Skip to main content
Springer Nature Link
Log in

Trajectory-following methods for large-scale degenerate convex quadratic programming

  • Full Length Paper
  • Published:
Mathematical Programming Computation Aims and scope Submit manuscript

Abstract

We consider a class of infeasible, path-following methods for convex quadratric programming. Our methods are designed to be effective for solving both nondegerate and degenerate problems, where degeneracy is understood to mean the failure of strict complementarity at a solution. Global convergence and a polynomial bound on the number of iterations required is given. An implementation, CQP, is available as part of GALAHAD. We illustrate the advantages of our approach on the CUTEr and Maros–Meszaros test sets.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

Notes

  1. Available from http://galahad.rl.ac.uk/galahad-www/. A Matlab interface is also provided.

  2. When \(\ell \le 2\), the exact line minimizer of \(\phi \big (v_{k,\ell }(\alpha );\tau \big )\) is found by calculus.

References

  1. Anitescu, M., Lesaja, G., Potra, F.A.: Equivaence between different formulations of the linear complementarity problem. Optim. Methods Softw. 7(3–4), 265–290 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  2. Billups, S.C., Ferris, M.C.: Convergence of an infeasible interior-point algorithm from arbitrary positive starting points. SIAM J. Optim. 6(2), 316–325 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cartis, C., Gould, N.I.M.: Finding a point in the relative interior of a polyhedron. Technical Report RAL-TR-2006-016, Rutherford Appleton Laboratory. Chilton (2006)

  4. Colombo, M., Gondzio, J.: Further development of multiple centrality correctors for interior point methods. Comput. Optim. Appl. 41(3), 277–305 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Conn, A.R., Gould, N.I.M., Toint, Ph.L.: Trust-Region Methods. MPS-SIAM Series on Optimization. SIAM publications, Philadelphia (2000)

  6. Dolan, E.D., Moré, J.J.: Benchmarking optimization software with performance profiles. Math. Program. 91(2), 201–213 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  7. Graña Drummond, L.M., Svaiter, B.F.: On well definedness of the central path. J. Optim. Theory Appl. 102(2), 223–237 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  8. Duff, I.S., Reid, J.K.: The design of MA48: a code for the direct solution of sparse unsymmetric linear systems of equations. Trans. ACM Math. Softw. 22(2), 187–226 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dunaway, D.K.: Calculation of zeros of a real polynomial through factorization using Euclid’s algorithm. SIAM J. Numer. Anal. 11(6), 1087–1104 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  10. Gill, P.E., Murray, W., Wright, M.H.: Practical Optimization. Academic Press, London (1981)

    MATH  Google Scholar 

  11. Gondzio, J.: Multiple centrality corrections in a primal-dual method for linear programming. Comput. Optim. Appl. 6(2), 137–156 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gould, N.I.M., Orban, D., Toint, Ph.L.: \(\sf CUTEr\) (and \(\sf SifDec \)), a constrained and unconstrained testing environment, revisited. Trans. ACM Math. Softw. 29(4), 373–394 (2003)

  13. Gould, N.I.M., Orban, D., Toint, Ph.L.: \(\sf GALAHAD \)—a library of thread-safe fortran 90 packages for large-scale nonlinear optimization. Trans. ACM Math. Softw. 29(4), 353–372 (2003)

  14. Gould, N.I.M., Robinson, D.P.: A second derivative SQP method: global convergence. SIAM J. Optim. 20(4), 2023–2048 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gould, N.I.M., Robinson, D.P.: A second derivative SQP method: local convergence and practical issues. SIAM J. Optim. 20(4), 2049–2079 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Gould, N.I.M., Toint, Ph.L.: Preprocessing for quadratic programming. Math. Program. B 100(1), 95–132 (2004)

    Google Scholar 

  17. Gupta, A.: WSMP: Watson sparse matrix package part I—direct solution of symmetric sparse system. Research Report RC 21886. IBM T. J. Watson Research Center, Yorktown Heights (2010)

