Advertisement

Interior-Point Solver for Large-Scale Quadratic Programming Problems with Bound Constraints

  • S. Cafieri
  • M. D’Apuzzo
  • M. Marino
  • A. Mucherino
  • G. Toraldo
Article

Abstract

In this paper, we present an interior-point algorithm for large and sparse convex quadratic programming problems with bound constraints. The algorithm is based on the potential reduction method and the use of iterative techniques to solve the linear system arising at each iteration. The global convergence properties of the potential reduction method are reassessed in order to take into account the inexact solution of the inner system. We describe the iterative solver, based on the conjugate gradient method with a limited-memory incomplete Cholesky factorization as preconditioner. Furthermore, we discuss some adaptive strategies for the fill-in and accuracy requirements that we use in solving the linear systems in order to avoid unnecessary inner iterations when the iterates are far from the solution. Finally, we present the results of numerical experiments carried out to verify the effectiveness of the proposed strategies. We consider randomly generated sparse problems without a special structure. Also, we compare the proposed algorithm with the MOSEK solver.

Keywords

Bound-constrained quadratic programming potential reduction method conjugate gradient method incomplete Cholesky factorization 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    BYRD, R. H., HRIBAR, M. E., and NOCEDAL, J., An Interior-Point Algorithm for Large-Scale Nonlinear Programming, SIAM Journal on Optimization, Vol. 9, No. 4, pp. 877–900, 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    BYRD, R. H., GILBERT, J. C., and NOCEDAL J., A Trust-Region Method Based on Interior-Point Techniques for Nonlinear Programming, Mathematical Programming, Vol. 89A, No. 1, pp. 149–185, 2000.CrossRefMathSciNetGoogle Scholar
  3. 3.
    BYRD, R. H., NOCEDAL J., and WALTZ, R. A., Feasible Interior Methods Using Slacks for Nonlinear Optimization, Computational Optimization and Applications, Vol. 26, No. 1, pp. 35–61, 2003.zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    D’APUZZO, M., and MARINO, M., Parallel Computational Issues of an Interior-Point Method for Solving Large Bound-Constrained Quadratic Programming Problems, Parallel Computing, Vol. 29, No. 4, pp. 467–483, 2003.CrossRefMathSciNetGoogle Scholar
  5. 5.
    GONDZIO, J., and SARKISSIAN, R., Parallel Interior-Point Solver for Structured Linear Programs. Mathematical Programming, Vol. 96, No. 3, pp. 561–584, 2003.zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    PARDALOS, P. M., and RESENDE, M. G. C., Interior-Point Methods for Global Optimization, Interior-Point Methods in Mathematical Programming, Edited by T. Terlaky. Kluwer Academic Publishers, Dordrecht, Netherlands, pp. 467–500, 1996.Google Scholar
  7. 7.
    SHANNO, D. F., and VANDERBEI, R. J., An Interior-Point Algorithm for Nonconvex Nonlinear Programming, Computational Optimization and Applications, Vol. 13, No. 1–3, pp. 231–252, 1999.zbMATHMathSciNetGoogle Scholar
  8. 8.
    SHANNO, D. F., and VANDERBEI, R. J., Interior-Point Methods for Nonconvex Nonlinear Programming: Orderings and Higher-Order Methods, Mathematical Programming, Vol. 87, No. 2, pp. 303–316, 2000.zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    BENSON, H. Y., SHANNO, D. F., and Vanderbei, R. J., Interior-Point Methods for Nonconvex Nonlinear Programming: Jamming and Numerical Testing, Mathematical Programming, Vol. 99, No. 6, pp. 35–48, 2004.zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    MORALES, J. L., and NOCEDAL, J., Assessing the Potential of Interior Methods for Nonlinear Optimization, Large-Scale PDE-Constrained Optimization, Edited by L. T. Biegler et al., Springer Verlag, Berlin, Germany, Vol. 30, pp. 167–183, 2003.Google Scholar
  11. 11.
    ANDERSEN, E. D., and ANDERSEN, K. D., The MOSEK Interior-Point Optimizer for Linear Programming: An Implementation of the Homogeneous Algorithm, High Performance Optimization, Edited by H. Frenk, K. Roos, T. Terlaky, and S. Zhang, Kluwer Academic Publishers, Dordrecht, Netherlands, pp. 197–232, 2000.Google Scholar
  12. 12.
    VANDERBEI, R. J., LOQO: An Interior-Point Code for Quadratic Programming, Optimization Methods and Software, Vol. 12, No. 5, pp. 451–484, 1999.CrossRefMathSciNetGoogle Scholar
  13. 13.
