 

  13. Vandenberghe, L., Boyd, S.: Connections between semi-infinite and semidefinite programming. In: Reemtsen, R., Ruckmann, J.J. (eds.) Semi-Infinite Programming. Nonconvex Optimization and Its Applications, vol. 25. Springer, Boston (1998)

    Google Scholar 

  14. Wolkowicz, H., Saigal, R., Vandenberghe, L. (eds.): Handbook of Semidefinite Programming. Kluwer Academic Publishers, Boston (2000)

    Google Scholar 

Download references

Acknowledgements

The author would like to thank Professor Ken Kortanek for making the author aware of the references [7] and [8], which inspired the author to start working on this topic. The author would also like to express his gratitude to two anonymous referees for their valuable comments that greatly improve the presentation of the paper.

Author information

Authors and Affiliations

  1. Northern Michigan University, Marquette, 49855, MI, USA

    Qinghong Zhang

Authors
  1. Qinghong Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Qinghong Zhang.

Additional information

Communicated by Sonia Cafieri.

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, Q. Understanding Badly and Well-Behaved Linear Matrix Inequalities Via Semi-infinite Optimization. J Optim Theory Appl (2024). https://doi.org/10.1007/s10957-024-02405-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10957-024-02405-6

Keywords

Mathematics Subject Classification

Search

Navigation