Solving Monotone Inclusions Research Articles

A Modified Forward-Backward Splitting Method for Solving Monotone Inclusions and Fixed Points Problems

A Modified Forward-Backward Splitting Method for Solving Monotone Inclusions and Fixed Points Problems

Forward-reflected-backward and shadow-Douglas–Rachford with partial inverse for solving monotone inclusions

Forward-reflected-backward and shadow-Douglas–Rachford with partial inverse for solving monotone inclusions

The forward-backward-forward algorithm with extrapolation from the past and penalty scheme for solving monotone inclusion problems and applications

The forward-backward-forward algorithm with extrapolation from the past and penalty scheme for solving monotone inclusion problems and applications

Modified proximal gradient methods involving double inertial extrapolations for monotone inclusion

In this work, we propose a novel class of forward‐backward‐forward algorithms for solving monotone inclusion problems. Our approach incorporates a self‐adaptive technique to eliminate the need for explicitly selecting Lipschitz assumptions and utilizes two‐step inertial extrapolations to enhance the convergence rate of the algorithm. We establish a weak convergence theorem under mild assumptions. Furthermore, we conduct numerical tests on image deblurring and data classification as practical applications. The experimental results demonstrate that our algorithm surpasses some existing methods in the literature which shows its superior performance and effectiveness.

A Relaxed Inertial Method for Solving Monotone Inclusion Problems with Applications

We study a relaxed inertial forward–backward–half-forward splitting approach with variable step size to solve a monotone inclusion problem involving a maximal monotone operator, a cocoercive operator, and a monotone Lipschitz operator. The convergence of the sequence of iterations generated by the discretisations of a continuous-time dynamical system is established under suitable conditions. Given the challenges associated with computing the resolvent of the composite operator, the proposed method is employed to tackle the composite monotone inclusion problem. Additionally, a convergence analysis is conducted under certain conditions. To demonstrate the effectiveness of the algorithm, numerical experiments are performed on the image deblurring problem.

Forward-backward-half forward splitting algorithm with deviations

ABSTRACT In this paper, we present a forward-backward-half forward splitting algorithm with deviations for solving the structured monotone inclusion problem composed of a maximally monotone operator, a maximally monotone and Lipschitz continuous operator and a cocoercive operator. The weak and linear convergence of the proposed algorithms is established. The aim of introducing deviations is to improve the performance of the forward-backward-half forward splitting algorithm through suitable choices of the deviations. A numerical example is given to show that the two-step inertial variant of the forward-backward-half forward splitting algorithms with deviations outperforms the existing methods in Briceño-Arias and Davis [Forward-backward-half forward algorithm for solving monotone inclusions, SIAM J Optimiz, 2017;28(4):2839–2871. doi: 10.1137/17M1120099], Zong et al. [An accelerated forward-backward-half forward splitting algorithm for monotone inclusion with applications to image restoration, Optimization, 2022;73(2):401–428. doi: 10.1080/02331934.2022.2107926] and Fan et al. [Convergence of an inertial shadow Douglas–Rachford splitting algorithm for monotone inclusions, Numer Func Anal Optim, 2021;42(14):1627–1644. doi: 10.1080/01630563.2021.2001749].

A generalized viscosity forward-backward splitting scheme with inertial terms for solving monotone inclusion problems and its applications

<p>Our aim was to establish a novel generalized viscosity forward-backward splitting scheme that incorporates inertial terms for addressing monotone inclusion problems within a Hilbert space. By incorporating appropriate control conditions, we achieved strong convergence. The significance of this theorem lies in its applicability to resolve convex minimization problems. To demonstrate the efficacy of our proposed algorithm, we conducted a comparative analysis of its convergence behavior against other algorithms. Finally, we showcased the performance of our proposed method through numerical experiments aimed at addressing image restoration problems.</p>

A new self-adaptive method for solving resolvent of sum of two monotone operators in Banach spaces

We introduce a Tseng extragradient method for solving monotone inclusion problem in Banach space. A strong convergence result of an Halpern inertial extrapolation method for solving the resolvent of sum of two monotone operators without the knowledge of the Lipschitz constant was established. Lastly, we illustrate some numerical behavior of our iterative scheme to showcase the performance of the proposed method compared to other related results in the literature.

Two-step inertial forward–reflected–anchored–backward splitting algorithm for solving monotone inclusion problems

The main purpose of this paper is to propose and study a two-step inertial anchored version of the forward–reflected–backward splitting algorithm of Malitsky and Tam in a real Hilbert space. Our proposed algorithm converges strongly to a zero of the sum of a set-valued maximal monotone operator and a single-valued monotone Lipschitz continuous operator. It involves only one forward evaluation of the single-valued operator and one backward evaluation of the set-valued operator at each iteration; a feature that is absent in many other available strongly convergent splitting methods in the literature. Finally, we perform numerical experiments involving image restoration problem and compare our algorithm with known related strongly convergent splitting algorithms in the literature.

