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The Bound-Constrained Conjugate Gradient Method for Non-negative Matrices

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Abstract

Existing conjugate gradient (CG)-based methods for convex quadratic programs with bound constraints require many iterations for solving elastic contact problems. These algorithms are too cautious in expanding the active set and are hampered by frequent restarting of the CG iteration. We propose a new algorithm called the Bound-Constrained Conjugate Gradient method (BCCG). It combines the CG method with an active-set strategy, which truncates variables crossing their bounds and continues (using the Polak–Ribière formula) instead of restarting CG. We provide a case with n=3 that demonstrates that this method may fail on general cases, but we conjecture that it always works if the system matrix A is non-negative. Numerical results demonstrate the effectiveness of the method for large-scale elastic contact problems.

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Acknowledgements

Thanks to my colleagues Jok Tang, Martin van Gijzen, Kees Oosterlee, and Jing Zhao for valuable discussions on this work. Thanks to Jing Zhao for her work on the test problem. Thanks to VORtech (Mark Roest) for supporting this work financially. Thanks also to the reviewers for their comments and suggestions on this work.

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Authors and Affiliations

  1. DIAM, Delft University of Technology, P.O. Box 5031, 2600 GA, Delft, The Netherlands

    Edwin A. H. Vollebregt

  2. VORtech BV, P.O. Box 260, 2600 AG, Delft, The Netherlands

    Edwin A. H. Vollebregt

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  1. Edwin A. H. Vollebregt
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Correspondence to Edwin A. H. Vollebregt.

Additional information

Communicated by Johannes O. Royset.

Appendix: MATLAB Code

Appendix: MATLAB Code

The Matlab code below implements the Enhanced BCCG algorithm. Plain BCCG is obtained by setting use_plain=1; in the fourth line, the NORM+CG algorithm is obtained when use_normcg=1;.

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Vollebregt, E.A.H. The Bound-Constrained Conjugate Gradient Method for Non-negative Matrices. J Optim Theory Appl 162, 931–953 (2014). https://doi.org/10.1007/s10957-013-0499-x

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