Proximal-based regularization methods and successive approximation of variational inequalities in Hilbert spaces

Control and Cybernetics

Volumn 31, Issue 3, 2002, Pages 521-544

  Kaplan, Alexander ^a   Tichatschke, Rainer ^a  

^a UNIVERSITY OF TRIER   (Germany)

Author keywords

Bregman function; Monotone operators; Proximal point methods; Regularization; Variational inequalities

Indexed keywords

APPROXIMATION THEORY; FUNCTIONS; MATHEMATICAL OPERATORS; OPTIMIZATION;

HILBERT SPACES;

CYBERNETICS;

EID: 0036925961     PISSN: 03248569     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Conference Paper

References (35)

1. ALBER, Y., BURACHIK, R., and IUSEM, A. (1997) A proximal method for non-smooth convex optimization problems in Banach spaces. Abstr. Appl. Anal., 2, 97-120.
2. AUSLENDER, A. and HADDOU, M. (1995) An interior-proximal method for convex linearly constrained problems and its extension to variational inequalities. Math. Programming, 71, 77-100.
3. AUSLENDER, A., TEBOULLE, M., and BEN-TIBA, S. (1999) Interior proximal and multiplier methods based on second order homogeneous kernels. Mathematics of Oper. Res., 24, 645-668.
4. BRØNDSTED, A. and ROCKAFELLAR, R. (1965) On the subdifferentiability of convex functions. Proc. Amer. Math. Soc., 16, 605-611.
5. BURACHIK, R. and IUSEM, A. (1998) A generalized proximal point algorithm for the variational inequality problem in a Hilbert space. SIAM J. Optim., 8, 197-216.
6. BURACHIK, R., IUSEM, A., and SVAITER, B. (1997) Enlargements of maximal monotone operators with application to variational inequalities. Set-Valued Analysis, 5, 159-180.
7. BURACHIK, R., SAGASTIZÁBAL, C., and SVAITER, B. (1999a) Bundle methods for maximal monotone operators. In: Thera, M. and Tichatschke, R., eds., Ill-posed Variational Problems and Regularization Techniques, LNEMS, 477, 49-64. Springer.
8. BURACHIK, R., SAGASTIZÁBAL, C., and SVAITER, B. (1999s) ε-enlargement of maximal monotone operators: Theory and applications. In: Fukushima, M., ed., Reformulation-Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, Kluwer, 25-43.
9. BUTNARIU, D. and IUSEM, A. (1997) On a proximal point method for convex optimization in Banach spaces. Numer. Funct. Anal. and Optimiz., 18, 723-744.
10. BUTNARIU, D. AND IUSEM, A. (2000) Totally Convex Functions for Fixed Points Computation and Infinite Dimensional Optimization. Kluwer, Dordrecht-Boston-London.
11. ECKSTEIN, J. (1993) Nonlinear proximal point algorithms using Bregman functions, with application to convex programming. Math. Oper. Res., 18, 202-226.
12. EKELAND, I. and TEMAM, R. (1976) Convex Analysis and Variational Problems. North-Holland, Amsterdam.
13. GLOWINSKI, R., LIONS, J.L., and TRÉMOLIÉRES, R. (1981) Numerical Analysis of Variational Inequalities. North-Holland, Amsterdam.
14. HLAVÁČEK, I., HASLINGER, J., NEČAS, I., and LOVIŠEK, J. (1988) Numerical Solution of Variational Inequalities. Springer Verlag, Berlin-Heidelberg-New York.
15. IOFFE, A.D., and TIKHOMIROV, V.M. (1975) Theory of Extremal Problems. North Holland Publ. Co., Amsterdam.
16. IUSEM, A. (1995) On some properties of generalized proximal point methods for quadratic and linear programming. JOTA, 85, 593-612.
17. IUSEM, A. (1998) On some properties of paramonotone operators. J. of Conv. Analysis, 5, 269-278.
18. KAPLAN, A. and TICHATSCHKE, R. (2002) Convergence analysis of non-quadratic proximal methods for variational inequalities in Hilbert spaces. J. of Global Optimization, 22, 119-136.
19. KAPLAN, A. and TICHATSCHKE, R. (2000a) Auxiliary problem principle and the approximation of variational inequalities with non-symmetric multivalued operators. CMS Conference Proc., 27, 185-209.
20. KAPLAN, A. and TICHATSCHKE, R. (2000b) Proximal point approach and approximation of variational inequalities. SIAM J. Control Optim., 39, 1136-1159.
21. KAPLAN, A. and TICHATSCHKE, R. (1997) Prox-regularization and solution of ill-posed elliptic variational inequalities. Applications of Mathematics, 42, 111-145.
22. KAPLAN, A. and TICHATSCHKE, R. (1994) Stable Methods for Ill-Posed Variational Problems - Prox-Regularization of Elliptical Variational Inequalities and Semi-Infinite Optimization Problems. Akademie Verlag, Berlin.
23. KIWIEL, K. (1999) A projection-proximal bundle method for convex nondifferentiable minimization. In: Théra, M. and Tichatschke, R., eds., Ill-posed Variational Problems and Regularization Techniques, LNEMS, 477, 137-150. Springer.
24. LIONS, J.L. (1969) Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires. Dunod, Gauthier-Villars, Paris.
25. MARTINET, B. (1970) Régularisation d'inéquations variationelles par approximations successives. RIRO, 4, 154-159.
26. POLYAK, B.T. (1987) Introduction to Optimization. Optimization Software, Inc. Publ. Division, New York.
27. POLYAK, R. and TEBOULLE, M. (1997) Nonlinear rescaling and proximal-like methods in convex optimization. Math. Programming, 76, 265-284.
28. RENAUD, A. and COHEN, G. (1997) An extension of the auxiliary problem principle to nonsymmetric auxiliary operators. ESAIM: Control, Optimization and Calculus of Variations, 2, 281-306.
29. ROCKAFELLAR, R.T. (1970) On the maximality of sums of nonlinear monotone operators. Trans. Amer. Math. Soc., 149, 75-88.
30. ROCKAFELLAR, R.T. (1976) Monotone operators and the proximal point algorithm. SIAM J. Control Optim., 14, 877-898.
31. ROTIN, S. (1999) Convergence of the proximal-point method for ill-posed control problems with partial differential equations. PhD Thesis, University of Trier.
32. SALMON, G., NGUYEN, V.H., and STRODIOT, J.J. (2000) A perturbed and inexact version of the auxiliary problem method for solving general variational inequalities with a multivalued operator. In: Nguyen, V. H., Strodiot, J. J. and Tossings, P., eds., Optimization, LNEMS, 481, 396-418. Springer.
33. SCHMITT, H. (1997) On the regularized Bingham problem. In: Gritzmann, P., Horst, R., Sachs, E. and Tichatschke, R., eds., Recent Advances in Optimization, LNEMS, 452, 298-314. Springer.
34. TEBOULLE, M. (1992) Entropic proximal mappings with application to nonlinear programming. Mathematics of Oper. Res., 17, 670-690.
35. TSENG, P. and BERTSEKAS, D. (1993) On the convergence of the exponential multiplier method for convex programming. Math. Programming, 60, 1-19.