Relaxed Tseng splitting method with double inertial steps for solving monotone inclusions and fixed point problems

Relaxed Tseng splitting method with double inertial steps for solving monotone inclusions and fixed point problems

On Proximal Algorithms with Inertial Effects Beyond Monotonicity

Inertial procedures attached to classical methods for solving monotone inclusion and optimization problems, which arise from an implicit discretization of second-order differential equations, have shown a remarkable acceleration effect with respect to these classical algorithms. Among these classical methods, one can mention steepest descent, alternate directions, and the proximal point methods. For the problem of finding zeroes of set-valued operators, the convergence analysis of all existing inertial-proximal methods requires the monotonicity of the operator. We present here a new inertial-proximal point algorithm for finding zeroes of set-valued operators, whose convergence is established for a relevant class of nonmonotone operators, namely the hypomonotone ones.

The variable metric three-operator algorithms for solving monotone inclusions

In this paper, we propose an inexact variable metric three-operator algorithm and a viscous version of it for finding a zero of the sum of three monotone operators in a real Hilbert space. Under some appropriate conditions, we establish the weak or strong convergence of the proposed algorithms. As a special case of the proposed problem, we present a variable metric Douglas–Rachford splitting algorithm and obtain a strong convergence result. The algorithms and the results in this paper are further extensions of the algorithms and results for solving the zero inclusion problems of monotone operators. To illustrate the efficiency of the proposed algorithms, we construct the numerical experiments and give some choices for the variable metric matrix.

A Modified Tseng Splitting Method with Double Inertial Steps for Solving Monotone Inclusion Problems

A Modified Tseng Splitting Method with Double Inertial Steps for Solving Monotone Inclusion Problems

Resolvent-Free Method for Solving Monotone Inclusions

In this work, we consider the monotone inclusion problem in real Hilbert spaces and propose a simple inertial method that does not include any evaluations of the associated resolvent and projection. Under suitable assumptions, we establish the strong convergence of the method to a minimal norm solution. Saddle points of minimax problems and critical points problems are considered as the applications. Numerical examples in finite- and infinite-dimensional spaces illustrate the performances of our scheme.

Convergence of Parameterized Variable Metric Three-Operator Splitting with Deviations for Solving Monotone Inclusions

In this paper, we propose a parameterized variable metric three-operator algorithm for finding a zero of the sum of three monotone operators in a real Hilbert space. Under some appropriate conditions, we prove the strong convergence of the proposed algorithm. Furthermore, we propose a parameterized variable metric three-operator algorithm with a multi-step inertial term and prove its strong convergence. Finally, we illustrate the effectiveness of the proposed algorithm with numerical examples.

Solving monotone inclusions involving the sum of three maximally monotone operators and a cocoercive operator with applications

Solving monotone inclusions involving the sum of three maximally monotone operators and a cocoercive operator with applications

Strong Convergence of Forward–Reflected–Backward Splitting Methods for Solving Monotone Inclusions with Applications to Image Restoration and Optimal Control

In this paper, we propose and study several strongly convergent versions of the forward–reflected–backward splitting method of Malitsky and Tam for finding a zero of the sum of two monotone operators in a real Hilbert space. Our proposed methods only require one forward evaluation of the single-valued operator and one backward evaluation of the set-valued operator at each iteration; a feature that is absent in many other available strongly convergent splitting methods in the literature. We also develop inertial versions of our methods and strong convergence results are obtained for these methods when the set-valued operator is maximal monotone and the single-valued operator is Lipschitz continuous and monotone. Finally, we discuss some examples from image restorations and optimal control regarding the implementations of our methods in comparisons with known related methods in the literature.

Iterative algorithm for solving monotone inclusion and fixed point problem of a finite family of demimetric mappings

<abstract><p>The goal of this study is to develop a novel iterative algorithm for approximating the solutions of the monotone inclusion problem and fixed point problem of a finite family of demimetric mappings in the context of a real Hilbert space. The proposed algorithm is based on the inertial extrapolation step strategy and combines forward-backward and Tseng's methods. We introduce a demimetric operator with respect to $ M $-norm, where $ M $ is a linear, self-adjoint, positive and bounded operator. The algorithm also includes a new step for solving the fixed point problem of demimetric operators with respect to the $ M $-norm. We study the strong convergence behavior of our algorithm. Furthermore, we demonstrate the numerical efficiency of our algorithm with the help of an example. The result given in this paper extends and generalizes various existing results in the literature.</p></abstract>

Accelerated preconditioning Krasnosel'skiĭ-Mann method for efficiently solving monotone inclusion problems

<abstract><p>In this article, we propose a strongly convergent preconditioning method for finding a zero of the sum of two monotone operators. The proposed method combines a preconditioning approach with the robustness of the Krasnosel'skiĭ-Mann method. We show the strong convergence result of the sequence generated by the proposed method to a solution of the monotone inclusion problem. Finally, we provide numerical experiments on the convex minimization problem.</p></abstract>

An alternated inertial algorithm with weak and linear convergence for solving monotone inclusions

An alternated inertial algorithm with weak and linear convergence for solving monotone inclusions