Algorithm For Variational Inequality Problems Research Articles

An improved projection algorithm for variational inequality problem with multivalued mapping

<p style='text-indent:20px;'>In this paper, we propose a new projection method (extragradient method) for solving the generalized variational inequality problem. Our method is well-defined and is proven to be globally convergent to a solution of the problem under the assumptions that the multivalued mapping is continuous and satisfies a condition that is strictly weaker than the pseudomonotonicity. Finally, to support our results and to illustrate the numerical behavior of the proposed algorithm, some fundamental experiments are provided and also compared with recent works.</p>

Single-step algorithm for variational inequality problems in 2-uniformly convex banach spaces

Single-step algorithm for variational inequality problems in 2-uniformly convex banach spaces

A shrinking projection algorithm for variational inequality problems with a Lipschitz monotone mapping and fixed point problems of relatively weak nonexpansive mappings

A shrinking projection algorithm for variational inequality problems with a Lipschitz monotone mapping and fixed point problems of relatively weak nonexpansive mappings

Strong convergence of an inertial iterative algorithm for variational inequality problem, generalized equilibrium problem, and fixed point problem in a Banach space

We propose and analyze an inertial iterative algorithm to approximate a common solution of generalized equilibrium problem, variational inequality problem, and fixed point problem in the framework of a 2-uniformly convex and uniformly smooth real Banach space. Further, we study the convergence analysis of our proposed iterative method. Finally, we give application and a numerical example to illustrate the applicability of the main algorithm.

Inertial Haugazeau's hybrid subgradient extragradient algorithm for variational inequality problems in Banach spaces

The variational inequality problem plays an important role in nonlinear analysis and optimization. It is a generalization of the nonlinear complementarity problem. For a variational inequality problem in a Hilbert space, the extragradient algorithm with inertial effects has been studied. For a variational inequality problem in a Banach space, Nakajo introduced Haugazeau's hybrid method and Liu introduced the Halpern subgradient extragradient method. In this paper, we construct a new inertial iterative method for solving variational inequality problems in Banach spaces based on the work we mentioned above. We propose a strong convergence theorem. As applications, our result can be used to solve constrained convex minimization problems.

Inertial hybrid algorithm for variational inequality problems in Hilbert spaces

For a variational inequality problem, the inertial projection and contraction method have been studied. It has a weak convergence result. In this paper, we propose a strong convergence iterative method for finding a solution of a variational inequality problem with a monotone mapping by projection and contraction method and inertial hybrid algorithm. Our result can be used to solve other related problems in Hilbert spaces.

Mann-type algorithms for variational inequality problems and fixed point problems

The purpose of this paper is to investigate the problem of finding a common element of the solution set of a variational inequality problem for a monotone, Lipschitz-continuous mapping and set of fixed point of a quasi-nonexpansive mapping. We introduce Mann-type Tseng's extragradient-like approximation method which is based on so-called inertial Tseng's extragradient method and Mann method. We establish two weak convergence theorems for two iterative sequences generated by these methods. Our results extend and improve some related results in the literatures.

Convergence analysis of a projection algorithm for variational inequality problems

In this paper, we propose a projection based Newton-type algorithm for solving the variational inequality problems. A comprehensive study is conducted to analyze both global and local convergence properties of the algorithm. In particular, the algorithm is shown to be of superlinear convergence when the solution is a regular point. In addition, when the Jacobian matrix of the underlying function is positive definite at the solution or the solution is a non-degenerate point, the algorithm still possesses its superlinear convergence. Compared to the relevant projection algorithms in literature, the proposed algorithm is of remarkable advantages in terms of its generalization and favorable convergence properties under relaxed assumptions.

A strong convergence theorem for Tseng’s extragradient method for solving variational inequality problems

In this paper, we introduce a new algorithm for solving variational inequality problems with monotone and Lipschitz-continuous mappings in real Hilbert spaces. Our algorithm requires only to compute one projection onto the feasible set per iteration. We prove under certain mild assumptions, a strong convergence theorem for the proposed algorithm to a solution of a variational inequality problem. Finally, we give some numerical experiments illustrating the performance of the proposed algorithm for variational inequality problems.

A new algorithm for variational inequality problems with alpha-inverse strongly monotone maps and common fixed points for a countable family of relatively weak nonexpansive maps, with applications

Let $E$ be a $2$-uniformly convex and uniformly smooth real Banach space with dual space $E^*$. Let $C$ be a nonempty closed and convex subset of $E.$ Let $A:C\to E^*$ and $T_i:C\rightarrow E$, $i=1,2,\cdots,$ be an $\alpha$-inverse strongly monotone map and a {\it countable family} of relatively weak nonexpansive maps, respectively. Assume that the intersection of the set of solutions of the variational inequality problem, $VI(C,A)$, and the set of common fixed points of $\{T_i\}_{i=1}^{\infty}$, $\cap_{i=1}^{\infty}F(T_i)$, is nonempty. A generalized projection algorithm is constructed and proved to converge {\it strongly} to some $x^*\in VI(C,A)\cap \Big(\cap_{i=1}^{\infty}F(T_i)\Big)$. Our theorem is a significant improvement of recent important results, in particular, the results of Zegeye and Shahzad (Nonlinear Anal. 70 (7) (2009), 2707-2716), Liu (Appl. Math. Mech. -Engl. Ed. 30 (7) (2009), 925-932), and Zhang {\it et al.} (Appl. Math. and Informatics 29 (1-2) (2011), 87-102) and a host of other results. Finally, applications of our theorem to convex optimization problems, zeros of $\alpha$-inverse strongly monotone maps and complementarity problems are presented.

