Solutions Of Variational Inequality Problem Research Articles

Strong Convergent Inertial Two-subgradient Extragradient Method for Finding Minimum-norm Solutions of Variational Inequality Problems

AbstractIn 2012, Censor et al. (Extensions of Korpelevich’s extragradient method for the variational inequality problem in Euclidean space. Optimization 61(9):1119–1132, 2012b) proposed the two-subgradient extragradient method (TSEGM). This method does not require computing projection onto the feasible (closed and convex) set, but rather the two projections are made onto some half-space. However, the convergence of the TSEGM was puzzling and hence posted as open question. Very recently, some authors were able to provide a partial answer to the open question by establishing weak convergence result for the TSEGM though under some stringent conditions. In this paper, we propose and study an inertial two-subgradient extragradient method (ITSEGM) for solving monotone variational inequality problems (VIPs). Under more relaxed conditions than the existing results in the literature, we prove that proposed method converges strongly to a minimum-norm solution of monotone VIPs in Hilbert spaces. Unlike several of the existing methods in the literature for solving VIPs, our method does not require any linesearch technique, which could be time-consuming to implement. Rather, we employ a simple but very efficient self-adaptive step size method that generates a non-monotonic sequence of step sizes. Moreover, we present several numerical experiments to demonstrate the efficiency of our proposed method in comparison with related results in the literature. Finally, we apply our result to image restoration problem. Our result in this paper improves and generalizes several of the existing results in the literature in this direction.

Strong Convergence Results for Variational Inequality and Equilibrium Problem in Hadamard Spaces

The main purpose of this paper is to introduce and study a viscosity type algorithm in a Hadamard space which comprises of a demimetric mapping, a finite family of inverse strongly monotone mappings and an equilibrium problem for a bifunction. Strong convergence of the proposed algorithm to a common solution of variational inequality problem, fixed point problem and equilibrium problem is established in Hadamard spaces. Nontrivial Applications and numerical examples were given. Our results compliment some results in the literature.

Characterizations of the existence of solutions for variational inequality problems in Hilbert spaces

Characterizations of the existence of solutions for variational inequality problems in Hilbert spaces

A Halpern-type algorithm for a common solution of nonlinear problems in Banach spaces

Abstract In this article, we propose a Halpern-type subgradient extragradient algorithm for solving a common element of the set of solutions of variational inequality problems for continuous monotone mappings and the set of f-fixed points of continuous f-pseudocontractive mappings in reflexive real Banach spaces. In addition, we prove a strong convergence theorem for the sequence generated by the algorithm. As a consequence, we obtain a scheme that converges strongly to a common f-fixed point of continuous f-pseudocontractive mappings and a scheme that converges strongly to a common zero of continuous monotone mappings in Banach spaces. Furthermore, we provide a numerical example to illustrate the implementability of our algorithm.

Modified inertial extragradient methods for finding minimum-norm solution of the variational inequality problem with applications to optimal control problem

In order to discover the minimum-norm solution of the pseudomonotone variational inequality problem in a real Hilbert space, we provide two variants of the inertial extragradient approach with a novel generalized adaptive step size. Two of the suggested algorithms make use of the projection and contraction methods. We demonstrate several strong convergence findings without requiring the prior knowledge of the Lipschitz constant of the mapping. Finally, we give a number of numerical examples that highlight the benefits and effectiveness of the suggested algorithms and how they may be used to solve the optimal control problem.

Self-Adaptive Extragradient Methods for Solving Variational Inequalities and Fixed Point Problems in 2-Uniformly Convex and Uniformly Smooth Banach Spaces

In this paper, we mainly introduce a new and different self adaptive extragradient algorithm with an inertial technique for solving variational inequality problems of pseudomonotone operators and fixed point problems of relatively nonexpansive mappings in 2-uniformly convex and uniformly smooth Banach spaces. Under some appropriate assumptions imposed on the parameters and operators, we prove that the iterative sequence generated by our proposed algorithm converges strongly to a common solution of variational inequality problems and fixed point problems. Moreover, we give some numerical examples to show the effectiveness and feasibility of our algorithm.

A new modified extragradient method with line-search process for solving pseudomonotone variational inequality in Hilbert spaces

ABSTRACT In this paper, we mainly introduce a new algorithm with a different line-search process for solving variational inequality problem of pseudomonotone and non-Lipschitz operators in real Hilbert spaces. Under some appropriate restrictions imposed on the parameters, we prove a strong convergence theorem for finding an element of solutions of variational inequality problem. At the same time, we give some numerical examples to illustrate the effectiveness of our proposed algorithm. The main results obtained in this paper extend and improve many recent ones in the literature.

