A generalized proximal point algorithm for the variational inequality problem in a Hilbert space

SIAM Journal on Optimization

Volumn 8, Issue 1, 1998, Pages 197-216

  Burachik, Regina S ^a   Iusem, Alfredo N ^b  

^a FEDERAL UNIVERSITY OF RIO DE JANEIRO   (Brazil)

^b IMPA   (Brazil)

Author keywords

Convex optimization; Monotone operators; Proximal point methods; Variational inequalities

Indexed keywords

EID: 0032351233     PISSN: 10526234     EISSN: None     Source Type: Journal    
DOI: 10.1137/S1052623495286302     Document Type: Article

References (22)

1. L. M. BREGMAN, The relaxation method of finding the common points of convex sets and its application to the solution of problems in convex programming, USSR Computational Mathematics and Mathematical Physics, 7 (1967), pp. 200-217.
2. H. BRÉZIS AND A. HARAUX, Image d'une somme d'opérateurs monotones et applications, Israel J. Math., 23 (1976), pp. 165-186.
3. H. BRÉZIS, Opérateurs Monotones Maximaux et Semigroups de Contractions dans les Espaces de Hilbert, Mathematics Studies 5. North-Holland, New York, 1973.
4. F. E. BROWDER, Nonlinear Operators and Nonlinear Equations of Evolution in Banach Spaces, Proc. Sympos. Pure Math. 18, American Mathematical Society, Providence, RI, 1976.
5. R. S. BURACHIK, Generalized Proximal Point Methods for the Variational Inequality Problem, Ph.D. thesis, Instituto de Matemática Pura e Aplicada, Rio de Janeiro, 1995.
6. Y. CENSOR AND S. ZENIOS, The proximal minimization algorithm with D-functions, J. Optim. Theory Appl., 73 (1992), pp. 451-464.
7. Y. CENSOR, A. N. IUSEM, AND S. A. ZENIOS, An interior point method with Bregman functions for the variational inequality problem with paramonotone operators, Math. Programming, to appear.
8. G. CHEN AND M. TEBOULLE, Convergence analysis of a proximal-like minimization algorithm using Bregman functions, SIAM J. Optim., 3 (1993), pp. 538-543.
9. J. ECKSTEIN, Nonlinear proximal point algorithms using Bregman functions, with applications to convex programming, Math. Oper. Res., 18 (1993), pp. 202-226.
10. P. P. B. EGGERMONT, Multiplicative iterative algorithms for convex programming, Linear Algebra Appl., 130 (1990), pp. 25-42.
11. O GÜLER, On the convergence of the proximal point algorithm for convex minimization, SIAM J. Control Optim., 29 (1991), pp. 403-419.
12. P. T. HARKER AND J. S. PANG, Finite dimensional variational inequalities and nonlinear complementarity problems: A survey of theory, algorithms and applications, Math. Programming, 48 (1990), pp. 161-220.
13. A. N. IUSEM, On some properties of generalized proximal point methods for quadratic and linear programming. J. Optim. Theory Appl., 85 (1995), pp. 513-612.
14. A. N. IUSEM, On some properties of paramonotone operators, manuscript.
15. S. KARAMARDIAN, Complementarity over cones with monotone and pseudomonotone maps, J. Optim. Theory Appl., 18 (1976), pp. 445-454.
16. D. KINDERLEHRER AND G. STAMPACCHIA, An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, 1980.
17. K. KIWIEL, Proximal minimization methods with generalized Bregman functions, SIAM J. Control Optim., 35 (1997), pp. 1142-1168.
18. B. LEMAIRE, The proximal algorithm, in Internat. Ser. Numer. Math., J. P. Penot, ed., Birkhäuser, Basel, 87 (1989), pp. 73-87.
19. D. PASCALI AND S. SBURLAN, Nonlinear Mappings of Monotone Type, Editura Academiei, Bucharest, 1978.
20. R. T. ROCKAFELLAR, Convex Analysis, Princeton University Press, Princeton, NJ, 1970.
21. R. T. ROCKAFELLAR, On the maximality of sums of nonlinear monotone operators, Trans. Amer. Math. Soc., 149 (1970), pp. 75-88.
22. R. T. ROCKAFELLAR, Monotone operators and the proximal point algorithm, SIAM J. Control Optim., 14 (1976), pp. 877-898.