Variational Inclusion Problems Research Articles

Nonmonotone Spectral Analysis for Variational Inclusions

Gradient descent algorithms perform well in convex optimization but can get tied for finding local minima in non-convex optimization. A robust method that combines a spectral approach with nonmonotone line search strategy for solving variational inclusion problems is proposed. Spectral properties using eigenvalues information are used for accelerating the convergence. Nonmonotonic behaviour is exhibited to relax descent property and escape local minima. Nonmonotone spectral conditions leverage adaptive search directions and global convergence for the proposed spectral subgradient algorithm.

Refined Iterative Method for a Common Variational Inclusion and Common Fixed-Point Problem with Practical Applications

This paper introduces a novel parallel method for solving common variational inclusion and common fixed-point (CVI-CFP) problems. The proposed algorithm provides a strong convergence theorem established under specific conditions associated with the CVI-CFP problem. Numerical simulations demonstrate the algorithm’s efficacy in the context of signal recovery problems involving various types of blurred filters. The results highlight the algorithm’s potential for practical applications in image processing and other fields.

On split equality monotone variational inclusion and fixed point problems in reflexive Banach spaces

In this paper, motivated by the works of Akbar and Shahrosvand [Filomat {\bf 32} (2018), no.\ 11, 3917-3932], Ogbuisi and Izuchukwu [Numer. Funct. Anal. Optim. {\bf 41} (2020), no.\ 2, 322-343], and some other related results in the literature, we introduce a Halpern iterative algorithm and employ a Bregman distance approach for approximating a solution of split equality monotone variational inclusion problem and fixed point problem of Bregman relatively nonexpansive mapping in reflexive Banach spaces. Under suitable condition, we state and prove a strong convergence result for approximating a common solution of the aforementioned problems. Furthermore, we give an application of our main result to variational inequality problems and provide some numerical examples to illustrate the convergence behavior of our result. The result presented in this paper extends and complements many related results in literature.

Modi ed inertia Halpern method for split null point problem in Banach spaces

Abstract. In this paper, we study split null point problem in reflexive Banach spaces. Using the Bregman technique together with a modified inertial Halpern method, we approximate a solution of split null point problem. Also, we establish a strong convergence result for approximating the solution of the aforementioned problems. It is worth mentioning that the iterative algorithm employ in this study is design in such a way that it does not require prior knowledge of operator norm. We display some numerical examples to illustrate the performance of the proposed iterative method. The result discuss in this paper extends and complements many related results in literature. Mathematics Subject Classification (2010): 47H06, 47H09, 47J05, 47J25. Keywords: Monotone variational inclusion problem, split feasibility problem, firmly nonexpansive-type mapping, fixed point problem, inertial method.

Parallel inertial forward–backward splitting methods for solving variational inequality problems with variational inclusion constraints

The inertial forward–backward splitting algorithm can be considered as a modified form of the forward–backward algorithm for variational inequality problems with monotone and Lipschitz continuous cost mappings. By using parallel and inertial techniques and the forward–backward splitting algorithm, in this paper, we propose a new parallel inertial forward–backward splitting algorithm for solving variational inequality problems, where the constraints are the intersection of common solution sets of a finite family of variational inclusion problems. Then, strong convergence of proposed iteration sequences is showed under standard assumptions imposed on cost mappings in a real Hilbert space. Finally, some numerical experiments demonstrate the reliability and benefits of the proposed algorithm.

Enhanced Double Inertial Forward–Backward Splitting Algorithm for Variational Inclusion Problems: Applications in Mathematical Integrated Skill Prediction

Enhanced Double Inertial Forward–Backward Splitting Algorithm for Variational Inclusion Problems: Applications in Mathematical Integrated Skill Prediction

Generalized Yosida inclusion problem involving multi-valued operator with XOR operation

Abstract In this article, we study a generalized Yosida variational inclusion problem involving multi-valued operator with XOR operation. It is shown that the generalized Yosida variational inclusion problem involving multi-valued operator with XOR operation is equivalent to a fixed point equation. We have proved that the generalized Yosida approximation operator is Lipschitz continuous. Finally, we prove an existence and convergence result for our problem.

Solving split equality equilibrium and fixed point problems in Banach spaces

In this paper, we introduce a new algorithm for approximating a common solution of Split Equality Generalised Mixed Equilibrium Problem (SEGMEP) and Split Equality Fixed Point Problem (SEFPP) for two infinite families of closed uniformly Li-Lipschitz continuous and Ki-Lipschitz continuous and uniformly quasi-ϕ-asymptotically nonexpansive mappings (i ∈ N and ϕ is the Lyapunov functional) in Banach spaces. Under standard and mild assumption of monotonicity and lower semicontinuity of the SEGMEP associated mappings, we establish the strong convergence of the scheme without imposing any compactness type conditions on either the operators or the spaces considered. We apply our result to approximate the solution of Split Equality Convex Minimization Problem (SECMP) and Split Equality Variational Inclusion Problem (SEVIP). A numerical example is presented to illustrate the performance and implementability of our method. Our results extend, generalize and complement several related works in the literature.

