Global and superlinear convergence of inexact Uzawa methods for saddle point problems with nondifferentiable mappings

SIAM Journal on Numerical Analysis

Volumn 35, Issue 3, 1998, Pages 1130-1148

  Chen, Xiaojun ^a  

^a UNIVERSITY OF NEW SOUTH WALES   (Australia)

Author keywords

Convergence; Inexact; Inner outer; Newton; Nonsmooth; Saddle point; Uzawa

Indexed keywords

EID: 0008141650     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036142995295789     Document Type: Article

References (35)

1. O. AXELSSON AND E. KAPORIN, On the Solution of Nonlinear Equations for Nondifferentiable Mapping, preprint, Department of Mathematics, University of Nijmegen, Nijmegen, 1994.
2. R. E. BANK, B. D. WELFERT, AND H. YSERENTANT, A class of iterative methods for solving saddle point problems, Numer. Math., 56 (1990), pp. 645-666.
3. J. BRAMBLE, J. PASCIAK, AND A. VASSILEV, Analysis of the inexact Uzawa algorithm for saddle point problems, SIAM J. Numer. Anal., 34 (1997), pp. 1072-1092.
4. 1 convex constrained optimization, SIAM J. Control Optim., 34 (1996), pp. 2051-2063.
5. X. CHEN, A verification method for solutions of nonsmooth equations, Computing, 58 (1997), pp. 281-294.
6. X. CHEN, M. Z. NASHED, AND L. QI, Convergence of Newton's method for singular smooth and nonsmooth equations using outer inverses, SIAM J. Optim., 7 (1997), pp. 445-462.
7. X. CHEN AND R. S. WOMERSLEY, A parallel inexact Newton method for stochastic programs with recourse, Ann. Oper. Res., 64 (1996), pp. 113-141.
8. P. G. CIARLET, Introduction to Numerical Linear Algebra and Optimization, Cambridge University Press, Cambridge, 1989.
9. F. H. CLARKE, Optimization and Nonsmooth Analysis, John Wiley, New York, 1983.
10. J. E. DENNIS AND R. B. SCHNABEL, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice-Hall, Englewood Cliffs, NJ, 1983.
11. H. C. ELMAN AND G. H. GOLUB, Inexact and preconditioned Uzawa algorithms for saddle point problems, SIAM J. Numer. Anal., 31 (1994), pp. 1645-1661.
12. 1 functions and the Maratos effect, Oper. Res. Lett., 17 (1995), pp. 131-137.
13. R. FLETCHER, Practical Methods of Optimization, 2nd ed., John Wiley, Chichester, 1987.
14. S. A. FUNKEN AND E. P. STEPHAN, Fast Solvers for Nonlinear FEM-BEM Equations, preprint, Institut für Angewandte Mathematik, University of Hannover, Hannover, 1995.
15. V. GIRAULT AND P. A. RAVIART, Finite Element Methods for Navier-Stokes Equations, Springer-Verlag, Berlin, Heidelberg 1986.
16. R. T. GREGORY AND D. L. KARNEY, A Collection of Matrices for Testing Computational Algorithms, Wiley-Interscience, New York, 1969.
17. S. P. HAN, J. S. PANG, AND N. RANGARAJ, Globally convergent Newton methods for nonsmooth equations, Math. Oper. Res., 17 (1992), pp. 586-607.
18. M. HEINKENSCHLOSS, C. T. KELLEY, AND H. T. TRAN, Fast algorithms for nonsmooth compact fixed point problems, SIAM J. Numer. Anal., 29 (1992), pp. 1769-1792.
19. H. JIANG AND L. QI, Local uniqueness and Newton-type methods for nonsmooth variational inequalities, J. Math. Anal. Appl., 196 (1996), pp. 314-331.
20. J. M. MARTINEZ AND L. QI, Inexact Newton methods for solving nonsmooth equations, J. Comput. Appl. Math., 60 (1995), pp. 127-145.
21. R. MIFFLIN, Semismooth and semiconvex functions in constrained optimization, SIAM J. Control Optim., 15 (1977), pp. 957-972.
22. J. M. ORTEGA AND W. C. RHEINBOLDT, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970.
23. J. S. PANG, Newton's method for B-differentiate equations, Math. Oper. Res., 15 (1990), pp. 311-341.
24. J. S. PANG AND L. QI, Nonsmooth equations: Motivation and algorithms, SIAM J. Optim., 3 (1993), pp. 443-465.
25. L. QI, Convergence analysis of some algorithms for solving nonsmooth equations, Math. Oper. Res., 18 (1993), pp. 227-244.
26. L. QI AND X. CHEN, A globally convergent successive approximation method for severely nonsmooth equations, SIAM J. Control Optim., 33 (1995), pp. 402-418.
27. L. QI AND J. SUN, A nonsmooth version of Newton's method, Math. Programming, 58 (1993), pp. 353-368.
28. D. RALPH, Global convergence of damped Newton's method for nonsmooth equations, via the path search, Math. Oper. Res., 19 (1994), pp. 352-389.
29. R. T. ROCKAFELLAR, Convex Analysis, Princeton University Press, Princeton, NJ, 1970.
30. R. T. ROCKAFELLAR AND R. J.-B. WETS, Linear-Quadratic Problems with Stochastic Penalties: The Finite Generation Algorithm, Lecture Notes in Control and Information Sciences 81, Springer-Verlag, Berlin, 1987, pp. 545-560.
31. T. RUSTEN AND R. WINTHER, A preconditioned iterative method for saddle point problems, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 887-904.
32. J. W. SCHMIDT, W. HOYER, AND C. HAUFE, Consistent approximations in Newton-type decomposition methods, Numer. Math., 47 (1985), pp. 413-425.
33. P. TSENG, Applications of a splitting algorithm to decomposition in convex programming and variational inequalities, SIAM J. Control Optim., 29 (1991), pp. 119-138.
34. A. J. WATHEN AND E. P. STEPHAN, Convergence of Preconditioned Minimum Residual Iteration for Coupled Finite Element/Boundary Element Computations, Report AM-94-03, Department of Mathematics, University of Bristol, UK.
35. B. D. WELFERT, Convergence of inexact Uzawa algorithms for saddle point problems, Department of Mathematics, Arizona State University, Tempe, AZ.