Publication Cover
Optimization
A Journal of Mathematical Programming and Operations Research
Volume 64, 2015 - Issue 7
317
Views
11
CrossRef citations to date
0
Altmetric
Articles

A variant of forward-backward splitting method for the sum of two monotone operators with a new search strategy

&
Pages 1471-1486 | Received 08 May 2013, Accepted 24 Dec 2013, Published online: 07 Feb 2014

References

  • Eckstein J. Splitting methods for monotone operators, with applications to parallel optimization [PhD thesis]. Cambridge (MA): Massachusetts Institute of Techonology; 1989. Report LIDS-TH-1877, Laboratory for Information and Decision Systems, M.I.T.
  • Eckstein J, Svaiter BF. A family of projective splitting methods for the sum of two maximal monotone operators. Math. Program. 2008;111:173–199.
  • Tseng P. A modified forward-backward splitting method for maximal monotone mappings. SIAM J. Control Optim. 2000;38:431–446.
  • Douglas J, Rachford HH. On the numerical solution of heat conduction problems in two or three space variables. Trans. Amer. Math. Soc. 1956;82:421–439.
  • Lions PL, Mercier B. Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal. 1979;16:964–979.
  • Passty GB. Ergodic convergence to a zero of the sum of monotone operators in Hilbert space. J. Math. Anal. Appl. 1979;72:383–390.
  • Minty G. On the maximal domain of a “monotone” function. Michigan Math. J. 1961;8:135–137.
  • Hartman P, Stampacchia G. On some non-linear elliptic differential-functional equations. Acta Math. 1966;115:271–310.
  • Kinderlehrer D, Stampacchia G. An introduction to variational inequalities and their applications. New York (NY): Academic Press; 1980.
  • Facchinei F, Pang JS. Finite-dimensional variational inequalities and complementarity problems. Berlin: Springer; 2003.
  • Bello Cruz JY, Iusem AN. A strongly convergent direct method for monotone variational inequalities in Hilbert spaces. Numer. Func. Anal. Optim. 2009;30:23–36.
  • Iusem AN, Svaiter BF. A variant of Korpelevich’s method for variational inequalities with a new search strategy. Optimization. 1997;42:309–321.
  • Solodov MV, Svaiter BF. A new projection method for variational inequality problems. SIAM J. Control Optim. 1999;37:765–776.
  • Combettes PL, Pesquet J-C. Proximal splitting methods in signal processing. Fixed-point algorithms for inverse problems in science and engineering. Springer Optim. Appl. 2011;10:185–212.
  • Zaraytonelo EH. Projections on convex sets in Hilbert space and spectral theory. In: Zarantonello E, editor. Contributions to nonlinear functional analysis. New York (NY): Academic Press; 1971. p. 237–424.
  • Bauschke HH, Burke JV, Deutsch FR, Hundal HS, Vanderwerff JD. A new proximal point iteration that converges weakly but not in norm. Proc. Amer. Math. Soc. 2005;133:1829–1835.
  • Burachik RS, Iusem AN. Set-valued mappings and enlargements of monotone operators. Berlin: Springer; 2008.
  • Bello Cruz JY, Iusem AN. Convergence of direct methods for paramonotone variational inequalities. Comput. Optim. Appl. 2010;46:247–263.
  • Minty G. Monotone (nonlinear) operators in Hilbert Space. Duke Math. J. 1962;29:341–346.
  • Browder FE. Convergence theorems for sequences of nonlinear operators in Banach spaces. Math. Z. 1967;100:201–225.
  • Iusem AN, Svaiter BF, Teboulle M. Entropy-like proximal methods in convex programming. Math. Oper. Res. 1994;19:790–814.
  • Bauschke HH, Borwein JM. On projection algorithms for solving convex feasibility problems. SIAM Rev. 1996;38:367–426.
  • Bauschke HH, Combettes PL. Convex analysis and monotone operator theory in Hilbert spaces. New York (NY): Springer; 2011.
  • Bello Cruz JY, Iusem AN. A strongly convergent method for nonsmooth convex minimization in Hilbert spaces. Numer. Func. Anal. Optim. 2011;32:1009–1018.
  • Solodov MV, Svaiter BF. Forcing strong convergence of proximal point iterations in a Hilbert space. Math. Prog. 2000;87:189–202.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.