Advertisement

Journal of Global Optimization

, Volume 45, Issue 2, pp 211–227 | Cite as

On the convergence of the entropy-exponential penalty trajectories and generalized proximal point methods in semidefinite optimization

  • O. P. Ferreira
  • P. R. Oliveira
  • R. C. M. Silva
Article
  • 92 Downloads

Abstract

The convergence of primal and dual central paths associated to entropy and exponential functions, respectively, for semidefinite programming problem are studied in this paper. It is proved that the primal path converges to the analytic center of the primal optimal set with respect to the entropy function, the dual path converges to a point in the dual optimal set and the primal-dual path associated to this paths converges to a point in the primal-dual optimal set. As an application, the generalized proximal point method with the Kullback-Leibler distance applied to semidefinite programming problems is considered. The convergence of the primal proximal sequence to the analytic center of the primal optimal set with respect to the entropy function is established and the convergence of a particular weighted dual proximal sequence to a point in the dual optimal set is obtained.

Keywords

Generalized proximal point methods Bregman distances Central path Semidefinite programming 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aulender A., Teboulle M.: Interior gradient and proximal methods for convex and conic optimization. SIAM J. Optim. 16(3), 697–725 (2006)CrossRefGoogle Scholar
  2. 2.
    Aulender A., Héctor Ramírez C.: Penalty and barrier methods for convex semidefinite programming. Math. Methods Oper. Res. 63, 195–219 (2006)CrossRefGoogle Scholar
  3. 3.
    Bolte J., Daniilidis A., Lewis A.: The Lojasiewicz inequality for nonsmooth subanalytic functions with applications to subgradient dynamical systems. SIAM J. Optim. 17(4), 1205–1223 (2007)CrossRefGoogle Scholar
  4. 4.
    Cominetti R., San Martín J.: Asymptotic analysis of the exponential penalty trajectory in linear programming. Math. Program. 67(2), 169–187 (1994)CrossRefGoogle Scholar
  5. 5.
    da Cruz Neto J.X., Ferreira O.P., Monteiro R.D.C.: asymptotic behavior of the central path for a special class of degenerate SDP problems. Math. Program. 103(3), 487–514 (2005)CrossRefGoogle Scholar
  6. 6.
    da Cruz Neto J.X., Iusem A.N., Ferreira O.P., Monteiro R.D.C: Dual convergence of the proximal point method with Bregman distances for linear programming. Optim. Methods Softw. Engl. 22, 339–360 (2007)CrossRefGoogle Scholar
  7. 7.
    Dieudonné J.A.: Foundations of Modern Analysis. AcademicPress, New York (1960)Google Scholar
  8. 8.
    Doljansky M., Teboulle M.: An interior proximal algorithm and the exponential multiplier method for semidefinite programming. SIAM J. Optim. 9(1), 1–13 (1998)CrossRefGoogle Scholar
  9. 9.
    Dym, H.: Linear algebra in action. Graduate Studies in Mathematics 78, AMS 2007, Providence, RIGoogle Scholar
  10. 10.
    Halická M., de Klerk E., Roos C.: On the convergence of the central path in semidefinite optimization. SIAM J. Optim. 12(4), 1090–1099 (2002)CrossRefGoogle Scholar
  11. 11.
    Halická M., de Klerk E., Roos C.: Limiting behavior of the central path in semidefinite optimization. Optim. Methods Softw. 20(1), 99–113 (2005)CrossRefGoogle Scholar
  12. 12.
    Horn R.A., Johnson C.R.: Matrix Analysis. Cambridge University Press, New York (1985)Google Scholar
  13. 13.
    Iusem A.: Augmented Lagrangian methods and proximal point methods for convex optimization. Investigación Operativa 8, 11–49 (1999)Google Scholar
  14. 14.
    Iusem A.N., Monteiro R.D.C.: On dual convergence of the generalized proximal point method with Bregman distances. Math. Oper. Res. 25(4), 606–624 (2000)CrossRefGoogle Scholar
  15. 15.
    Iusem A.N., Svaiter B.F., da Cruz Neto J.X.: Central paths, generalized proximal point methods, and cauchy trajectories in Riemannian manifolds. SIAM J. Control Optim. 37(2), 566–588 (1999)CrossRefGoogle Scholar
  16. 16.
    Jensen D.L., Polyak R.A.: The convergence of a modified barrier method for convex programming. IBM J. Res. Dev. 38(3), 307–321 (1994)CrossRefGoogle Scholar
  17. 17.
    de Klerk E.: Aspects of Semidefinite Programming: Interior Point Algorithms and Selected Applications, Applied Optimization, Vol. 65. Kluwer Academic Publishers, Dordrecht (2002)Google Scholar
  18. 18.
    Kojima M., Meggido N., Noma T., Yoshise A.: A Unified Approach to Interior Point Algorithms for Linear Complementarity Problems, Lecture Notes in Computer Science, Vol. 538. Springer Verlag, Berlin, Germany (1991)Google Scholar
  19. 19.
    Krantz S., Parks H.R.: A Primer of Real Analytic Functions. Birkhäuser Verlag, Boston (1992)Google Scholar
  20. 20.
    Kurdyka K., Mostowski T., Parusinski A.: Proof of the gradient conjecture of R.Thom. Annals Math. II Ser. 152(3), 763–792 (2000)CrossRefGoogle Scholar
  21. 21.
    Lojasiewicz, S.: Ensembles Semi-analitiques. I.H.E.S., Bures-sur-Yvette (1965)Google Scholar
  22. 22.
    Mosheyev L., Zibulevski M.: Penalty/barrier multiplier algorithm for semidefinite programming. Optim. Methods Softw. 13, 235–261 (2000)CrossRefGoogle Scholar
  23. 23.
    Papa Quiroz E.A., Roberto Oliveira P.: A new barrier for a class of semidefinte problems. RAIRO-Oper. Res. 40, 303–323 (2006)CrossRefGoogle Scholar
  24. 24.
    Polyak R., Teboulle M.: Nonlinear rescaling and proximal-like methods in convex optimization. Math. Program. 76(2), 265–284 (1997)CrossRefGoogle Scholar
  25. 25.
    Powell M.J.D.: Some convergence properties of the modified log barrier method for linear programming. SIAM J. Optim. 5(4), 695–739 (1995)CrossRefGoogle Scholar
  26. 26.
    Shiota M.: Geometry of Subanalytic and Semialgebraics Sets. Birkhäuser, Boston (1997)Google Scholar
  27. 27.
    Sporre, G., Forsgren, A.: Characterization of the limit point of the central path in semidefinite programming. Technical Report TRITA-MAT-2002-OS12, Departament of Mathematics, Royal Institute of Technology, SE-100 44 Stockholm, Sweden, June 2002Google Scholar
  28. 28.
    Todd M.J.: Semidefinite optimization. Acta Numerica 10, 515–560 (2001)CrossRefGoogle Scholar
  29. 29.
    Tseng P., Bertsekas D.P.: On the convergence of the exponential multiplier method for convex programming. Math. Program. 60(1), 1–19 (1993)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC. 2008

Authors and Affiliations

  • O. P. Ferreira
    • 1
  • P. R. Oliveira
    • 2
  • R. C. M. Silva
    • 3
  1. 1.IMEUniversidade Federal de GoiásGoianiaBrazil
  2. 2.COPPE-SistemasUniversidade Federal do Rio de JaneiroRio de JaneiroBrazil
  3. 3.DM/ICEUniversidade Federal do AmazonasManausBrazil

Personalised recommendations