Scaled forward-backward algorithm and the modified superiorized version for solving the split monotone variational inclusion problem

In this paper, we propose an inexact scaled forward-backward algorithm to solve the split monotone variational inclusion problem in a real Hilbert space and prove the strong convergence of it under appropriate conditions. Based on this, we discuss the bounded perturbation resilience of the exact algorithm for introducing the corresponding superiorized version and the superiorization algorithm with restarted perturbations. The numerical experiments illustrate that the proposed algorithms perform well and the superiorization version with restarted perturbations has advantage in decreasing the number of the iterations.

Generalized viscosity method for approximating solutions of countable families of certain nonlinear mappings in real Hilbert space

The purpose of this paper is to introduce a three step iterative algorithm which include a general viscosity explicit method for approximating a common solution of fixed point problem of an infinite family of k_i-demimetric mapping and a directed operator in the framework of real Hilbert space. Furthermore, we prove a strong convergence theorem for approximating a common solution of the aforementioned problems. We also show that our iterative algorithm holds for an infinite family of L-Lipschitzian and quasi-pseudocontractive mapping together with a directed operator. The iterative algorithm presented in this article is design in such a way that it solves some variational inequality problem and no compactness condition is impose on our scheme and mapping. Finally, we give applications of our main result to variational inclusion and equilibrium problems. Our result complements and extends some related result in literature.

Inertial iterative algorithms for common solution of variational inequality and system of variational inclusion problems

Inertial iterative algorithms for common solution of variational inequality and system of variational inclusion problems

Relaxed Inertial Projective Forward-backward Splitting Algorithms for Regularized Least Square Problems

This study presents flexible conditions of the inertial extrapolation parameter for easy implementation that is added to the algorithm in faster convergence. Modified inertial forward-backward splitting algorithms for solving variational inclusion problems are introduced to apply to solve the LASSO problem for image restoration and signal recovery, and the elastic net model for classification problem. Projection methods are used in the final step for narrowing down the search field, resulting in a better solution.

Solvability of a Class of Generalized System of Variational Inclusion Problems Involving \(\oplus\) Operation

Solvability of a Class of Generalized System of Variational Inclusion Problems Involving \(\oplus\) Operation

Relaxed Inertial Method for Solving Split Monotone Variational Inclusion Problem with Multiple Output Sets Without Co-coerciveness and Lipschitz Continuity

Relaxed Inertial Method for Solving Split Monotone Variational Inclusion Problem with Multiple Output Sets Without Co-coerciveness and Lipschitz Continuity

Correction: Convergence theorem for fixed point and split generalized variational inclusion problems with multiple output sets in Banach spaces

Correction: Convergence theorem for fixed point and split generalized variational inclusion problems with multiple output sets in Banach spaces

Self-adaptive Technique with Double Inertial Steps for Inclusion Problem on Hadamard Manifolds

AbstractIn this article, we investigate monotone and Lipschitz continuous variational inclusion problem in the settings of Hadamard manifolds. We propose a forward–backward method with a self-adaptive technique for solving variational inclusion problem. To increase the rate of convergence of our proposed method, we incorporate our iterative method with double inertial steps and establish a convergence result of our iterative method under some mild conditions. Finally, in order to illustrate the computational effectiveness of our method, some numerical examples are also discussed. The result present in this article is new in this space and extends many related results in the literature.

Two inertial projective Mann forward-backward algorithm for variational inclusion problems and application to stroke prediction

Two inertial projective Mann forward-backward algorithm for variational inclusion problems and application to stroke prediction

Strong convergence of split equality variational inequality, variational inclusion, and multiple sets fixed point problems in Hilbert spaces with application

This paper introduces an innovative inertial simultaneous cyclic iterative algorithm designed to address a range of mathematical problems within the realm of split equality variational inequalities. Specifically, the algorithm accommodates finite families of split equality variational inequality problems, infinite families of split equality variational inclusion problems, and multiple-sets split equality fixed point problems involving demicontractive operators in infinite-dimensional Hilbert spaces. The algorithm integrates well-established methods, including the cyclic method, the inertial method, the viscosity approximation method, and the projection method. We establish the strong convergence of this proposed algorithm, demonstrating its applicability in various scenarios and unifying disparate findings from existing literature. Additionally, a numerical example is presented to validate the primary convergence theorem.

An inertial shrinking projection self-adaptive algorithm for solving split variational inclusion problems and fixed point problems in Banach spaces

Abstract In this article, we study the split variational inclusion and fixed point problems using Bregman weak relatively nonexpansive mappings in the p p -uniformly convex smooth Banach spaces. We introduce an inertial shrinking projection self-adaptive iterative scheme for the problem and prove a strong convergence theorem for the sequences generated by our iterative scheme under some mild conditions in real p p -uniformly convex smooth Banach spaces. The algorithm is designed to select its step size self-adaptively and does not require the prior estimate of the norm of the bounded linear operator. Finally, we provide some numerical examples to illustrate the performance of our proposed scheme and compare it with other methods in the literature.

On some fast iterative methods for split variational inclusion problem and fixed point problem of demicontractive mappings

On some fast iterative methods for split variational inclusion problem and fixed point problem of demicontractive mappings