Skip to main content
SpringerLink
Log in

A note on strong duality in convex semidefinite optimization: necessary and sufficient conditions

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

Abstract

A strong duality which states that the optimal values of the primal convex problem and its Lagrangian dual problem are equal (i.e. zero duality gap) and the dual problem attains its maximum is a corner stone in convex optimization. In particular it plays a major role in the numerical solution as well as the application of convex semidefinite optimization. The strong duality requires a technical condition known as a constraint qualification (CQ). Several CQs which are sufficient for strong duality have been given in the literature. In this note we present new necessary and sufficient CQs for the strong duality in convex semidefinite optimization. These CQs are shown to be sharper forms of the strong conical hull intersection property (CHIP) of the intersecting sets of constraints which has played a critical role in other areas of convex optimization such as constrained approximation and error bounds.

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. Balakrishnan V. and Vandenberghe L. (2003). Semidefinite programming duality and linear time-invariant systems. IEEE Trans. Autom. Control 48: 30–41

    Article  MathSciNet  Google Scholar 

  2. Borwein J.M. and Lewis A.S. (1992). Partially finite convex programming, part I: Quasi relative interiors and duality theory. Math. Progr. 57: 15–48

    Article  MATH  MathSciNet  Google Scholar 

  3. Boyd, S., El Ghaoui, L., Feron, E., Balakrishnan, V.: Linear matrix inequalities in systems and control theory. In: SIAM Stud. Appl. Math., vol. 15. SIAM, Philadelphia (1994)

  4. Boyd S. and Vandenberghe L. (2004). Convex Optimization. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  5. Burachik R.S. and Jeyakumar V. (2005). A simple closure condition for the normal cone intersection formula. Proc. Am. Math. Soc. 133: 1741–1748

    Article  MATH  MathSciNet  Google Scholar 

  6. Burachik R.S. and Jeyakumar V. (2005). A new geometric condition for Fenchel’s duality in infinite dimensional spaces. Math. Program. Ser. B 104: 229–233

    Article  MATH  MathSciNet  Google Scholar 

  7. Burachik R.S., Jeyakumar V. and Wu Z.Y. (2006). Necessary and sufficient conditions for stable conjugate duality. Nonlin. Anal. 64: 1998–2006

    Article  MATH  MathSciNet  Google Scholar 

  8. Deutsch F. (1998). The role of conical hull intersection property in convex optimization and approximation. In: Chui, C.K. and Schumaker, L.L. (eds) Approximation Theory, vol. IX, pp. Vanderbilt University Press, Nashville

    Google Scholar 

  9. Deutsch F., Li W. and Swetits J. (1999). Fenchel duality and the strong conical hull intersection property. J. Optim. Theory Appl. 102: 681–695

    Article  MATH  MathSciNet  Google Scholar 

  10. Guignard M. (1969). Generalized Kuhn-Tucker conditions for mathematical programming problems in a Banach space. SIAM J. Control 7: 232–241

    Article  MATH  MathSciNet  Google Scholar 

  11. Helmberg C. (2002). Semidefinite programming. Eur. J. Oper. Res. 137: 461–482

    Article  MATH  MathSciNet  Google Scholar 

  12. Hiriart-Urruty J-B. and Lemarechal C. (1993). Convex Analysis and Minimization Algorithms vols. I, II. Springer, Heidelberg

    Google Scholar 

  13. Jeyakumar V. (2006). The strong conical hull intersection property for convex programming. Math. Progr. Ser. A 106: 81–92

    Article  MATH  MathSciNet  Google Scholar 

  14. Jeyakumar V. and Nealon M. (2000). Complete characterizations of optimality for convex semidefinite programming. Can. Math. Soc. Conf. Proc 27: 165–173

    MathSciNet  Google Scholar 

  15. Jeyakumar V. and Wolkowicz H. (1992). Generalizations of Slater’s constraint qualification for infinite convex programs. Math. Progr. 57: 85–102

    Article  MATH  MathSciNet  Google Scholar 

  16. Jeyakumar V., Lee G.M. and Dinh N. (2003). New sequential Lagrange multiplier conditions characterizing optimality without constraint qualification for convex programs. SIAM J. Optim. 14: 534–547

    Article  MATH  MathSciNet  Google Scholar 

  17. De Oliveira M.C. (2004). Investigating duality on stability conditions. Syst. Control Lett. 52: 1–6

    Article  MATH  MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  19. Ramana M.V., Tunçcel L. and Wolkowicz H. (1997). Strong duality for semidefinite programming. SIAM J. Optim. 7: 644–662

    Article  MathSciNet  Google Scholar 

  20. Shapiro, A.: On duality theory of conic linear problems, semi-infinite programming (Alicante, 1999). In: Nonconvex Optim. Appl. vol. 57, pp. 135–165. Kluwer, Dordrecht (2001)

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  23. Wolkowicz, H., Saigal, R., Vandenberghe, L.: Handbook of semidefinite programming. In: International Series in Operations Research and Management Science, vol. 27. Kluwer, Dordrecht (2000)

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, University of New South Wales, Sydney, Australia

    V. Jeyakumar

Authors
  1. V. Jeyakumar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. Jeyakumar.

Additional information

Research was partially supported by the Australian Research Council. The author is grateful to the referees for their helpful comments

Rights and permissions

Reprints and permissions

About this article

Cite this article

Jeyakumar, V. A note on strong duality in convex semidefinite optimization: necessary and sufficient conditions. Optimization Letters 2, 15–25 (2008). https://doi.org/10.1007/s11590-006-0038-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-006-0038-x

Keywords

Search

Navigation