W-convergence of the proximal point algorithm in complete CAT(0) metric spaces

Document Type : Research Paper


Department of Mathematics‎, ‎College of Sciences‎, ‎Higher Education Center of Eghlid‎, ‎Eghlid‎, ‎Iran.


‎In this paper‎, ‎we generalize the proximal point algorithm to complete CAT(0) spaces and show‎ ‎that the sequence generated by the proximal point algorithm‎ $w$-converges to a zero of the maximal‎ ‎monotone operator‎. ‎Also‎, ‎we prove that if $f‎: ‎X\rightarrow‎ ‎]-\infty‎, +‎\infty]$ is a proper‎, ‎convex and lower semicontinuous‎ ‎function on the complete CAT(0) space $X$‎, ‎then the proximal point algorithm $w$-converges to a zero of the subdifferential of $f$‎, ‎i.e., a minimizer of $f$‎. ‎Some strong convergence results (convergence in metric) are also presented with additional assumptions on the monotone operator and‎ ‎the convex function $f$.


Main Subjects

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Volume 43, Issue 3
June 2017
Pages 817-834
  • Receive Date: 30 January 2016
  • Revise Date: 21 February 2016
  • Accept Date: 04 March 2016
  • First Publish Date: 01 June 2017