Weak convergence of a relaxed and inertial hybrid projection-proximal point algorithm for maximal monotone operators in Hilbert space

SIAM Journal on Optimization

Volumn 14, Issue 3, 2004, Pages 773-782

  Alvarez, Felipe ^a  

^a UNIVERSITY OF CHILE   (Chile)

Author keywords

Hilbert space; Inexact iteration; Maximal monotone operator; Orthogonal projection; Proximal point; Relative error; Relaxation; Separating hyperplane; Weak convergence

Indexed keywords

EID: 4043116347     PISSN: 10526234     EISSN: None     Source Type: Journal    
DOI: 10.1137/S1052623403427859     Document Type: Article

References (25)

1. F. ALVAREZ, On the minimizing property of a second order dissipative system in Hilbert spaces, SIAM J. Control Optim., 38 (2000), pp. 1102-1119.
2. F. ALVAREZ AND H. ATTOUCH, An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping, in Wellposedness in Optimization and Related Topics (Gargnano, 1999), Set-Valued Anal., 9 (2001), pp. 3-11.
3. A. S. ANTIPIN, Minimization of convex functions on convex sets by means of differential equations, Differential Equations, 30 (1994), pp. 1365-1375.
4. H. ATTOUCH, X. GOUDOU, AND P. REDONT, The heavy ball with friction method. I. The continuous dynamical system, Commun. Contemp. Math., 2 (2000), pp. 1-34.
5. φ(u) ∋ 0, J. Funct. Anal., 28 (1978), pp. 369-376.
6. D. P. BERTSEKAS, Constrained Optimization and Lagrange Multiplier Methods, Academic Press, New York, 1982.
7. H. BREZIS, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, Mathematics Studies 5, North-Holland, Amsterdam, 1973.
8. R. E. BRUCK, Asymptotic convergence of nonlinear contraction semigroups in Hilbert space, J. Funct. Anal., 18 (1975), pp. 15-26.
9. J. ECKSTEIN AND D. P. BERTSEKAS, On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators, Math. Program., 55 (1992), pp. 293-318.
10. J. ECKSTEIN AND M. C. FERRIS, Operator-splitting methods for monotone affine variational inequalities, with a parallel application to optimal control, INFORMS J. Comput., 10 (1998), pp. 218-235.
11. O. R. GÁRCIGA, A. IUSEM, AND B. F. SVAITER, On the need for hybrid steps in hybrid proximal point methods, Oper. Res. Lett., 29 (2001), pp. 217-220.
12. E. G. GOL'SHTEIN AND N. V. TRET'YAKOV, Modified Lagrangians in convex programming and their generalizations, Math. Program. Stud., 10 (1979), pp. 86-97.
13. O. GÜLER, On the convergence of the proximal point algorithm for convex minimization, SIAM J. Control Optim., 29 (1991), pp. 403-419.
14. F. JULES AND P. E. MAINGÉ, Numerical approach to a stationary solution of a second order dissipative dynamical system, Optimization, 51 (2002), pp. 235-255.
15. B. MARTINET, Régularisation d'inéquations variationnelles par approximations successives, Rev. Française Informat. et Recherche Opérationnelle, 4 (1970), pp. 154-158.
16. B. MARTINET, Determination approchée d'un point fixe d'une application pseudo-contractante, C. R. Acad. Sci. Paris Ser. A-B, 274 (1972), pp. 163-165.
17. G. MINTY, Monotone (nonlinear) operators in Hilbert space, Duke Math. J., 29 (1962), pp. 341-346.
18. J. J. MOREAU, Proximité et dualité dans un espace hilbertien, Bull. Soc. Math. France, 93 (1965), pp. 273-299.
19. A. MOUDAFI, Second-order differential proximal methods for equilibrium problems, JIPAM J. Inequal. Pure Appl. Math., 14 (2003).
20. Z. OPIAL, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc., 73 (1967), pp. 591-597.
21. B. T. POLYAK, Some methods of speeding up the convergence of iterative methods, Zh. Vychisl. Mat. Mat. Fiz., 4 (1964), pp. 1-17.
22. R. T. ROCKAFELLAR, Monotone operators and the proximal point algorithm, SIAM J. Control Optim., 14 (1976), pp. 877-898.
23. M. V. SOLODOV AND B. F. SVAITER, A hybrid projection-proximal point algorithm, J. Convex Anal., 6 (1999), pp. 59-70.
24. M. V. SOLODOV AND B. F. SVAITER, Forcing strong convergence of proximal point iterations in a Hilbert space, Math. Program., 87 (2000), pp. 189-202.
25. F. ZIRILLI, F. ALUFFI AND V. PARISI, DAFNE: A Differential Equations Algorithm for Non-linear Equations, ACM Trans. Math. Software, 10 (1984), pp. 317-324.