Semismooth newton and augmented lagrangian methods for a simplified friction problem

SIAM Journal on Optimization

Volumn 15, Issue 1, 2005, Pages 39-62

  Stadler, Georg ^a  

^a UNIVERSITY OF GRAZ   (Austria)

Author keywords

Augmented Lagrangians; Friction problem; Primaldual active set algorithm; Semismooth Newton method

Indexed keywords

ALGORITHMS; CONVERGENCE OF NUMERICAL METHODS; FRICTION; MATHEMATICAL MODELS; OPTIMIZATION; PROBLEM SOLVING; SET THEORY;

AUGMENTED LAGRANGIANS; FRICTION PROBLEMS; PRIMALDUAL ACTIVE SET ALGORITHMS; SEMISMOOTH NEWTON METHOD;

LAGRANGE MULTIPLIERS;

EID: 14944341127     PISSN: 10526234     EISSN: None     Source Type: Journal    
DOI: 10.1137/S1052623403420833     Document Type: Article

References (33)

1. M. BERGOUNIOUX, M. HADDOU, M. HINTERMÜLLER, AND K. KUNISCH, A comparison of a Moreau-Yosida-based active set strategy and interior point methods for constrained optimal control problems, SIAM J. Optim., 11 (2000), pp. 495-521.
2. M. BERGOUNIOUX, K. ITO, AND K. KUNISCH, Primal-dual strategy for constrained optimal control problems, SIAM J. Control Optim., 37 (1999), pp. 1176-1194.
3. D. P. BERTSEKAS, Constraint Optimization and Lagrange Multipliers, Academic Press, Paris, 1982.
4. X. CHEN, Z. NASHED, AND L. QI, Smoothing methods and semismooth methods for nondifferentiable operator equations, SIAM J. Numer. Anal., 38 (2000), pp. 1200-1216.
5. P. W. CHRISTENSEN, A. KLARBRING, J. S. PANG, AND N. STRÖMBERG, Formulation and comparison of algorithms for fractional contact problems, Internat. J. Numer. Methods Engrg., 42 (1998), pp. 145-173.
6. P. W. CHRISTENSEN AND J. S. PANG, Frictional contact algorithms baaed on semismooth Newton methods, in Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, Appl. Optim. 22, M. Fukushima and L. Qi, eds., Kluwer Academic Publishers, Dordrecht, The Netherlands, 1999, pp. 81-116.
7. P. W. CHRISTENSEN, A nonsmooth Newton method for elastoplaatic problems, Comput. Methods Appl. Mech. Engrg., 191 (2002), pp. 1189-1219.
8. G. DUVAUT AND J.-L. LIONS, Inequalities in Mechanics and Physics, Springer-Verlag, Berlin, 1976.
9. I. EKELAND AND R. TÉMAM, Convex Analysis and Variational Problems, Classics Appl. Math. 28, SIAM, Philadelphia, 1999.
10. F. FACCHINEI AND J.-S. PANG, Finite-Dimensional Variational Inequalities and Complementarity Problems, Vol. 1 and 2, Springer Ser. Oper. Res., Springer-Verlag, New York, 2003.
11. R. GLOWINSKI, J.-L. LIONS, AND R. TRÉMOLIÉRES, Numerical Analysis of Variational Inequalities, North-Holland, Amsterdam, 1981.
12. R. GLOWINSKI, Numerical Methods for Nonlinear Variational Problems, Springer-Verlag, New York, 1984.
13. W. HAN, A regularization procedure for a simplified friction problem, Math. Comput. Modelling, 15 (1991), pp. 65-70.
14. W. HAN, On the numerical approximation of a frictional contact problem with normal compliance, Numer. Funct. Anal. Optim., 17 (1996), pp. 307-321.
15. J. HASLINGER, Z. DOSTÁL, AND R. KUČERA, On a splitting type algorithm for the numerical realization of contact problems with Coulomb friction, Comput. Methods Appl. Mech. Engrg., 191 (2002), pp. 2261-2281.
16. M. HINTERMÜLLER, K. ITO, AND K. KUNISCH, The primal-dual active set strategy as a semismooth Newton method, SIAM J. Optim., 13 (2003), pp. 865-888.
17. M. HINTERMÜLLER AND G. STADLER, A semi-smooth Newton method for constrained linear-quadratic control problems, ZAMM Z. Angew. Math. Mech., 83 (2003), pp. 219-237.
18. M. HINTERMÜLLER, A primal-dual active set algorithm for bilaterally control constrained optimal control problems, Quart. Appl. Math., 61 (2003), pp. 131-160.
19. I. HLAVÁČEK, J. HASLINGER, J. NEČAS, AND J. LOVÍŠEK, Solution of Variational Inequalities in Mechanics, Appl. Math. Sci. 66, Springer-Verlag, New York, 1988.
20. H. C. HUANG, W. HAN, AND J. S. ZHOU, The regularization method for an obstacle problem, Numer. Math., 69 (1994), pp. 155-166.
21. K. ITO AND K. KUNISCH, Augmented Lagrangian methods for nonsmooth, convex optimization in Hilbert spaces, Nonlinear Anal., 41 (2000), pp. 591-616.
22. K. ITO AND K. KUNISCH, Semi-smooth Newton methode for variational inequalities of the first kind, M2AN Math. Model. Numer. Anal., 37 (2003), pp. 41-62.
23. K. ITO AND K. KUNISCH, The primal-dual active set method for nonlinear optimal control problems with bilateral constraints, SIAM J. Control Optim., 43 (2004), pp. 357-376.
24. N. KIKUCHI AND J. T. ODEN, Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods, SIAM Stud. Appl. Math. 8, SIAM, Philadelphia, 1988.
25. R. KORNHUBER, Adaptive Monotone Multigrid Methods for Nonlinear Variational Problems, Adv. Numer. Math., Teubner, Stuttgart, 1997.
26. R. KORNHUBER, Monotone multigrid methods for elliptic variational inequalities. II, Numer. Math., 72 (1996), pp. 481-499.
27. B. KUMMER, Generalized Newton and NCP-methods: Convergence, regularity, actions, Discuss. Math. Differ. Incl. Control Optim., 20 (2000), pp. 209-244.
28. C.-Y. LEE AND J. T. ODEN, A priori error estimation of hp-finite element approximations of frictional contact problems with normal compliance, Internat. J. Engrg. Sci., 31 (1993), pp. 927-952.
29. A. Y. T. LEUNG, G. CHEN, AND W. CHEN, Smoothing Newton method for solving two- and three-dimensional frictional contact problems, Internat. J. Numer. Methods Engrg., 41 (1998), pp. 1001-1027.
30. L. QI AND J. SUN, A nonsmooth version of Newton's method, Math. Programming, 58 (1993), pp. 353-367.
31. L. QI, Convergence analysis of some algorithms for solving nonsmooth equations, Math. Oper. Res., 18 (1993), pp. 227-244.
32. G. SAVARÉ, Regularity and perturbation results for mixed second order elliptic problems, Comm. Partial Differential Equations, 22 (1997), pp. 869-899.
33. M. ULBRICH, Semismooth Newton methods for operator equations in function spaces, SIAM J. Optim., 13 (2003), pp. 805-841.