Skip to main content
SpringerLink
Log in

A Polynomial Time Constraint-Reduced Algorithm for Semidefinite Optimization Problems

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

Abstract

We present an infeasible primal-dual interior point method for semidefinite optimization problems, making use of constraint reduction. We show that the algorithm is globally convergent and has polynomial complexity, the first such complexity result for primal-dual constraint reduction algorithms for any class of problems. Our algorithm is a modification of one with no constraint reduction due to Potra and Sheng (1998) and can be applied whenever the data matrices are block diagonal. It thus solves as special cases any optimization problem that is a linear, convex quadratic, convex quadratically constrained, or second-order cone problem.

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

Notes

  1. The Matlab package is available in http://www.math.nus.edu.sg/~mattohkc/sdpt3.html.

  2. Refer to Fujisawa, Kojima, and Nakata [36] to see how to exploit the sparsity of \({{\mathbf {A}}}_{ij}\).

  3. Rank-1 modification of Cholesky factor is implemented by “schud.f” and “dchud.f” in LINPACK. See Gill et al. [41] and LINPACK documentation [42].

References

  1. Potra, F.A., Sheng, R.: A superlinearly convergent primal-dual infeasible-interior-point algorithm for semidefinite programming. SIAM J. Optim. 8(4), 1007–1028 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  2. Dantzig, G.B., Ye, Y.: A Build-Up Interior-Point Method for Linear Programming: Affine Scaling Form. Tech. Rep. Stanford University, Stanford (1991)

    Google Scholar 

  3. Tone, K.: An active-set strategy in an interior point method for linear programming. Math. Progr. 59(3), 345–360 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  4. Kaliski, J.A., Ye, Y.: A decomposition variant of the potential reduction algorithm for linear programming. Manag. Sci. 39, 757–776 (1993)

    Article  Google Scholar 

  5. Hertog, D.D., Roos, C., Terlaky, T.: Adding and deleting constraints in the path-following method for LP. In: Advances in Optimization and Approximation, vol. 1, pp. 166–185. Springer, New York (1994)

  6. Tits, A.L., Absil, P.A., Woessner, W.P.: Constraint reduction for linear programs with many inequality constraints. SIAM J. Optim. 17(1), 119–146 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Winternitz, L.B., Nicholls, S.O., Tits, A.L., O’Leary, D.P.: A constraint-reduced variant of Mehrotra’s predictor-corrector algorithm. Comput. Optim. Appl. 51(3), 1001–1036 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. Jung, J.H., O’Leary, D.P., Tits, A.L.: Adaptive constraint reduction for training support vector machines. Electron. Trans. Numer. Anal. 31, 156–177 (2008)

    MathSciNet  MATH  Google Scholar 

  9. Williams, J.A.: The Use of Preconditioning for Training Support Vector Machines, Master’s Thesis, Applied Mathematics and Scientific Computing Program. University of Maryland, College Park (2008)

    Google Scholar 

  10. Jung, J.H., O’Leary, D.P., Tits, A.L.: Adaptive constraint reduction for convex quadratic programming. Comput. Optim. Appl. 51(1), 125–157 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. Helmberg, C., Rendl, F., Vanderbei, R.J., Wolkowicz, H.: An interior-point method for semidefinite programming. SIAM J. Optim. 6(2), 342–361 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kojima, M., Shindoh, S., Hara, S.: Interior-point methods for the monotone semidefinite linear complementarity problem in symmetric matrices. SIAM J. Optim. 7(1), 86–125 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  13. Monteiro, R.D.C.: Primal-dual path-following algorithms for semidefinite programming. SIAM J. Optim. 7(3), 663–678 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  14. Alizadeh, F., Haeberly, J.P.A., Overton, M.L.: Primal-dual interior-point methods for semidefinite programming. Tech. rep., Manuscript presented at the Math. Programming Symposium, Ann Arbor, MI (1994)

  15. Nesterov, Y.E., Todd, M.J.: Self-scaled barriers and interior-point methods for convex programming. Math. Oper. Res. 22(1), 1–42 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  16. Nesterov, Y.E., Todd, M.J.: Primal-dual interior-point methods for self-scaled cones. SIAM J. Optim. 8(2), 324–364 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  17. Monteiro, R.D.C., Zhang, Y.: A unified analysis for a class of long-step primal-dual path-following interior point algorithms for semidefinite programming. Math. Progr. 81(3), 281–299 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  18. Zhang, Y.: On extending some primal-dual interior-point algorithms from linear programming to semidefinite programming. SIAM J. Optim. 8(2), 365–386 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  19. Alizadeh, F., Haeberly, J.P.A., Overton, M.L.: Primal-dual interior-point methods for semidefinite programming: convergence rates, stability and numerical results. SIAM J. Optim. 8(3), 746–768 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  20. Monteiro, R.D.C.: Polynomial convergence of primal-dual algorithms for semidefinite programming based on Monteiro and Zhang family of directions. SIAM J. Optim. 8(3), 797–812 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kojima, M., Shida, M., Shindoh, S.: Local convergence of predictor-corrector infeasible-interior-point algorithms for SDPs and SDLCPs. Math. Progr. 80(2), 129–160 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  22. Potra, F.A., Sheng, R.: Superlinear convergence of interior-point algorithms for semidefinite programming. J. Optim. Theory Appl. 99(1), 103–119 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kojima, M., Shida, M., Shindoh, S.: A predictor-corrector interior-point algorithm for the semidefinite linear complementarity problem using the Alizadeh–Haeberly–Overton search direction. SIAM J. Optim. 9(2), 444–465 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  24. Potra, F.A., Sheng, R.: Superlinear convergence of a predictor-corrector method for semidefinite programming without shrinking central path neighborhood. Tech. Rep. 91, Reports on Computational Mathematics, Department of Mathematics, University of Iowa (1996)

