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Augmented Lagrangian Methods for Contact Problems, Optimal Control and Image Restoration

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From Convexity to Nonconvexity

Part of the book series: Nonconvex Optimization and Its Applications ((NOIA,volume 55))

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Abstract

The aim of this contribution is to review the available results on mathematical aspects of augmented Lagrangian methods and their applications to contact problems without and with friction, parameter estimation, optimal control and image restoration.

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References

  1. Hestens, M.R., Multiplier and gradient methods, J. Optim. Theory Appl.,4, 303–320.

    Article  Google Scholar 

  2. Powell, M.J.D., A method for nonlinear constraints in minimization problems, in: Optimization, ed. by R. Fletcher, Academic Press, New York, pp. 283–298.

    Google Scholar 

  3. Rockafellar, R.T., Convex Analysis, Princeton University Press, Princeton.

    MATH  Google Scholar 

  4. Fichera, G., Boundary value problems of elasticity with unilateral consraints, in: Handbuch der Physik, Band 6a/2, Springer-Verlag, Berlin.

    Google Scholar 

  5. Rockafellar, R.T., The multiplier method of Hestens and Powell applied to convex programming, J. Optim. Theory Appl., 12, 555–562.

    Article  MathSciNet  MATH  Google Scholar 

  6. Rockafellar, R.T., A dual approch to solving nonlinear programming problems by unconstrained optimization, Math. Programming,5, 354–373.

    Article  MathSciNet  MATH  Google Scholar 

  7. Polyak, B.T., Tretyakov, N.V., The method of penalty estimates for conditional extremum problems, Zh. Vychisl. Mat. i Mat. Fiz., 13, 34–46, in Russian

    MATH  Google Scholar 

  8. Nakayama, H., Sayama, H., Sawaragi, Y., A generalized Lagrangian function and multiplier method, J. Optim. Theory Appl., 17, 211–227.

    Article  MathSciNet  MATH  Google Scholar 

  9. Rockafellar, R.T., Augmented Lagrangians and applications of the proximal point algorithm in convex programming, Math. Res, Oper., 12, 97–116

    Article  Google Scholar 

  10. Rockafellar, R.T., Proximal subgradients, marginal values and augmented Lagrangians in nonconvex optimization, Math. Oper. Res., 6, 424–436.

    Article  MathSciNet  MATH  Google Scholar 

  11. Bertsekas, D.P., Constrained Optimization and Lagrangian Multiplier Methods, Academic Press, New York.

    Google Scholar 

  12. Le Tallec, P., Augmented Lagrangians in finite elasticity and in plasticity, in: Computational Aspects of Penetration Mechanics, Lecture Notes in Engineering, vol. 3, Springer-Verlag, Berlin.

    Google Scholar 

  13. Landers, J.A., Taylor, R.L., An augmented Lagrangian formulation for the finite element solution of contact problems, Department of Civil Engineering, University of California, Berkeley, California, Report No UCB/SESM-85/09.

    Google Scholar 

  14. Simo, J., Wriggers, P., Taylor, R., A perturbed Lagrangian formulation for the finite element solution of contact problems, Comp. Meth. Mech. Eng.,50, 163–180.

    Article  MathSciNet  MATH  Google Scholar 

  15. Blanchard, D., Le Tallec, P., Numerical analysis of the equations of small strains quasistatic elastoviscoplasicity, Numer. Math., 50, 147–169.1986

    Article  MathSciNet  MATH  Google Scholar 

  16. Le Tallec, P., Numerical solution of viscoplastic flow problems by augmented Lagrangians, IMA J. Numer. Anal., 6, 185–219.

    Article  MathSciNet  MATH  Google Scholar 

  17. Kikuchi, N., Oden, J.T., Contact Problems in Elasticity, SIAM, Philadel-phia.

    Google Scholar 

  18. Alart, P., Multiplicateurs “augmentés” et méthode de Newton généralisée pour contact avec frottement, Document, Departement de Mécanique, Ecole Polytechnique Fédérale de Lausanne.

    Google Scholar 

  19. Rocicafellar, R.T., First-and second - order epi-differentiability in non-linear programming, Trans. Amer. Math. Soc., 307, 75–108.

    Article  Google Scholar 

  20. Glowinski, R., Le Tallec, P., Augmented Lagrangian and Operator - Splitting Methods in Nonlinear Mechanics, SIAM, Philadelphia.

    Google Scholar 

  21. Gwinner, J., A penalty approximation for unilateral contact problem in non-linear elasticity, Mat. Meth. in the Appl. Sci., 11, 447–458.

    Article  MathSciNet  MATH  Google Scholar 

  22. Ito, K., Kunisch, K., An augmented Lagrangian technique for variational inequalities, Appl. Math. Optim., 21 223–241.