Inertial Extragradient Method for Solving Variational Inequality and Fixed Point Problems of a Bregman Demigeneralized Mapping in a Reflexive Banach Spaces

In this paper, by employing a Bregman distance approach, we introduce a self-adaptive inertial extragradient method for finding a common solution of variational inequality problem involving a pseudo-monotone operator and fixed point problem of a Bregman demigeneralized mapping in a reflexive Banach spaces. Using a Bregman distance approach, we establish a strong convergence result for approximating the common solution of the aforementioned problems under some mild assumptions. We display some numerical examples to show the performance of our iterative method. The result presented in this paper extends and complements many related results in literature.

An inertial parallel CQ subgradient extragradient method for variational inequalities application to signal-image recovery

In this paper, we introduce an inertial parallel CQ subgradient extragradient method for finding a common solutions of variational inequality problems. The novelty of this paper is using linesearch methods to find unknown L constant of L-Lipschitz continuous mappings. Strong convergence theorem has been proved under some suitable conditions in Hilbert spaces. Finally, we show applications to signal and image recovery, and show the good efficiency of our proposed algorithm when the number of subproblems is increasing.

Inertial self-adaptive parallel extragradient-type method for common solution of variational inequality problems

ABSTRACT In this paper, we introduce a new inertial self-adaptive parallel subgradient extragradient method for finding common solution of variational inequality problems with monotone and Lipschitz continuous operators. The stepsize of the algorithm is updated self-adaptively at each iteration and does not involve a line search technique nor a prior estimate of the Lipschitz constants of the cost operators. Also, the algorithm does not required finding the farthest element of the finite sequences from the current iterate which has been used in many previous methods. We prove a strong convergence result and provide some applications of our result to other optimization problems. We also give some numerical experiments to illustrate the performance of the algorithm by comparing with some other related methods in the literature.

Commutation principles for optimization problems on spectral sets in Euclidean Jordan algebras

The commutation principle of Ramírez et al. (SIAM J Optim 23:687–694, 2013) proved in the setting of Euclidean Jordan algebras says that when the sum of a real valued function h and a spectral function $$\Phi $$ is minimized/maximized over a spectral set E, any local optimizer a at which h is Fréchet differentiable operator commutes with the derivative $$h^{\prime }(a)$$ . In this note, we describe some analogs of the above result by assuming the existence of a subgradient in place of the derivative (of h) and obtaining strong operator commutativity relations. We show, for example: if a solves the problem $$\underset{E}{\max }\,(h+\Phi )$$ , then a strongly operator commutes with every element in the subdifferential of h at a; If E and h are convex and a solves the problem $$\underset{E}{\min }\,h$$ , then a strongly operator commutes with the negative of some element in the subdifferential of h at a. These results improve known operator commutativity relations for linear h and for solutions of variational inequality problems. We establish these results via a geometric commutation principle that is valid not only in Euclidean Jordan algebras, but also in a broader setting.

Inertial normal S-type Tseng’s extragradient algorithm for solution of variational inequality problems

In this paper, we introduce inertial Tseng’s extragradient algorithms combined with normal-S iteration process for solving variational inequality problems involving pseudo-monotone and Lipschitz continuous operators. Under mild conditions, we establish the weak convergence results in Hilbert spaces. Numerical examples are also presented to show that faster behaviour of the proposed method.

Convergence Theorems for the Variational Inequality Problems and Split Feasibility Problems in Hilbert Spaces

In this paper, we establish an iterative algorithm by combining Yamada’s hybrid steepest descent method and Wang’s algorithm for finding the common solutions of variational inequality problems and split feasibility problems. The strong convergence of the sequence generated by our suggested iterative algorithm to such a common solution is proved in the setting of Hilbert spaces under some suitable assumptions imposed on the parameters. Moreover, we propose iterative algorithms for finding the common solutions of variational inequality problems and multiple-sets split feasibility problems. Finally, we also give numerical examples for illustrating our algorithms.

