Method For Variational Inequalities Research Articles

A reconsideration on convergence of the extra-gradient method for solving quasimonotone variational inequalities

ABSTRACT In this paper, we continue to investigate convergence of the extra-gradient method for solving quasimonotone variational inequalities in Hilbert spaces. Under some suitable conditions, we prove that the iterative sequence generated by the extra-gradient method converges weakly to a solution of quasimonotone variational inequalities. Our result corrects and improves the main result in Salahuddin [The extra-gradient method for quasimonotone variational inequalities. Optimization. 2022;71:2519–2528].

Approximation of an Inertial Iteration Method for a Set-Valued Quasi Variational Inequality

Approximation of an Inertial Iteration Method for a Set-Valued Quasi Variational Inequality

A novel method for hierarchical variational inequality with split common fixed point constraint

A novel method for hierarchical variational inequality with split common fixed point constraint

Perseus: a simple and optimal high-order method for variational inequalities

Perseus: a simple and optimal high-order method for variational inequalities

A Modified Form of Inertial Viscosity Projection Methods for Variational Inequality and Fixed Point Problems

This paper aims to introduce an iterative algorithm based on an inertial technique that uses the minimum number of projections onto a nonempty, closed, and convex set. We show that the algorithm generates a sequence that converges strongly to the common solution of a variational inequality involving inverse strongly monotone mapping and fixed point problems for a countable family of nonexpansive mappings in the setting of real Hilbert space. Numerical experiments are also presented to discuss the advantages of using our algorithm over earlier established algorithms. Moreover, we solve a real-life signal recovery problem via a minimization problem to demonstrate our algorithm’s practicality.

Strong convergence of modified inertial extragradient methods for non-Lipschitz continuous variational inequalities and fixed point problems

Strong convergence of modified inertial extragradient methods for non-Lipschitz continuous variational inequalities and fixed point problems

A generalized extragradient method for variational inequalities of the second kind

A generalized extragradient method for variational inequalities of the second kind

Modified Halpern and viscosity methods for hierarchical variational inequalities on Hadamard manifolds

Modified Halpern and viscosity methods for hierarchical variational inequalities on Hadamard manifolds

Adaptive Methods for Variational Inequalities with Relatively Smooth and Reletively Strongly Monotone Operators

Adaptive Methods for Variational Inequalities with Relatively Smooth and Reletively Strongly Monotone Operators

Multigrid methods for non coercive variational inequalities

In this study, our examination centers around the numerical resolution of non-coercive issues using a multi-grid approach. Our particular emphasis is directed towards employing multi-grid methodologies to tackle non-linear variational inequalities. Our primary goal involves confirming the consistent convergence of the multi-grid algorithm. To attain this objective, we make use of fundamental sub-differential calculus and glean insights from the convergence principles of non-linear multi-grid techniques.

Linearized Douglas–Rachford method for variational inequalities with Lipschitz mappings

Linearized Douglas–Rachford method for variational inequalities with Lipschitz mappings

Modified Inertial-Type Subgradient Extragradient Methods for Variational Inequalities and Fixed Points of Finite Bregman Relatively Nonexpansive and Demicontractive Mappings

In this paper, we design two inertial-type subgradient extragradient algorithms with the linear-search process for resolving the two pseudomonotone variational inequality problems (VIPs) of and the common fixed point problem (CFPP) of finite Bregman relatively nonexpansive operators and Bregman relatively demicontractive operators in Banach spaces of both p-uniform convexity and uniform smoothness, which are more general than Hilbert ones. By the aid of suitable restrictions, it is shown that the sequences fabricated by the suggested schemes converge weakly and strongly to a solution of a pair of VIPs with a CFPP constraint, respectively. Additionally, the illustrative instance is furnished to back up the practicability and implementability of the suggested methods. This paper reveals the competitive advantage of the proposed algorithms over the existing algorithms; that is, the existing hybrid projection method for a single VIP with an FPP constraint is extended to develop the modified inertial-type subgradient extragradient method for a pair of VIPs with an CFPP constraint.

A fully adaptive method for variational inequalities with quasi-monotonicity

This paper presents a fully adaptive iterative method without linesearch procedure to solve variational inequalities in the setting of quasi-monotone which is locally Lipschitz. Our method involves one functional evaluation alongside one projection per iteration and our stepsizes are not restricted to be nonincreasing as opposed to relevant methods without linesearch procedure but with nonincreasing stepsizes. We give weak and strong convergence analysis of the sequences generated by our proposed method under some mild conditions. We also give R-linear rate convergence under an error bound condition and give some numerical implementations of our method. Numerical results show that our proposed method is competitive with other related methods in the literature.