    WRIGHT, M. H., III-Conditioning and Computational Error in Interior Methods for Nonlinear Programming, SIAM Journal on Optimization, Vol. 9, No. 1, pp. 84–111, 1998.zbMATHCrossRefGoogle Scholar
  14. 14.
    O’LEARY, D. P., Symbiosis between Linear Algebra and Optimization, Journal of Computational and Applied Mathematics, Vol. 123, Nos. 1–2, pp. 447–465. 2000.zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    WALTZ, R. A., and NOCEDAL, J., KNITRO User’s Manual, Report 2003/05, Optimization Technology Center, Northwestern University, Evanston, Illinois, 2003.Google Scholar
  16. 16.
    WANG, W., and O’LEARY, D. P., Adaptive Use of Iterative Methods in Predictor-Corrector Interior-Point Methods for Linear Programming, Numerical Algorithms, Vol. 25, Nos. 1–4, pp. 387–406, 2000.zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    SAAD, Y., Iterative Methods for Sparse Linear Systems, 2nd Edition, SIAM, Philadelphia, Pennsylvania, 2003.zbMATHGoogle Scholar
  18. 18.
    LIN, C. J., and MORÉ, J. J., Incomplete Cholesky Factorizations with Limited Memory, SIAM Journal on Statistical and Scientific Computing, Vol. 21, No. 1, pp. 24–45, 1999.zbMATHCrossRefGoogle Scholar
  19. 19.
    BERGAMASCHI, L., GONDZIO, J., and ZILLI, G., Preconditioning Indefinite Systems in Interior-Point Methods for Optimization, Computational Optimization and Applications, Vol. 28, No. 2, pp. 149–171, 2004.zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    DURAZZI, C., and RUGGIERO, V., Indefinitely Preconditioned Conjugate Gradient Method for Large Sparse Equality and Inequality Constrained Quadratic Problems, Numerical Linear Algebra with Applications, Vol. 10, No. 8, pp. 673–688, 2003.zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    HRIBAR, M. E., GOULD, N. I. M., and NOCEDAL, J., On the Solution of Equality Constrained Quadratic Programming Problems Arising in Optimization, SIAM Journal on Statistical and Scientific Computing, Vol. 23, No. 4, pp. 1375–1394, 2001.MathSciNetGoogle Scholar
  22. 22.
    KELLER, C., GOULD, N. I. M., and WATHEN, A. J., Constraint Preconditioning for Indefinite Linear Systems, SIAM Journal on Matrix Analysis and Applications, Vol. 21, No. 4, pp. 1300–1317, 2000.zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    HAN, C. G., PARDALOS, P. M., and YE, Y., Computational Aspects of an Interior-Point Algorithm for Quadratic Programming Problems with Box Constraints, Large-Scale Numerical Optimization, Edited by T. Coleman and Y. Li, SIAM, Philadelphia, Pennsylvania, pp. 92–112, 1990.Google Scholar
  24. 24.
    TODD, M. J., and YE, Y., A Centered Projective Algorithm for Linear Programming. Mathematics of Operational Research, Vol. 15, No. 3, pp. 508–529, 1990.zbMATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    YE, Y., An O(n 3 L) Potential Reduction Algorithm for Linear Programming, Mathematical Programming, Vol. 50, No. 2, pp. 239–258, 1991.zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    KOJIMA, M., MIZUNO, S., and YOSHISE, A., An \(O(\sqrt{n}L)\) Iteration Potential Reduction Algorithm for Linear Complementarity Problems, Mathematical Programming, Vol. 50, No. 3, pp. 331–342, 1991.Google Scholar
  27. 27.
    LIN, C. J., and MORÉ, J. J., Newton’s Method for Large Bound-Constrained Optimization Problems, SIAM Journal on Optimization, Vol. 9, No. 4, pp. 1100–1127, 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    DONGARRA, J. J., DU CROZ, J., DUFF, I. S., and HAMMARLING, S., A Set of Level 3 Basic Linear Algebra Subprograms, ACM Transactions on Mathematical Software, Vol. 16, No. 1, pp. 1–17, 1990.zbMATHCrossRefGoogle Scholar
  29. 29.
    MORÉ, J. J., and TORALDO, G., On the Solution of Large Quadratic Programming Problems with Bound Constraints, SIAM Journal on Optimization, Vol. 1, No. 1, pp. 93–113, 1991.zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    MORÉ, J. J., and TORALDO, G., Algorithms for Bound-Constrained Quadratic Programming Problems, Numerische Mathematik, Vol. 55, No. 4, pp. 377–400, 1989.zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    IMB EDITOR, ESSL for AIX V4. 1: Guide and References, IBM Pubblications, 2003.Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • S. Cafieri
    • 1
  • M. D’Apuzzo
    • 1
  • M. Marino
    • 2
  • A. Mucherino
    • 1
  • G. Toraldo
    • 2
  1. 1.Department of MathematicsSecond University of NaplesCasertaItaly
  2. 2.Department of Agricultural Engineering and AgronomyUniversity of Naples Federico IINaplesItaly

Personalised recommendations