Finite Termination of Inexact Proximal Point Algorithms in Hilbert Spaces.

In the present paper, we study the finite termination of sequences generated by inexact proximal point algorithms for finding zeroes of a maximal monotone (set-valued) operator $$T$$ on a Hilbert space. Under some mild conditions, we get that a sequence generated by inexact proximal point algorithm stops after a finite number of iterations. Our results extend the corresponding results in Rockafellar (SIAM J Control Optim 14:877-898, ). In particular, for optimization problems, our results improve corresponding results in Ferris (Math Progr 50:359-366, ). As applications, we obtain finite termination of projected gradient method. [ABSTRACT FROM AUTHOR]