Inertial Tseng's extragradient method for solving variational inequality problems of pseudo-monotone and non-Lipschitz operators

<p style='text-indent:20px;'>In this paper, we propose a new inertial Tseng's extragradient iterative algorithm for solving variational inequality problems of pseudo-monotone and non-Lipschitz operator in real Hilbert spaces. We prove that the sequence generated by proposed algorithm converges strongly to an element of solutions of variational inequality problem under some suitable assumptions imposed on the parameters. Finally, we give some numerical experiments for supporting our main results. The main results obtained in this paper extend and improve some related works in the literature.</p>

Nonlinear Viscosity Algorithm with Perturbation for Nonexpansive Multi-Valued Mappings

The viscosity iterative algorithms for finding a common element of the set of fixed points for nonlinear operators and the set of solutions of variational inequality problems have been investigated by many authors. The viscosity technique allow us to apply this method to convex optimization, linear programming and monoton inclusions. In this paper, based on viscosity technique with perturbation, we introduce a new nonlinear viscosity algorithm for finding an element of~the set of~fixed points of nonexpansive multi-valued mappings in a Hilbert spaces. Furthermore, strong convergence theorems of~this algorithm were established under suitable assumptions imposed on~parameters. Our results can be viewed as a generalization and improvement of various existing results in the current literature. Moreover, some numerical examples that show the efficiency and implementation of our algorithm are presented.

New Tseng\u2019s extragradient methods for pseudomonotone variational inequality problems in Hadamard manifolds

We propose Tseng’s extragradient methods for finding a solution of variational inequality problems associated with pseudomonotone vector fields in Hadamard manifolds. Under standard assumptions such as pseudomonotone and Lipschitz continuous vector fields, we prove that any sequence generated by the proposed methods converges to a solution of variational inequality problem, whenever it exits. Moreover, we give some numerical experiments to illustrate our main results.

A new inertial-projection algorithm for approximating common solution of variational inequality and fixed point problems of multivalued mappings

<p style='text-indent:20px;'>In this paper, we present a new modified self-adaptive inertial subgradient extragradient algorithm in which the two projections are made onto some half spaces. Moreover, under mild conditions, we obtain a strong convergence of the sequence generated by our proposed algorithm for approximating a common solution of variational inequality problem and common fixed point of a finite family of demicontractive mappings in a real Hilbert space. The main advantages of our algorithm are: strong convergence result obtained without prior knowledge of the Lipschitz constant of the related monotone operator, the two projections made onto some half-spaces and the inertial technique which speeds up rate of convergence. Finally, we present an application and a numerical example to illustrate the usefulness and applicability of our algorithm.</p>

Advanced Algorithms and Common Solutions to Variational Inequalities

The paper aims to present advanced algorithms arising out of adding the inertial technical and shrinking projection terms to ordinary parallel and cyclic hybrid inertial sub-gradient extra-gradient algorithms (for short, PCHISE). Via these algorithms, common solutions of variational inequality problems (CSVIP) and strong convergence results are obtained in Hilbert spaces. The structure of this problem is to find a solution to a system of unrelated VI fronting for set-valued mappings. To clarify the acceleration, effectiveness, and performance of our parallel and cyclic algorithms, numerical contributions have been incorporated. In this direction, our results unify and generalize some related papers in the literature.

A Parallel-Viscosity-Type Subgradient Extragradient-Line Method for Finding the Common Solution of Variational Inequality Problems Applied to Image Restoration Problems

In this paper, we study a modified viscosity type subgradient extragradient-line method with a parallel monotone hybrid algorithm for approximating a common solution of variational inequality problems. Under suitable conditions in Hilbert spaces, the strong convergence theorem of the proposed algorithm to such a common solution is proved. We then give numerical examples in both finite and infinite dimensional spaces to justify our main theorem. Finally, we can show that our proposed algorithm is flexible and has good quality for use with common types of blur effects in image recovery.

Linear conditioning, weak sharpness and finite convergence for equilibrium problems

The present paper first provides sufficient conditions and characterizations for linearly conditioned bifunction associated with an equilibrium problem. It then introduces the notion of weak sharp solution for equilibrium problems which is analogous to the linear conditioning notion. This new notion generalizes and unifies the notion of weak sharp minima for optimization problems as well as the notion of weak sharp solutions for variational inequality problems. Some sufficient conditions and characterizations of weak sharpness are also presented. Finally, we study the finite convergence property of sequences generated by some algorithms for solving equilibrium problems under linear conditioning and weak shapness assumptions. An upper bound of the number of iterations by which the sequence generated by proximal point algorithm converges to a solution of equilibrium problems is also given.