Advertisement

Testing Regularity on Linear Semidefinite Optimization Problems

  • Eloísa MacedoEmail author
Part of the CIM Series in Mathematical Sciences book series (CIMSMS, volume 4)

Abstract

This paper presents a study of regularity of Semidefinite Programming (SDP) problems. Current methods for SDP rely on assumptions of regularity such as constraint qualifications (CQ) and well-posedness. In the absence of regularity, the characterization of optimality may fail and the convergence of algorithms is not guaranteed. Therefore, it is important to have procedures that verify the regularity of a given problem before applying any (standard) SDP solver. We suggest a simple numerical procedure to test within a desired accuracy if a given SDP problem is regular in terms of the fulfilment of the Slater CQ. Our procedure is based on the recently proposed DIIS algorithm that determines the immobile index subspace for SDP. We use this algorithm in a framework of an interactive decision support system. Numerical results using SDP problems from the literature and instances from the SDPLIB suite are presented, and a comparative analysis with other results on regularity available in the literature is made.

Notes

Acknowledgements

The author would like to thank the anonymous referee for the valuable comments that have helped to improve the paper. This work was supported by Portuguese funds through the CIDMA – Center for Research and Development in Mathematics and Applications (University of Aveiro), and the Portuguese Foundation for Science and Technology (“FCT – Fundação para a Ciência e a Tecnologia”), within project UID/MAT/04106/2013.

