A family of projective splitting methods for the sum of two maximal monotone operators

Mathematical Programming

Volumn 111, Issue 1-2, 2008, Pages 173-199

  Eckstein, Jonathan ^a   Svaiter, B F ^b  

^a RUTGERS UNIVERSITY   (United States)

^b IMPA   (Brazil)

Author keywords

[No Author keywords available]

Indexed keywords

ALGORITHMS; CONVERGENCE OF NUMERICAL METHODS; HILBERT SPACES; ITERATIVE METHODS; MATHEMATICAL MODELS; PROBLEM SOLVING;

BOUNDARY CASES; MONOTONE OPERATORS; MONOTONICITY TECHNIQUES; SPLITTING METHOD;

MATHEMATICAL OPERATORS;

EID: 34347211821     PISSN: 00255610     EISSN: 14364646     Source Type: Journal    
DOI: 10.1007/s10107-006-0070-8     Document Type: Conference Paper

References (23)

1. Bauschke, H.H., Borwein, J.M.: On projection algorithms for solving convex feasibility problems. SIAM Rev. 38(3), 367-426 (1996)
2. Bauschke, H.H., Combettes, P.L., Reich, S.: The asymptotic behavior of the composition of two resolvents. Nonlin. Anal. 60(2), 283-301 (2005)
3. Browder, F.E.: Multi-valued monotone nonlinear mappings and duality mappings in Banach spaces. Trans. Am. Math. Soc. 118, 338-351. (1965)
4. Browder, F.E.: Nonlinear maximal monotone operators in Banach space. Math. Ann. 175, 89-113(1968)
5. Cimmino, G.: Calcolo approssimato per le soluzioni dei sistemi di equazioni lineari. Ric. Sci. Progr. Teen. Econ. Naz. 1, 326-333 (1938)
6. Combettes, P.L.: Solving monotone inclusions via compositions of nonexpansive averaged operators. Optimization 53(5-6), 475-504 (2004)
7. Eckstein, J.: Some saddle-function splitting methods for convex programming. Optim. Meth. Softw. 4(1), 75-83 (1994)
8. Eckstein, J., Bertsekas, D.P.: On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators. Math. Program. 55(3), 293-318 (1992)
9. Eckstein, J., Ferris, M.C.: Operator-splitting methods for monotone affine variational inequalities, with a parallel application to optimal control. INFORMS J. Comput. 10(2), 218-235 (1998)
10. Gabay, D.: Applications of the method of multipliers to variational inequalities. In: Fortin, M., Glowinski, R. (eds.) Augmented Lagrangian Methods: Applications to the Solution of Boundary Value Problems, chap. IX, pp. 299-340. North-Holland, Amsterdam (1983)
11. Kaczmarz, S.: Angenäherte auflösung von systemen linearer gleichungen. Bull. Int. Acad. Pol. Sci. A. 1937, 355-357 (1937)
12. Kaczmarz, S.: Approximate solution of systems of linear equations. Int. J. Control. 57(6), 1269-1271 (1993) (Translated from the German)
13. Lawrence, J., Spingarn, J.E.: On fixed points of nonexpansive piecewise isometric mappings. Proc. Lond. Math. Soc. 55(3), 605-624 (1987)
14. Lions, P.-L.: Une méthode itérative de résolution d'une inéquation variationnelle. Isr. J. Math. 31(2), 204-208 (1978)
15. Lions, P.-L., Mercier, B.: Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal. 16(6), 964-979 (1979)
16. Minty, G.J.: Monotone (nonlinear) operators in Hubert space. Duke Math. J. 29, 341-346 (1962)
17. Passty, G.B.: Ergodic convergence to a zero of the sum of monotone operators in Hilbert space. J. Math. Anal. Appl. 72(2), 383-390 (1979)
18. Rockafellar, R.T.: On the maximality of sums of nonlinear monotone operators. Trans. Am. Math. Soc. 149, 75-88 (1970)
19. Solodov, M.V., Svaiter, B.F.: A hybrid approximate extragradient-proximal point algorithm using the enlargement of a maximal monotone operator. Set-Valued Anal. 7(4), 323-345 (1999)
20. Solodov, M.V., Svaiter, B.F.: A hybrid projection-proximal point algorithm. J. Convex Anal. 6(1), 59-70 (1999)
21. Solodov, M.V., Svalter, B.F.: Forcing strong convergence of proximal point iterations in a Hilbert space. Math. Program. 87(1), 189-202 (2000)
22. Spingarn, J.E.: Partial inverse of a monotone operator. Appl. Math. Optim. 10(3), 247-265 (1983)
23. Tseng, P.: A modified forward-backward splitting method for maximal monotone mappings. SIAM J. Control Optim. 38(2), 431-446 (2000)