Modified Tseng’s extragradient algorithms for variational inequality problems

In this work, our interest is in investigating the monotone variational inequality problems in the framework of real Hilbert spaces. For solving this problem, we introduce two modified Tseng’s extragradient methods using the inertial technique. The weak convergence theorems are established under the standard assumptions imposed on cost operators. Finally, numerical results are reported to illustrate the behavior of the new algorithms and also to compare with others.

A new algorithm for variational inequality problems with a generalized phi-strongly monotone map over the set of common fixed points of a finite family of quasi-phi-nonexpansive maps, with applications

Let E be a uniformly convex and uniformly smooth real Banach space with dual space $$E^*$$ and C be a nonempty, closed and convex subset of E. Let $$A:E\rightarrow E^*$$ be a generalized $$\Phi $$ -strongly monotone and bounded map and let $$T_i:C\rightarrow E, i=1,2,3,\ldots , N$$ be a finite family of quasi- $$\phi $$ -nonexpansive maps such that $$\cap _{i=1}^{N} F(T_{i})\ne \emptyset $$ . Suppose $$VI(A,\cap _{i=1}^{N} F(T_{i}))\ne \emptyset $$ . A new iterative algorithm that converges strongly to a point in $$VI(A,\cap _{i=1}^{N} F(T_{i}))$$ is constructed. Results obtained are applied to a convex optimization problem. Furthermore, the theorems proved complement, improve and unify several recent important results. Finally, we consider a family $$\{T_i\}_{i=1}^N$$ of maps where for each $$i,\, T_i $$ maps E into its dual space $$E^*$$ and prove a strong convergence theorem for $$VI(A, \cap _{i=1}^{N} F(T_{i}))$$ , where $$F_J(T_i)$$ is the set of J-fixed points introduced by Chidume and Idu.

Modified subgradient extragradient algorithms for variational inequality problems and fixed point problems

The subgradient extragradient method can be considered as an improvement of the extragradient method for variational inequality problems for the class of monotone and Lipschitz continuous mappings. In this paper, we propose two new algorithms as combination between the subgradient extragradient method and Mann-like method for finding a common element of the solution set of a variational inequality and the fixed point set of a demicontractive mapping.

Viscosity iterative algorithm for variational inequality problems and fixed point problems of strict pseudo-contractions in uniformly smooth Banach spaces

The purpose of this paper is to study a new viscosity iterative algorithm based on a generalized contraction for finding a common element of the set of solutions of a general variational inequality problem for finite inversely strongly accretive mappings and the set of common fixed points for a countable family of strict pseudo-contractions in uniformly smooth Banach spaces. We prove some strong convergence theorems under some suitable conditions. The results obtained in this paper improve and extend the recent ones announced by many others in the literature.

Viscosity iterative algorithm for variational inequality problems and fixed point problems in a real q-uniformly smooth Banach space

The purpose of this paper is to study a viscosity iterative algorithm for finding a common element of the set of solutions of a general variational inequality problem for two inverse strongly accretive operators and the set of fixed points of a δ-strict pseudocontraction in a real q-uniformly smooth Banach space. Some strong convergence theorems are obtained under appropriate conditions. As an application, we prove some strong convergence theorems for fixed point problems and variational inequality problems or equilibrium problems in Hilbert spaces. These results improve and extend the corresponding results announced by many others.

A class of nonlinear proximal point algorithms for variational inequality problems

This work is motivated by a recent work on an extended linear proximal point algorithm (PPA) [B.S. He, X.L. Fu, and Z.K. Jiang, Proximal-point algorithm using a linear proximal term, J. Optim. Theory Appl. 141 (2009), pp. 299–319], which aims at relaxing the requirement of the linear proximal term of classical PPA. In this paper, we make further contributions along the line. First, we generalize the linear PPA-based contraction method by using a nonlinear proximal term instead of the linear one. A notable superiority over traditional PPA-like methods is that the nonlinear proximal term of the proposed method may not necessarily be a gradient of any functions. In addition, the nonlinearity of the proximal term makes the new method more flexible. To avoid solving a variational inequality subproblem exactly, we then propose an inexact version of the developed method, which may be more computationally attractive in terms of requiring lower computational cost. Finally, we gainfully employ our new methods to solve linearly constrained convex minimization and variational inequality problems.

New iterative algorithm for variational inequality problem and fixed point problem in Hilbert spaces

New iterative algorithm for variational inequality problem and fixed point problem in Hilbert spaces

A tri-dimensional filter SQP algorithm for variational inequality problems

In this paper, the filter technique proposed for solving optimization problem is applied to solve variational inequality problem. It avoids the use of a merit function which often arises in solving VI problems by an optimization method. The presented method is a tri-dimensional filter sequential quadratic programming method. Specially, a differentiable quadratic function is employed as the objective function in implementing the optimality measure in filter components. Under some mild conditions, global convergent is derived. Numerical results show efficiency of this method.

Gap Functions and Algorithms for Variational Inequality Problems

We solve several kinds of variational inequality problems through gap functions, give algorithms for the corresponding problems, obtain global error bounds, and make the convergence analysis. By generalized gap functions and generalized D-gap functions, we give global bounds for the set-valued mixed variational inequality problems. And through gap function, we equivalently transform the generalized variational inequality problem into a constraint optimization problem, give the steepest descent method, and show the convergence of the method.

Finite Convergence of the Proximal Point Algorithm for Variational Inequality Problems

In this paper, we establish sufficient conditions for guaranteeing finite termination of an arbitrary algorithm for solving a variational inequality problem in a Banach space. Applying these conditions, it shows that sequences generated by the proximal point algorithm terminate at solutions in a finite number of iterations.