  18. Hogg, J.D., Scott, J.A.: A note on the solve phase of a multicore solver. Technical Report RAL-TR-2010-007, Rutherford Appleton Laboratory, Chilton (2010)

  19. HSL. A collection of Fortran codes for large-scale scientific computation (2011). http://www.hsl.rl.ac.uk

  20. Ji, J., Potra, F.A., Huang, S.: Predictor–corrector method for linear complementarity problems with polynomial complexity and superlinear convergence. J. Optim. Theory Appl. 85(1), 187–199 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kojima, M., Mizuno, S., Noma, T.: Limiting behavior of trajectories generated by a continuation method for monotone complementarity problems. Math. Oper. Res. 15(4), 662–675 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  22. Liu, X., Potra, F.A.: Predictor-corrector methods for sufficient linear complementarity problems in a wide neighborhood of the central path. Optim. Methods Softw. 20(1), 145–168 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  23. Liu, X., Potra, F.A.: Corrector–predictor methods for sufficient linear complementarity problems in a wide neighborhood of the central path. SIAM J. Optim. 17(3), 871–890 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. Lustig, I.J., Marsten, R.E., Shanno, D.F.: Computational experience with a primal–dual interior point method for linear programming. Linear Algebra Appl. 152, 191–222 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  25. Madsen, K., Reid, J.K.: Fortran subroutines for finding polynomial zeros. Technical Report AERE-R 7986, AERE Harwell Laboratory, Harwell (1975)

  26. Maros, I., Meszaros, C.: A repository of convex quadratic programming problems. Optim. Methods Softw. 11–12, 671–681 (1999)

    Article  MathSciNet  Google Scholar 

  27. McCormick, G.P., Witzgall, C.: Logarithmic SUMT limits in convex programming. Math. Program. 90(1), 113–145 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  28. Megiddo, N.: Pathways to the optimal set in linear programming. In: Megiddo, N. (ed.) Progress in Mathematical Programming, pp. 131–158. Springer, New-York (1989)

    Chapter  Google Scholar 

  29. Mehrotra, S.: On the implementation of a primal–dual interior point method. SIAM J. Optim. 2, 575–601 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  30. Mizuno, S.: A superlinearly convergent infeasible-interior-point algorithm for geometrical LCPs without a strictly complementary condition. Math. Oper. Res. 21(2), 382–400 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  31. Mizuno, S., Todd, M.J., Ye, Y.: On adaptive-step primal–dual interior-point algorithms for linear programming. Math. Oper. Res. 18, 964–981 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  32. Monteiro, R.D.C., Tsuchiya, T.: Limiting behavior of the derivatives of certain trajectories associated with a monotone horizontal linear complementarity problem. Math. Oper. Res. 21(4), 793–814 (1996)

    Google Scholar 

  33. Potra, F.A.: A superlinearly convergent predictor–corrector method for degenerate LCP in a wide neighborhood of the central path with \({O}(\sqrt{n}{L})\)-iteration complexity. Math. Program. 100(2), 317–337 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  34. Potra, F.A.: Primal–dual affine scaling interior point methods for linear complementarity problems. SIAM J. Optim. 19(1), 114–143 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  35. Potra, F.A., Sheng, R.: Superlinearly convergent infeasible-interior-point algorithm for degenerate lcp. J. Optim. Theory Appl. 97(2), 249–269 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  36. Potra, F.A., Stoer, J.: On a class of superlinearly convergent polynomial time interior point methods for sufficient LCP. SIAM J. Optim. 20(3), 1333–1363 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  37. Schenk, O., Gärtner, K.: On fast factorization pivoting methods for symmetric indefinite systems. Electron. Trans. Numer. Anal. 23, 158–179 (2006)

    MathSciNet  MATH  Google Scholar 

  38. Siegel, C.L.: Topics in Complex Function Theory. Elliptic Functions and Uniformization Theory, vol. 1. Wiley, Chichester (1988)

