An abstract proximal point algorithm

Abstract

The proximal point algorithm is a widely used tool for solving a variety of convex optimization problems such as finding zeros of maximally monotone operators, fixed points of nonexpansive mappings, as well as minimizing convex functions. The algorithm works by applying successively so-called "resolvent" mappings associated to the original object that one aims to optimize. In this paper we abstract from the corresponding resolvents employed in these problems the natural notion of jointly firmly nonexpansive families of mappings. This leads to a streamlined method of proving weak convergence of this class of algorithms in the context of complete CAT(0) spaces (and hence also in Hilbert spaces). In addition, we consider the notion of uniform firm nonexpansivity in order to similarly provide a unified presentation of a case where the algorithm converges strongly. Methods which stem from proof mining, an applied subfield of logic, yield in this situation computable and low-complexity rates of convergence.

Effective strong convergence of the proximal point algorithm in CAT(0) spaces

“…This is what we do in Section 3. We mention that such a rate for a quite general uniformity condition (though not including the case examined here) was previously obtained in the paper [15].…”

Abstract strongly convergent variants of the proximal point algorithm

“…Moreover, from this formulation given by (3) one immediately obtains, using (2), that a self-mapping of a CAT(0) space satisfying property (P 2 ) is nonexpansive. Following [34,48], if T and U are self-mappings of X and λ, µ > 0, we say that T and U are (λ, µ)-mutually firmly nonexpansive if for all x, y ∈ X and all α, β ∈…”

“…It is clear that a family (T γ ) is jointly firmly nonexpansive if and only if for every (γ n ) n∈N ⊆ (0, ∞), (T γn ) n∈N is jointly firmly nonexpansive with respect to (γ n ). In [34] it was shown that examples of jointly firmly nonexpansive families of mappings are furnished by resolvent-type mappings used in convex optimization -specifically, by:…”

“…Corollary 2.6 (cf. [34,Corollary 3.11]). Any two mutually (P 2 ) self-mappings of X have the same fixed points.…”

“…It has been observed by Eckstein [15] that the arguments used to prove the convergence of the usual proximal point algorithm "hinge primarily on the firmly nonexpansive properties of the resolvents". Inspired by this remark, the author, together with L. Leuştean and A. Nicolae, has introduced in [34] the concept of jointly firmly nonexpansive family of mappings, in the context of CAT(0) spaces, which allows for a highly abstract proof of the proximal point algorithm's convergence, encompassing virtually all known variants in the literature. (It was not accidental that we have already presented resolvents above in a quite abstract way.)…”