Related papers: On the nonexpansive operators based on arbitrary m…
We present a generalization of the proximal operator defined through a convex combination of convex objectives, where the coefficients are updated in a minimax fashion. We prove that this new operator is Bregman firmly nonexpansive with…
The Moore-Penrose algorithm provides a generalized notion of an inverse, applicable to degenerate matrices. In this paper, we introduce a covariant extension of the Moore-Penrose method that permits to deal with general relativity involving…
In this note, we consider a framework for the analysis of iterative algorithms which can described in terms of a structured set-valued operator. More precisely, at each point in the ambient space, we assume that the value of operator can be…
The convergence of a new general variable metric algorithm based on compositions of averaged operators is established. Applications to monotone operator splitting are presented.
The demiclosedness principle is one of the key tools in nonlinear analysis and fixed point theory. In this note, this principle is extended and made more flexible by two mutually orthogonal affine subspaces. Versions for finitely many…
We consider a Hilbert space that is a product of a finite number of Hilbert spaces and operators that are represented by "componental operators" acting on the Hilbert spaces that form the product space. We attribute operatorial properties…
Proximal operators are now ubiquitous in non-smooth optimization. Since their introduction in the seminal work of Moreau, many papers have shown their effectiveness on a wide variety of problems, culminating in their use to construct…
Averaged operators have played an important role in fixed point theory in Hilbert spaces. They emerged as a necessity to obtain solutions to fixed point problems where the underlying operator is not contractive and thus renders Banach fixed…
We focus on the convergence analysis of averaged relaxations of cutters, specifically for variants that---depending upon how parameters are chosen---resemble \emph{alternating projections}, the \emph{Douglas--Rachford method}, \emph{relaxed…
The concept of asymptotically nonexpansive mappings is an important generalization of the class of nonexpansive mappings. Implicit midpoint procedures are extremely fundamental for solving equations involving nonlinear operators. This paper…
In this paper we present a systematic study of regular sequences of quasi-nonexpansive operators in Hilbert space. We are interested, in particular, in weakly, boundedly and linearly regular sequences of operators. We show that the type of…
In this paper, we present a convergence rate analysis for the inexact Krasnosel'skii-Mann iteration built from nonexpansive operators. Our results include two main parts: we first establish global pointwise and ergodic iteration-complexity…
In this paper we introduce an interlacing condition on the elements of a family of operators that allows us to gather together a number of results on fixed points and common fixed points for single and families of mappings defined on metric…
In general there exists no relationship between the fixed point sets of the composition and of the average of a family of nonexpansive operators in Hilbert spaces. In this paper, we establish an asymptotic principle connecting the cycles…
Within convex analysis, a rich theory with various applications has been evolving since the proximal average of convex functions was first introduced over a decade ago. When one considers the subdifferential of the proximal average, a…
Frugal resolvent splittings are a class of fixed point algorithms for finding a zero in the sum of the sum of finitely many set-valued monotone operators, where the fixed point operator uses only vector addition, scalar multiplication and…
We introduce regularity notions for averaged nonexpansive operators. Combined with regularity notions of their fixed point sets, we obtain linear and strong convergence results for quasicyclic, cyclic, and random iterations. New convergence…
Our work presents a new iterative scheme to approximate the fixed points of nonexpansive mapping. The proposed algorithm is constructed to enhance convergence efficiency while preserving theoretical robustness. Under appropriate assumptions…
Decentralized optimization is a powerful paradigm that finds applications in engineering and learning design. This work studies decentralized composite optimization problems with non-smooth regularization terms. Most existing gradient-based…
We introduce and investigate a new generalized convexity notion for functions called prox-convexity. The proximity operator of such a function is single-valued and firmly nonexpansive. We provide examples of (strongly) quasiconvex, weakly…