References

  1. 1.
    Anjos, M.F., Lasserre, J.B. (eds.): Handbook of Semidefinite, Conic and Polynomial Optimization: Theory, Algorithms, Software and Applications. International Series in Operational Research and Management Science, vol. 166. Springer, New York (2012)Google Scholar
  2. 2.
    Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)zbMATHCrossRefGoogle Scholar
  3. 3.
    Borchers, B.: SDPLIB 1.2, a library of semidefinite programming test problems. Optim. Methods Softw. 11(1), 683–690 (1999)Google Scholar
  4. 4.
    Cheung, Y., Schurr, S., Wolkowicz, H.: Preprocessing and Reduction for Degenerate Semidefinite Programs. Research Report CORR 2011-02. http://www.optimization-online.org/DB_FILE/2011/02/2929.pdf (2013). Revised in January 2013
  5. 5.
    Dontchev, A.L., Zolezzi, T.: Well-Posed Optimization Problems. Lecture Notes in Mathematics, vol. 1543. Springer, Berlin (1993)Google Scholar
  6. 6.
    Freund, R.M.: Complexity of an Algorithm for Finding an Approximate Solution of a Semi-Definite Program, with no Regularity Condition (1995). Technical Report OR 302-94, Op. Research Center, MIT, Revised in December 1995Google Scholar
  7. 7.
    Freund, R.M., Sun, J.: Semidefinite Programming I: Introduction and minimization of polynomials, System Optimization. Available at http://www.myoops.org/cocw/mit/NR/rdonlyres/Sloan-School-of-Management/15-094Systems-Optimization--Models-and-ComputationSpring2002/1B59FD11-A822-4C80-9301-47B127500648/0/lecture22.pdf (2002)
  8. 8.
    Freund, R.M., Ordóñez, F., Toh, K.C.: Behavioral Measures and their Correlation with IPM Iteration Counts on Semi-Definite Programming Problems. Math. Program. 109(2), 445–475 (2007). Springer, New YorkGoogle Scholar
  9. 9.
    Fujisawa, K., Futakata, Y., Kojima, M., Matsuyama, S., Nakamura, S., Nakata, K., Yamashita, M.: SDPA-M (SemiDefinite Programming Algorithm in MATLAB) User’s Manual-V6.2.0, Series B: OR Department of Mathematical and Computing Sciences (2005)Google Scholar
  10. 10.
    Gärtner, B., Matoušek, J.: Interior Point Methods, Approximation Algorithms and Semidefinite Programming. Available at http://www.ti.inf.ethz.ch/ew/courses/ApproxSDP09/ (2009)
  11. 11.
    Helmberg, C.: Semidefinite Programming for Combinatorial Optimization. ZIB Report, Berlin (2000)Google Scholar
  12. 12.
    Hernández-Jiménez, B., Rojas-Medar, M.A., Osuna-Gómez, R., Beato-Moreno, A.: Generalized convexity in non-regular programming problems with inequality-type constraints. J. Math. Anal. Appl. 352(2), 604–613 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Jansson, C., Chaykin, D., Keil, C.: Rigorous error bounds for the optimal value in semidefinite programming. SIAM J. Numer. Anal. archive 46(1), 180–200 (2007)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Klatte, D.: First order constraint qualifications. In: Floudas, C.A., Pardalos, P.M. (eds.) Encyclopedia of Optimization, 2nd edn, pp. 1055–1060. Springer, US (2009)CrossRefGoogle Scholar
  15. 15.
    Klerk, E. de: Aspects of Semidefinite Programming – Interior Point Algorithms and Selected Applications. Applied Optimization, vol. 65. Kluwer, Boston (2004)Google Scholar
  16. 16.
    Kojima, M.: Introduction to Semidefinite Programs (Semidefinite Programming and Its Application), Institute for Mathematical Sciences National University of Singapore (2006)Google Scholar
  17. 17.
    Kolman, B., Beck, R.E.: Elementary Linear Programming with Applications, 2nd edn, Academic Press, San Diego (1995)zbMATHGoogle Scholar
  18. 18.
    Konsulova, A.S., Revalski, J.P.: Constrained convex optimization problems – well-posedness and stability. Numer. Funct. Anal. Optim. 15(7–8), 889–907 (1994)zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Kostyukova, O.I., Tchemisova, T.V.: Optimality criterion without constraint qualification for linear semidefinite problems. J. Math. Sci. 182(2), 126–143 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Luo, Z., Sturm, J., Zhang, S.: Duality results for conic convex programming, Econometric Institute Report No. 9719/A (1997)Google Scholar
  21. 21.
    Mitchell, J., Krishnan, K.: A unifying framework for several cutting plane methods for semidefinite programming, Technical Report, Department of Computational and Applied Mathematics, Rice University (2003)Google Scholar
  22. 22.
    Moré, J.J.: The Levenberg-Marquardt algorithm: implementation and theory. In: Watson, G.A. (ed.) Numerical Analysis. Lecture Notes in Mathematics, vol. 630, pp. 105–116. Springer, Berlin/Heidelberg (1977)Google Scholar
  23. 23.
    Nocedal, J., Wright, S.J.: Numerical Optimization. Springer, New York (1999)zbMATHCrossRefGoogle Scholar
  24. 24.
    Pataki, G.: Bad semidefinite programs: they all look the same, Technical report, Department of Operations Research, University of North Carolina (2011)Google Scholar
  25. 25.
    Pedregal, P.: Introduction to Optimization. Springer, New York (2004)zbMATHCrossRefGoogle Scholar
  26. 26.
    Polik, I.: Semidefinite programming Feasibility and duality. Available at http://imre.polik.net/wp-content/uploads/IE496/POLIK_IE496_04_duality.pdf (2009)
  27. 27.
    Renegar, J.: Some perturbation-theory for linear-programming. Math. Program. 65(1), 73–91 (1994)zbMATHMathSciNetCrossRefGoogle Scholar
  28. 28.
    Sturm, J., Zhang, S.: On sensitivity of central solutions in semidefinite programming. Math. Program. 90(2), 205–227 (1998). SpringerGoogle Scholar
  29. 29.
    Tikhonov, A.N., Arsenin, V.Y.: Solutions of ill-posed problems. John Wiley and Sons, New York (1977)zbMATHGoogle Scholar
  30. 30.
    Todd, M.J.: Semidefinite optimization. Acta Numer. 10, 515–560 (2001). Cambridge University PressGoogle Scholar
  31. 31.
    Vandenberghe, L., Boyd, S.: Semidefinite Programming. SIAM Rev. 38(1), 49–95 (1996)zbMATHMathSciNetCrossRefGoogle Scholar
  32. 32.
    Wolkowicz, H.: Duality for semidefinite programming. In: Floudas, C.A., Pardalos, P.M. (eds.) Encyclopedia of Optimization, 2nd edn, pp. 811–814. Springer, US (2009)CrossRefGoogle Scholar
  33. 33.
    Wolkowicz, H., Saigal, R., Vandenberghe, L.: Handbook of Semidefinite Programming: Theory, Algorithms, and Applications. Kluwer, Boston (2000)CrossRefGoogle Scholar
  34. 34.
    Zhang, Y.: Semidefinite Programming, Lecture 2. Available at http://rutcor.rutgers.edu/~alizadeh/CLASSES/95sprSDP/NOTES/lecture2.ps (1995)

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.University of AveiroAveiroPortugal

Personalised recommendations