Related papers: Resolvent of the parallel composition and proximit…

We introduce the resolvent composition, a monotonicity-preserving operation between a linear operator and a set-valued operator, as well as the proximal composition, a convexity-preserving operation between a linear operator and a function.…

In this paper, we propose a new general and stable fixed-point approach to compute the resolvents of the composition of a set-valued maximal monotone operator with a linear bounded mapping. Weak, strong and linear convergence of the…

Using the theory of Hilbert direct integrals, we introduce and study a monotonicity-preserving operation, termed the integral resolvent mixture. It combines arbitrary families of monotone operators acting on different spaces and linear…

We propose a flexible approach for computing the resolvent of the sum of weakly monotone operators in real Hilbert spaces. This relies on splitting methods where strong convergence is guaranteed. We also prove linear convergence under…

Resolvent compositions were recently introduced as monotonicity-preserving operations that combine a set-valued monotone operator and a bounded linear operator. They generalize in particular the notion of a resolvent average. We analyze the…

This paper presents an in-depth analysis of a parametrized version of the resolvent composition, an operation that combines a set-valued operator and a linear operator. We provide new properties and examples, and show that resolvent…

Monotone operators, especially in the form of subdifferential operators, are of basic importance in optimization. It is well known since Minty, Rockafellar, and Bertsekas-Eckstein that in Hilbert space, monotone operators can be understood…

The averaged alternating modified reflections algorithm is a projection method for finding the closest point in the intersection of closed convex sets to a given point in a Hilbert space. In this work, we generalize the scheme so that it…

In this paper we propose a resolvent splitting with minimal lifting for finding a zero of the sum of $n\ge 2$ maximally monotone operators involving the composition with a linear bounded operator. The resolvent of each monotone operator,…

We give in this paper a convergence result concerning parallel asynchronous algorithm with bounded delays to solve a nonlinear fixed point problems. This result is applied to calculate the solution of a strongly monotone operator. Special…

We focus on the linear convergence of generalized proximal point algorithms for solving monotone inclusion problems. Under the assumption that the associated monotone operator is metrically subregular or that the inverse of the monotone…

Recently, various high-order methods have been developed to solve the convex optimization problem. The auxiliary problem of these methods shares the general form that is the same as the high-order proximal operator proposed by Nesterov. In…

Resolvents of set-valued operators play a central role in various branches of mathematics and in particular in the design and the analysis of splitting algorithms for solving monotone inclusions. We propose a generalization of this notion,…

The aim of this article is to present two different primal-dual methods for solving structured monotone inclusions involving parallel sums of compositions of maximally monotone operators with linear bounded operators. By employing some…

Composite minimization involves a collection of smooth functions which are aggregated in a nonsmooth manner. In the convex setting, we design an algorithm by linearizing each smooth component in accordance with its main curvature. The…

Several aspects of the interplay between monotone operator theory and convex optimization are presented. The crucial role played by monotone operators in the analysis and the numerical solution of convex minimization problems is emphasized.…

We show that the set of fixed points of the average of two resolvents can be found from the set of fixed points for compositions of two resolvents associated with scaled monotone operators. Recently, the proximal average has attracted…

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.…

In this paper, under the monotonicity of pairs of operators, we propose some Generalized Proximal Point Algorithms to solve non-monotone inclusions using warped resolvents and transformed resolvents. The weak, strong, and linear convergence…

Properties of compositions and convex combinations of averaged nonexpansive operators are investigated and applied to the design of new fixed point algorithms in Hilbert spaces. An extended version of the forward-backward splitting…