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Quadratic Programming and Scalable Algorithms for Variational Inequalities

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Numerical Mathematics and Advanced Applications

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Abstract

We first review our recent results concerning optimal algorithms for the solution of bound and/or equality constrained quadratic programming problems. The unique feature of these algorithms is the rate of convergence in terms of bounds on the spectrum of the Hessian of the cost function. Then we combine these estimates with some results on the FETI method (FETI-DP, FETI and Total FETI) to get the convergence bounds that guarantee the scalability of the algorithms. i.e. asymptotically linear complexity and the time of solution inverse proportional to the number of processors. The results are confirmed by numerical experiments.

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References

  1. Avery, P., Rebel, G., Lesoinne, M., Farhat, C.: A umerically scalable dual-primal substructuring method for the solution of contact problems. I: the frictionless case. Comput. Methods Appl. Mech. Eng., 193, 2403–2426 (2004)

    Article  MATH  Google Scholar 

  2. Axelsson, O.: Iterative Solution Methods. Cambridge University Press, Cambridge (1994)

    MATH  Google Scholar 

  3. Bertsekas, D.P.: Nonlinear Optimization. Athena Scientific, Belmont (1999)

    Google Scholar 

  4. Conn, A.R., Gould, N.I.M., Toint, Ph.L.: A globally convergent augmented Lagrangian algorithm for optimization with general constraints and simple bounds. SIAM J. Num. Anal. 28, 545–572 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  5. Dostál, Z.: Box constrained quadratic programming with proportioning and projections. SIAM Journal on Optimization 7, 871–887 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  6. Dostál, Z.: A proportioning based algorithm for bound constrained quadratic programming with the rate of convergence. Numerical Algorithms, 34, 293–302 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  7. Dostál, Z.: Inexact semi-monotonic Augmented Lagrangians with optimal feasibility convergence for quadratic programming with simple bounds and equality constraints. SIAM Journal on Numerical Analysis, 43, 96–115 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  8. Dostál, Z.: An optimal algorithm for bound and equality constrained quadratic programming problems with bounded spectrum. Submitted

    Google Scholar 

  9. Dostál, Z.: Scalable FETI for numerical solution of variational inequalities. Submitted

    Google Scholar 

  10. Dostál, Z., Friedlander, A., Santos, S.A.: Solution of contact problems of elasticity by FETI domain decomposition. Contemporary Mathematics, 218, 82–93 (1998)

    Google Scholar 

  11. Dostál, Z., Friedlander, A., Santos, S.A.: Augmented Lagrangians with adaptive precision control for quadratic programming with simple bounds and equality constraints. SIAM Journal on Optimization, 13, 1120–1140 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  12. Dostál, Z., Gomes Neto, F.A.M., Santos, S. A.: Duality based domain decomposition with natural coarse space for variational inequalities. Journal of Computational and Applied Mathematics, 126, 397–415 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  13. Dostál, Z., Gomes Neto, F.A.M., Santos, S.A.: Solution of contact problems by FETI domain decomposition with natural coarse space projection. Computer Methods in Applied Mechanics and Engineering, 190, 1611–1627 (2000)

    Article  MATH  Google Scholar 

  14. Dostál, Z., Haslinger, J., Kučera, R.: Implementation of .xed point method for duality based solution of contact problems with friction. Journal of Computational and Applied Mathematics, 140, 245–256 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  15. Dostál, Z., Horák, D.: Scalability and FETI based algorithm for large discretized variational inequalities. Mathematics and Computers in Simulation, 61, 347–357 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  16. Dostál, Z., Horák, D.: Scalable FETI with Optimal Dual Penalty for Semicoercive Variational Inequalities. Contemporary Mathematics 329, 79–88 (2003)

    Google Scholar 

  17. Dostál, Z., Horák, D.: Scalable FETI with Optimal Dual Penalty for a Variational Inequality. Numerical Linear Algebra and Applications, 11, 455 - 472 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  18. Dostál, Z., Horák, D.: On scalable algorithms for numerical solution of variational inequalities based on FETI and semi-monotonic augmented Lagrangians. In: Kornhuber, R., Hoppe, R.H.W., Périaux, J., Pironneau, O., Widlund, O.B., Xu., J. (eds) Proceedings of DDM15. Springer, Berlin (2003)

    Google Scholar 

  19. Dostál, Z., Horák, D., Kučera, R.: Total FETI - an easier implementable variant of the FETI method for numerical solution of elliptic PDE. Submitted

    Google Scholar 

  20. Dostál, Z., Horák, D., Stefanica, D.: A Scalable FETI-DP Algorithm with Non-penetration Mortar Conditions on Contact Interface. Submitted

    Google Scholar 

  21. Dostál, Z., Horák, D., Stefanica, D.: A scalable FETI-DP algorithm for coercive variational inequalities. IMACS Journal Applied Numerical Mathematics, 54, 378–390 (2005)