  39. Stoer, J.: High order long-step methods for solving linear complemenarity problems. Ann. Oper. Res. 103 (2001)

  40. Stoer, J.: Analysis of interior-point paths. J. Res. Natl. Inst. Stand. Technol. 111(2) (2006)

  41. Stoer, J., Wechs, M.: The complexity of high-order predictor-corrector methods for solving sufficient linear complementarity problems. Optim. Methods Softw. 10(2), 393–417 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  42. Stoer, J., Wechs, M.: Infeasible-interior-point paths for sufficient linear complementarity problems and their analyticity. Math. Program. 83(1–3), 407–423 (1998)

    MathSciNet  MATH  Google Scholar 

  43. Stoer, J., Wechs, M.: On the analyticity properties of infeasible-interior-point paths for monotone linear complementarity problems. Numer. Math. 81, 631–645 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  44. Stoer, J., Wechs, M., Mizuno, S.: High order infeasible-interior-point methods for solving sufficient linear complementarity problems. Math. Oper. Res. 23(4), 832–862 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  45. Sturm, J.F.: Superlinear convergence of an algorithm for monotone linear complementarity problems, when no strictly complementary solution exists. Math. Oper. Res. 24(1), 72–94 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  46. Väliaho, H.: \(P_*\)-matrices are just sufficient. Linear Algebra Appl. 239, 103–108 (1996)

    MathSciNet  MATH  Google Scholar 

  47. Wright, M.H.: Interior methods for constrained optimization. Acta Numer. 1, 341–407 (1992)

    Article  Google Scholar 

  48. Wright, S.J.: Primal–Dual Interior-Point Methods. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1997)

  49. Wright, S.J., Orban, D.: Local convergence of the Newton/log-barrier method for degenerate problems. Math. Oper. Res. 27(3), 585–613 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  50. Ye, Y., Güler, O., Tapia, R.A., Zhang, Y.: A quadratically convergent \({O}(\sqrt{n}{L})\)-iteration algorithm for linear programming. Math. Program. 59, 151–162 (1993)

    Article  MATH  Google Scholar 

  51. Zhang, Y.: On the convergence of a class of infeasible interior-point methods for the horizontal linear complementarity problem. SIAM J. Optim. 4(1), 208–227 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  52. Zhang, Y., Zhang, D.: On polynomiality of the Mehrotra-type predictor–corrector interior-point algorithms. Math. Program. 68, 303–318 (1995)

    Article  MATH  Google Scholar 

  53. Zhao, G., Sun, J.: On the rate of local convergence of high-order-infeasible-path- following algorithms for \(P_*\)-linear complementarity problems. Comput. Optim. Appl. 14(3), 293–307 (1999)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

This work was supported by the EPSRC grants EP/E053351/1, EP/F005369/1 and EP/H026053/1 and NSERC Discovery Grant 299010-04. We are extremely grateful to a referee and associate editor for comments that lead to important clarifications of our numerical results.

Author information

Authors and Affiliations

  1. Scientific Computing Department, Rutherford Appleton Laboratory, Chilton, Oxfordshire, OX11 0QX, UK

    Nicholas I. M. Gould

  2. GERAD and Mathematics and Industrial Engineering Department, École Polytechnique de Montréal, C. P. 6079, Succ. Centre Ville, Montreal, QC, H3C 3A7, Canada

    Dominique Orban

  3. Department of Applied Mathematics and Statistics, Johns Hopkins University, 100 Whitehead Hall, 3400 N. Charles Street, Baltimore, MD, 21218, USA

    Daniel P. Robinson

Authors
  1. Nicholas I. M. Gould
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dominique Orban
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Daniel P. Robinson
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nicholas I. M. Gould.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (pdf 163 KB)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gould, N.I.M., Orban, D. & Robinson, D.P. Trajectory-following methods for large-scale degenerate convex quadratic programming. Math. Prog. Comp. 5, 113–142 (2013). https://doi.org/10.1007/s12532-012-0050-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12532-012-0050-3

Keywords

Mathematics Subject Classification (2000)

Search

Navigation