Numerical Algorithms

, Volume 69, Issue 1, pp 145–156 | Cite as

Regularized solution of LCP problems with application to rigid body dynamics

Original Paper

Abstract

For Linear Complementarity Problems (LCP) with a positive semidefinite matrix M, iterative solvers can be derived by a process of regularization. In [3] the initial LCP is replaced by a sequence of positive definite ones, with the matrices M + αI. Here we analyse a generalization of this method where the identity I is replaced by a positive definite diagonal matrix D. We prove that the sequence of approximations so defined converges to the minimal D-norm solution of the initial LCP. This extension opens the possibility for interesting applications in the field of rigid multibody dynamics.

Keywords

Linear complementarity problem Splitting method Regularization Weighted minimal norm solutions Rigid body dynamics 

Mathematics Subject Classifications (2010)

65F10 65J20 65F20 

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References

  1. 1.
    Boikanyo, O., Morosanu, G.: A proximal point algorithm converging strongly for general errors. Optimization Letters 4, 635–641 (2010)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Boikanyo, O., Morosanu, G.: Four parameter proximal point algorithms. Nonlinear Anal. 74, 544–555 (2011)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Cottle, R., Pang, J.S., Stone, R.: The linear complementarity problem. Academic Press Inc., New York (1992)MATHGoogle Scholar
  4. 4.
    Khatibzadeh, H., Ranjbar, S.: On the strong convergence of Halpern type proximal point algorithm. J. Optim. Theory Appl. 158, 385–396 (2013)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Lehdili, N., Moudafi, A.: Combining the proximal algorithm and Tikhonov regularization. Optimization 37(3), 239–252 (1996)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Rockafellar, R.: Monotone operators and proximal point algorithm. SIAM J. Control. Optim. 14(5), 877–989 (1976)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Rockafellar, R.: Monotone operators and augmented Lagrangian methods in nonlinear programming. In: Mangasarian, O., Meyer, R., Robinson, S. (eds.): Nonlinear Programming, Vol. 3, pp 1–25. Academic Press (1978)Google Scholar
  8. 8.
    Song, Y., Yang, C.: A note on a paper “A regularization method for the proximal point algorithm”. J. Glob. Optim. 43(1), 171–174 (2009)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Stewart, D.: Impact and friction solids. Structures and Machines, Birkhäuser, chapter Time-stepping methods and the mathematics of rigid body dynamics (2000)Google Scholar
  10. 10.
    Wang, F.: A note on the regularized proximal point algorithm. J. Glob. Optim. 50(3), 531–535 (2011)CrossRefMATHGoogle Scholar
  11. 11.
    Wang, F., Cui, H.: On the contraction-proximal point algorithms with multi-parameters. J. Glob. Optim. 54(3), 485–491 (2012)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Xu, H.K.: A regularization method for the proximal point algorithm. J. Glob. Optim. 36(1), 115–125 (2006)CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Constantin Popa
    • 1
    • 2
  • Tobias Preclik
    • 3
  • Ulrich Rüde
    • 3
  1. 1.Faculty of Mathematics and Computer ScienceOvidius UniversityConstantaRomania
  2. 2.Gheorghe Mihoc - Caius Iacob Institute of Statistical Mathematics and Applied Mathematics of the Romanian AcademyBucharestRomania
  3. 3.Department of Computer Science 10 (System Simulation)Friedrich-Alexander-Universität Erlangen-NürnbergErlangenGermany

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