Abstract strongly convergent variants of the proximal point algorithm

Inexact Variants of the Proximal Point Algorithm without Monotonicity

This paper studies convergence properties of inexact variants of the proximal point algorithm when applied to a certain class of nonmonotone mappings. The presented algorithms allow for constant relative errors, in the line of the recently proposed hybrid proximal-extragradient algorithm. The main convergence result extends a recent work of the second author, where exact solutions for the proxi...

A proximal point algorithm converging strongly for general errors

In this paper a proximal point algorithm (PPA) for maximal monotone operators with appropriate regularization parameters is considered. A strong convergence result for PPA is stated and proved under the general condition that the error sequence tends to zero in norm. Note that R.T. Rockafellar (1976) assumed summability for the error sequence to derive weak convergence of PPA in its initial for...

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

‎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 $f‎: ‎Xrightarrow‎ ‎]-infty‎, +‎infty]$ is a proper‎, ‎convex and lower semicontinuous‎ ‎function on the complete CAT(0) space $X$‎, ‎then the proximal...

Finite termination of the proximal point algorithm

where q~ is a c losed, convex func t ion def ined on R n, having values in ~ and S is a c losed, convex set in ~n. We write S for the op t ima l so lu t ion set o f (1), S : = arg minxes cb(x) and assume this set to be non-empty , in o rde r tha t a p ro jec t ion o p e r a t i o n on to this set is well def ined. In o rde r to s impl i fy our analysis , let us define , b ~ ( x ) := ¢~(x )+~, (...

The Three-Dimensional Gauss Algorithm Is Strongly Convergent Almost Everywhere