A proximal-projection method for finding zeros of set-valued operators

Dan Butnariu, Gábor Kassay

Research output: Contribution to journalArticlepeer-review


In this paper we study the convergence of an iterative algorithm for finding zeros with constraints for not necessarily monotone set-valued operators in a reflexive Banach space. This algorithm, which we call the proximal-projection method is, essentially, a fixed point procedure, and our convergence results are based on new generalizations of the Browder's demiclosedness principle. We show how the proximal-projection method can be applied for solving ill-posed variational inequalities and convex optimization problems with data given or computable by approximations only. The convergence properties of the proximal-projection method we establish also allow us to prove that the proximal point method (with Bregman distances), whose convergence was known to occur for maximal monotone operators, still converges when the operator involved in it is monotone with sequentially weakly closed graph.

Original languageEnglish
Pages (from-to)2096-2136
Number of pages41
JournalSIAM Journal on Control and Optimization
Issue number4
StatePublished - 2008
Externally publishedYes


  • Bregman distance
  • Bregman projection
  • D -nonexpansivity pole
  • D -resolvent
  • D-antiresolvent
  • D-firm operator
  • D-inverse strongly monotone operator
  • D-nonexpansive operator

ASJC Scopus subject areas

  • Control and Optimization
  • Applied Mathematics


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