Operator extrapolation method for variational inequalities in Banach spaces

Operator extrapolation method for variational inequalities in Banach spaces

Bi-step Method for Variational Inequalities on Riemannian Manifold of Non-negative Constant Curvature

Bi-step Method for Variational Inequalities on Riemannian Manifold of Non-negative Constant Curvature

An Approximation Method for Variational Inequality with Uncertain Variables

In this paper, a Stieltjes integral approximation method for uncertain variational inequality problem (UVIP) is studied. Firstly, uncertain variables are introduced on the basis of variational inequality. Since the uncertain variables are based on nonadditive measures, there is usually no density function. Secondly, the expected value model of UVIP is established after the expected value is discretized by the Stieltjes integral. Furthermore, a gap function is constructed to transform UVIP into an uncertain constraint optimization problem, and the optimal value of the constraint problem is proved to be the solution of UVIP. Finally, the convergence of solutions of the Stieltjes integral discretization approximation problem is proved.

A Unified Analysis of Variational Inequality Methods: Variance Reduction, Sampling, Quantization, and Coordinate Descent

In this paper, we present a unified analysis of methods for such a wide class of problems as variational inequalities, which includes minimization problems and saddle point problems. We develop our analysis on the modified Extra-Gradient method (the classic algorithm for variational inequalities) and consider the strongly monotone and monotone cases, which corresponds to strongly-convex-strongly-concave and convex-concave saddle point problems. The theoretical analysis is based on parametric assumptions about Extra-Gradient iterations. Therefore, it can serve as a strong basis for combining the already existing type methods and also for creating new algorithms. In particular, to show this we develop new robust methods, which include methods with quantization, coordinate methods, distributed randomized local methods, and others. Most of these approaches have never been considered in the generality of variational inequalities and have previously been used only for minimization problems. The robustness of the new methods is also confirmed by numerical experiments with GANs.

S-Iteration inertial subgradient extragradient method for variational inequality and fixed point problems

In developing iterative methods for approximating solutions of optimization problems, one of the major goals of researchers is to construct efficient iterative schemes. Over the years, different techniques have been devised by authors to achieve this goal. In this paper, we study a non-Lipschitz pseudomonotone variational inequality problem with common fixed points constraint of Bregman quasi-nonexpansive mappings. We introduce a new iterative method for approximating the solution of this problem in a more general framework of reflexive Banach spaces. Our method employs several techniques to achieve high level of efficiency. One of the techniques employed is the S-iteration process, a method known to be highly efficient in comparison with several of the well-known methods. Moreover, our proposed algorithm utilizes the inertial technique and a non-monotonic self-adaptive step size to guarantee high rate of convergence and easy implementation. Unlike several of the existing results on variational inequality problem with non-Lipschitz operator, the design of our method does not involve any linesearch technique. We obtain strong convergence result for the proposed algorithm without the sequentially weakly continuity condition often assumed by authors to guarantee convergence when solving pseudomonotone variational inequality problems. Furthermore, we apply our result to study utility-based bandwidth allocation problem and image restoration problem. Finally, we present several numerical experiments to demonstrate the efficiency of our proposed method in comparison with existing state-of-the-art methods. Our result extends and improves several of the recently announced results in this direction in the literature.

A prediction-correction based proximal method for monotone variational inequalities with linear constraints

<abstract><p>The monotone variational inequalities are being widely used as mathematical tools for studying optimal control problems and convex programming. In this paper, we propose a new prediction-correction method for monotone variational inequalities with linear constraints. The method consists of two procedures. The first procedure (prediction) utilizes projections to generate a predictor. The second procedure (correction) produces the new iteration via some minor computations. The main advantage of the method is that its main computational effort only depends on evaluating the resolvent mapping of the monotone operator, and its primal and dual step sizes can be enlarged. We prove the global convergence of the method. Numerical results are provided to demonstrate the efficiency of the method.</p></abstract>

On Mann-type accelerated projection methods for pseudomonotone variational inequalities and common fixed points in Banach spaces

<abstract><p>In this paper, we investigate two Mann-type accelerated projection procedures with line search method for solving the pseudomonotone variational inequality (VIP) and the common fixed-point problem (CFPP) of finitely many Bregman relatively nonexpansive mappings and a Bregman relatively asymptotically nonexpansive mapping in $ p $-uniformly convex and uniformly smooth Banach spaces. Under mild conditions, we show weak and strong convergence of the proposed algorithms to a common solution of the VIP and CFPP, respectively.</p></abstract>