    Article  MATH  Google Scholar 

  22. Dostál, Z., Horák, D., Stefanica, D.: Scalable FETI-DP Algorithm for Semicoercive Variational Inequalities. Submitted

    Google Scholar 

  23. Dostál, Z., Schöberl, J.: Minimizing quadratic functions over non-negative cone with the rate of convergence and finite termination. Computational Optimization and Applications, 30, 23–44 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  24. Dureisseix, D., Farhat, C.: A numerically scalable domain decomposition method for solution of frictionless contact problems. International Journal for Numerical Methods in Engineering, 50, 2643–2666 (2001)

    Article  MATH  Google Scholar 

  25. Farhat, C., Gérardin, M.: On the general solution by a direct method of a large scale singular system of linear equations: application to the analysis of floating structures. International Journal for Numerical Methods in Engineering, 41, 675–696 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  26. Farhat, C., Lesoinne, M., LeTallec, P., Pierson, K., Rixen, D.: FETI-DP: A dual-primal unified FETI method. I: A faster alternative to the two-level FETI method. Int. J. Numer. Methods Eng. 50, 1523–1544 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  27. Farhat, C., Mandel, J., Roux, F.X.: Optimal convergence properties of the FETI domain decomposition method. Computer Methods in Applied Mechanics and Engineering, 115, 365–385 (1994)

    Article  MathSciNet  Google Scholar 

  28. Farhat, C., Roux, F.X.: An unconventional domain decomposition method for an effcient parallel solution of large-scale finite element systems. SIAM Journal on Scientific Computing, 13, 379–396 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  29. Friedlander, A., Martýnez, J.M.: On the maximization of a concave quadratic function with box constraints. SIAM J. Optimiz., 4, 177–192 (1994)

    Article  MATH  Google Scholar 

  30. Friedlander, A., Martýnez, J.M., Santos, S.A.: A new trust region algorithm for bound constrained minimization. Applied Math. & Optimiz., 30, 235–266 (1994)

    Article  MATH  Google Scholar 

  31. Hlaváček, I., Haslinger, J., Nečas, J., Lovýšek J.: Solution of Variational Inequalities in Mechanics. Springer, Berlin (1988)

    MATH  Google Scholar 

  32. Klawonn, A., Widlund, O.B., Dryja, M.: Dual-Primal FETI Methods for Threedimensional Elliptic Problems with Heterogeneous Coeficients. SIAM J. Numer. Anal. 40, 159–179 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  33. Kornhuber, R.: Adaptive Monotone Multigrid Methods for Nonlinear Variational Problems. Teubner-Verlag, Stuttgart (1997)

    MATH  Google Scholar 

  34. Kornhuber, R., Krause, R.: Adaptive multigrid methods for Signorini's problem in linear elasticity. Computer Visualization in Science 4, 9–20 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  35. Mandel, J.: étude algébrique d'une méthode multigrille pour quelques problèmes de frontière libre (French). Comptes Rendus de l'Academie des Sciences I 298, 469–472 (1984)

    MathSciNet  Google Scholar 

  36. Mandel, J., Tezaur, R.: On the Convergence of a Dual-Primal Substructuring Method. Numerische Mathematik 88, 543–558 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  37. Schöberl, J.: Solving the Signorini problem on the basis of domain decomposition techniques. Computing, 60, 323–344 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  38. Schöberl, J.: Effcient contact solvers based on domain decomposition techniques. Comput. Math. Appl. 42, 1217–1228 (1998)

    Article  Google Scholar 

  39. Wohlmuth, B.: Discretization Methods and Iterative Solvers Based on Domain Decomposition. Springer, Berlin (2001)

    MATH  Google Scholar 

  40. Wohlmuth, B., Krause, R.: Monotone methods on nonmatching grids for nonlinear contact problems. SIAM J. Sci. Comput. 25 324–347 (2003)

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. FEI VŠB-Technical University Ostrava, CZ-70833, Ostrava, Czech Republic

    Zdeněek Dostál & David Horák

  2. Baruch College, City University of New York, 10010, NY, USA

    Dan Stefanica

Authors
  1. Zdeněek Dostál
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  2. David Horák
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  3. Dan Stefanica
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Editor information

Editors and Affiliations

  1. Departamento de Matemática Aplicada Facultade de Matemáticas, Universidade de Santiago de Compostela, 15706, Santiago de Compostela, Spain

    Alfredo Bermúdez de Castro , Dolores Gómez  & Peregrina Quintela ,  & 

  2. Departamento de Matématica Aplicada, Escola Politécnica Superior, 27002, Lugo, Spain

    Pilar Salgado

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Dostál, Z., Horák, D., Stefanica, D. (2006). Quadratic Programming and Scalable Algorithms for Variational Inequalities. In: de Castro, A.B., Gómez, D., Quintela, P., Salgado, P. (eds) Numerical Mathematics and Advanced Applications. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-34288-5_4

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