    Article  MathSciNet  MATH  Google Scholar 

  23. Ito, K., Kunisch, K., The augmented Lagrangian method for parameter estimation in elliptic systems, SIAM J. Control Optim., 28, 113–136.

    Article  MathSciNet  MATH  Google Scholar 

  24. Ito, K., Kunisch, K., The augmented Lagrangian Method for equality and inequality constraints in Hilbert spaces, Math. Programming, 46 341–360.

    Article  MathSciNet  MATH  Google Scholar 

  25. Alart, P., Curnier, A., A mixed formulation for frictional contact problems prone to Newton like solution methods, Comp. Meth. Appl. Mech. Eng., 92, 353–375.

    Article  MathSciNet  MATH  Google Scholar 

  26. Arora, J.S., Chahande, A.I., Paeng, J.K., Multiplier methods for engineering optimization, Int. J. Num. Meth. Eng., 32, 1485–1525.

    Article  MathSciNet  MATH  Google Scholar 

  27. De Saxcé, G., Feng, Z.Q., New inequality and functional for contact with friction: the implicit standard material approach, Mech. Struct. & Mach., 19, 301–325.

    Article  Google Scholar 

  28. Kunisch, K., Peichl, G., Estimation of a temporally and spatially varying diffusion coefficient in a parabolic system by an augmented Lagrangian technique, Numer. Math., 59, 473–509.

    Article  MathSciNet  MATH  Google Scholar 

  29. Zavarise, G., Problemi termomeccanici di contatto-aspetti fizici e com-putazionali, Ph.D. Thesis, Università di Padova, Facoltá, di Ingegneria.

    Google Scholar 

  30. Alduncin, G., Augmented Lagrangian methods for the quasistatic vis-coelastic two-body contact problem with friction, in: Proc. Int. Symp. Contact Mechanics, ed. by A. Curnier, pp. 337–359, Presses Polytech-niques et Universitaires Romandes, Lausanne.

    Google Scholar 

  31. Klarbring, A., Mathematical programming and augmented lagriangian methods for frictional contact oroblems, in: Proc. Int. Symp. Contact Mechanics, ed. by A. Curnier, pp. 409–422, Presses Polytechniques et Universitaires Romandes, Lausanne.

    Google Scholar 

  32. Laursen, T.A., Formulation and treatment of frictional contact problems using finite elements, SUDAM Report No 92–6.

    Google Scholar 

  33. Simo, J.C., Laursen, T.A., Augmented Lagrangean treatment of contact problems involwing friction, Comp. and Structures,42, 97–116.

    Article  MathSciNet  MATH  Google Scholar 

  34. Hager, W-W., Analysis and implementation of a dual algorithm for constrained optimization, J. Optim. Theory Applic., 79, 427–462.

    Article  MathSciNet  MATH  Google Scholar 

  35. Hiriart-Urruty, J.-B., Lemaréchal, C., Convex Analysis and Minirnization Algorithms, vol.I: Fundamentals, Springer-Verlag, Berlin.

    Google Scholar 

  36. Hiriart-Urruty, J.-B., Lemaréchal, C., Convex Analysis and Minirniza-tion Algorithms, vo1.II: Advanced Theory and Bundle Methods, Springer-Verlag, Berlin.

    Google Scholar 

  37. Laursen, T.A., Simo, J.C., Algorithmic symmetrization of Coulomb frictional problems using augmented Lagrangians, Comp. Meth. Appl. Mech. Eng., 108, l33–146.

    Article  MathSciNet  Google Scholar 

  38. Heegaard, J.-H., Curnier, A., An augmented Lagrangian method for discrete large-slip contact problems, Int. J. Num. Meth. Eng., 36, 569–593.

    Article  MathSciNet  MATH  Google Scholar 

  39. Wriggers, P., Zavarise, G., Application of augmented Lagrangian techniques for non-linear consttutive laws in contact interfaces, Comm. appl. numer. methods eng., 9, 815–824.

    Article  MATH  Google Scholar 

  40. Wriggers, P., Zavarise, G., Recent trends in computational methods for contact problems, in: Proc. of the XI Polish Conf. on Computer Methods in Mechanics, ed. by W. Gilewski and L. Chodor, vol.3, pp. 127–148, Kielce University of Technology, Poland.

    Google Scholar 

  41. Alduncin, G., Decomposition methods and mixed finite element approximations of adherence and contact problems in finite viscoelasticity, in: Contact Mechanics, ed. by M. Raous, M. Jean and J.J. Moreau, pp. 229–236., Plenum Press, New York.

    Chapter  Google Scholar 

  42. Auricchia, F., Sacco, E., An effective plate element for contact problems, in: Contact Mechanics, ed. by M. Raous, M. Jean and J.J. Moreau, pp. 237–241., Plenum Press, New York.