  25. Ji, J., Potra, F.A., Sheng, R.: On the local convergence of a predictor-corrector method for semidefinite programming. SIAM J. Optim. 10(1), 195–210 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  26. Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, New York (2004)

    Book  MATH  Google Scholar 

  27. Schrijver, A.: A comparison of the Delsarte and Loviász bounds. IEEE Trans. Inf. Theory 25(4), 425–429 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  28. de Klerk, E., Pasechnik, D.V., Sotirov, R.: On semidefinite programming relaxations of the traveling salesman problem. SIAM J. Optim. 19(4), 1559–1573 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  29. Bachoc, C., Vallentin, F.: New upper bounds for kissing number from semidefinite programming. J. Am. Math. Soc. 21(3), 909–924 (2007)

    Article  MathSciNet  Google Scholar 

  30. Zhao, Q., Karisch, S.E., Rendl, F., Wolkowicz, H.: Semidefinite programming relaxations for the quadratic assignment problem. J. Comb. Optim. 2(1), 71–109 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  31. Park, S.: Matrix Reduction in Numerical Optimization, Ph.D. Thesis, Computer Science Department, University of Maryland, College Park (2011). http://drum.lib.umd.edu/handle/1903/11751

  32. Toh, K.C., Todd, M.J., Tütüncü, R.H.: On the implementation and usage of SDPT3 - a MATLAB software package for semidefinite-quadratic-linear programming, version 4.0. In: Anjos, M.F., Lasserre, J.B. (eds.) Handbook on Semidefinite, Conic and Polynomial Optimization, vol. 166, pp. 715–754. Springer, New York (2012)

    Chapter  Google Scholar 

  33. Park, S., O’Leary, D.P.: A Polynomial Time Constraint Reduced Algorithm for Semidefinite Optimization Problems, with Convergence Proofs. Tech. rep., University of Maryland, College Park (2015). http://www.optimization-online.org/DB_HTML/2013/08/4011.html

  34. de Klerk, E.: Aspects of Semidefinite Programming: Interior Point Algorithms and Selected Applications. Kluwer Academic Publishers, Norwell (2002)

    Book  Google Scholar 

  35. Jansen, B.: Interior Point Techniques in Optimization. Kluwer Academic Publishers, Norwell (1997)

    Book  MATH  Google Scholar 

  36. Fujisawa, K., Kojima, M., Nakata, K.: Exploiting sparsity in primal-dual interior-point methods for semidefinite programming. Math. Progr. 79(1), 235–253 (1997)

    MathSciNet  MATH  Google Scholar 

  37. Hager, W.: Condition estimates. SIAM J. Sci. Stat. Comput. 5(2), 311–316 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  38. Higham, N.: A survey of condition number estimation for triangular matrices. SIAM Rev. 9(4), 575–596 (1987)

    Article  MathSciNet  Google Scholar 

  39. O’Leary, D.P.: Estimating matrix condition numbers. SIAM J. Sci. Stat. Comput. 1(2), 205–209 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  40. Stewart, P.G.W.: Efficient generation of random orthogonal matrices with an application to condition estimators. SIAM J. Numer. Anal. 17(3), 403–409 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  41. Gill, P.E., Golub, G.H., Murray, W., Saunders, M.A.: Methods for modifying matrix factorizations. Math. Comput. 28(126), 505–535 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  42. Dongarra, J.J., Moler, C.B., Bunch, J.R., Stewart, G.W.: LINPACK Users’ Guide. SIAM, Philadelphia, PA (1979)

    Book  Google Scholar 

Download references

Acknowledgments

We are very grateful to André Tits for careful reading of the manuscript, many suggestions, and insightful comments that helped shape the choice of active and inactive blocks, to Florian Potra for helpful discussions, and to anonymous referees for very careful reading and helpful suggestions.

Author information

Authors and Affiliations

  1. KCG holdings, 545 Washington BLVD, Jersey City, NJ, 07310, USA

    Sungwoo Park

  2. Computer Science Department and Institute for Advanced Computer Studies, University of Maryland, College Park, MD, 20742, USA

    Dianne P. O’Leary

Authors
  1. Sungwoo Park
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dianne P. O’Leary
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sungwoo Park.

Additional information

Communicated by Qianchuan Zhao.

This work was supported by the US Department of Energy under Grants DESC0002218 and DESC0001862.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Park, S., O’Leary, D.P. A Polynomial Time Constraint-Reduced Algorithm for Semidefinite Optimization Problems. J Optim Theory Appl 166, 558–571 (2015). https://doi.org/10.1007/s10957-015-0714-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-015-0714-z

Keywords

Mathematics Subject Classification

Search

Navigation