    Chapter  Google Scholar 

  43. Bergounioux, M., Kunisch, K., Augmented Lagrangian techniques for elliptic state constrained optimal control problems, Université d’Orléans, Département de mathématiques, URA 1803, Report No 95–12

    Google Scholar 

  44. Bille, J.-P., Cescotto, S., Habraken, A.M., Charlier, R., Numerical approach of contact using an augmented Lagrangian method, in: Contact Mechanics, ed. by M. Raous, M. Jean and J.J. Moreau, pp. 243–246, Plenum Press, New York.

    Chapter  Google Scholar 

  45. Cescotto, S., Modelling of frictional contact from mixed variational principles, in: Contact Mechanics, ed. by M. Raous, M. Jean and J.J. Moreau, pp. 217–227., Plenum Press, New York.

    Chapter  Google Scholar 

  46. Chabrand, P., Dubois, F., Numerical method for dealing with frictional contact problems in finite elastoplastic deformations, in: Contact Mechanics, ed. by M. Raous, M. Jean and J.J. Moreau, pp. 259–168., Plenum Press, New York.

    Google Scholar 

  47. Cuomo, M., Ventura, G., Augmented Lagrangian formulation for analysis of notension structures with unilateral supports, in: Contact Mechanics, ed. by M. Raous, M. Jean and J.J. Moreau, pp. 247–250., Plenum Press, New York.

    Chapter  Google Scholar 

  48. Heegaard, J.H., Geometric properties of unilateral contact constraints, in: Contact Mechanics, ed. by M. Raous, M. Jean and J.J. Moreau, pp. 173–182, Plenum Press, New York.

    Chapter  Google Scholar 

  49. Ito, K., Kunisch, K., Augmented Lagrangian formulation of nonsmooth, convex optimization in Hilbert spaces, in: Control of Partial Differential Equations and Applications, ed by E. Casas, Marcel Dekker, pp. 107–117.

    Google Scholar 

  50. Morel, J.-M., Solomini, S., Variatioonal Methods in Image Segmentation with seven image processing experiments, Birkhaüser, Boston.

    Google Scholar 

  51. Zavarise, G., Wriggers, P., Schrefler, B.A., On Augmented Lagrangian Algorithms for thermomechanical contact problems with friction, mt. J. Numer. Methods Eng., Vol. 38, 2929–2949.

    Article  MATH  Google Scholar 

  52. Eck, C., Existenz und Regularität der Lösungen fiir Kontaktprobleme mit Reibung, Ph. D. Thesis, Mathematisches Institut der Universität Stuttgart.

    Google Scholar 

  53. Golshtein, E.G., Tretyakov, N.V., Modified Lagrangians and Monotone Maps in Optimization, John Wiley & Sons, New York.

    MATH  Google Scholar 

  54. Hiptmair, R., Shiekofer, T., Wohlmuth, B., Multilevel preconditioned augmented Lagrangian techniques for 2nd order mixed problems, Computing, 57, 25–48

    Article  MathSciNet  MATH  Google Scholar 

  55. Ito, K., Kunisch, K., Augmented Lagrangian methods for nonsmooth, convex optimization in Hilbert spaces, Preprint, Institut für Mathematik, Technische Universität Graz, Austria.

    Google Scholar 

  56. Ito, K., Kunisch, K., Augmented Lagrangian- SQP-methods in Hilbert spaces and application to control in the coefficients problems, SIAM J. Optim., 6, 96–125.

    Article  MathSciNet  MATH  Google Scholar 

  57. Ito, K., Kunisch, K., Augmented Lagrangian- SQP-methods for nonlinear optimal control problems of tracking type, SIAM J. Control and Optim., 34, 874–891.

    Article  MathSciNet  MATH  Google Scholar 

  58. Boffi, D., Lovadina, C., Remarks on augmented Lagrangian formulations for mixed finite element schemes, Boll. Un. Mat. Ital., 11-A, 41–55.

    MathSciNet  Google Scholar 

  59. Ito, K., Kunisch, K., An active set strategy based on the augmented Lagrangian formulation for image restoration, Math Modelling Numer. Anal., in press.

    Google Scholar 

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Telega, J.J., Gałka, A. (2001). Augmented Lagrangian Methods for Contact Problems, Optimal Control and Image Restoration. In: Gilbert, R.P., Panagiotopoulos, P.D., Pardalos, P.M. (eds) From Convexity to Nonconvexity. Nonconvex Optimization and Its Applications, vol 55. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-0287-2_23

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  • DOI: https://doi.org/10.1007/978-1-4613-0287-2_23

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4613-7979-9

  • Online ISBN: 978-1-4613-0287-2

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