Method For Variational Inequality Problems Research Articles

A dynamic distributed conjugate gradient method for variational inequality problem over the common fixed-point constraints

A dynamic distributed conjugate gradient method for variational inequality problem over the common fixed-point constraints

A general inertial projected gradient method for variational inequality problems

A general inertial projected gradient method for variational inequality problems

Strong convergence of double-projection method for variational inequality problems

The paper proposes a numerical method for solving a variational inequality problem (VIP) involving a monotone and Lipschitz continuous operator in a Hilbert space. The method can be considered as a combination between an extragradient method and a regularization method. We prove that the iterative sequences generated by our method converge strongly to the smallest norm solution of problem VIP. Obtaining the strong convergence of the iterative sequences is based on regularization solutions when the regularization parameters are chosen suitably. Our numerical experiments are implemented to illustrate the behavior of the new method. The numerical results have shown the effectiveness and fast convergence of the method over existing methods.

Iterative method of extragradient type in Hilbert spaces

This article is concerned with hybrid implicit extragradient methods for variational inequality problems with constraints of a family of nonexpansive mappings, and a system of variational inequalities. One analyzes the convergence of the hybrid methods and obtains convergence theorems of solutions without the aid of compactness in Hilbert spaces.

A class of Benders decomposition methods for variational inequalities

We develop new variants of Benders decomposition methods for variational inequality problems. The construction is done by applying the general class of Dantzig–Wolfe decomposition of Luna et al. (Math Program 143(1–2):177–209, 2014) to an appropriately defined dual of the given variational inequality, and then passing back to the primal space. As compared to previous decomposition techniques of the Benders kind for variational inequalities, the following improvements are obtained. Instead of rather specific single-valued monotone mappings, the framework includes a rather broad class of multi-valued maximally monotone ones, and single-valued nonmonotone. Subproblems’ solvability is guaranteed instead of assumed, and approximations of the subproblems’ mapping are allowed (which may lead, in particular, to further decomposition of subproblems, which may otherwise be not possible). In addition, with a certain suitably chosen approximation, variational inequality subproblems become simple bound-constrained optimization problems, thus easier to solve.

On the convergence of inertial two-subgradient extragradient method for variational inequality problems

ABSTRACTIn 2012, Censor et al. proposed the so-called two-subgradient extragradient method (TSEGM) in which both projections have an explicit-formula because the projections are just executed onto half-spaces. However, the convergence of the TSEGM was posted as an open question. In this paper, we answer this open question affirmatively for the case where the closed convex set can be represented as the lower level set of a continuously differentiable convex function. Censor's open question is thus partially answered. More precisely, we combine the inertial technique with the TSEGM and establish the weak convergence of the proposed method under mild conditions.

New strong convergence theorem of the inertial projection and contraction method for variational inequality problems

In this paper, we introduce a new algorithm which combines the inertial projection and contraction method and the viscosity method for solving monotone variational inequality problems in real Hilbert spaces and prove a strong convergence theorem of our proposed algorithm under the standard assumptions imposed on cost operators. Finally, we give some numerical experiments to illustrate the proposed algorithm.

Extending the applicability of Newton’s method for variational inequality problems under Smale-Wang- γ criteria

Abstract The γ − theory for solving variational inequality problems using Newton’s method is extended using restricted convergence domains and the center γ − criterion.

Accelerated hybrid and shrinking projection methods for variational inequality problems

ABSTRACTIn this paper, we introduce several new extragradient-like approximation methods for solving variational inequalities in Hilbert spaces. Our algorithms are based on Tseng's extragradient method, subgradient extragradient method, inertial method, hybrid projection method and shrinking projection method. Strong convergence theorems are established under appropriate conditions. Our results extend and improve some related results in the literature. In addition, the efficiency of our algorithms is shown through numerical examples which are defined by the hybrid projection methods.

The new extragradient method extended to equilibrium problems

Malitsky and Semenov in (Cybern Syst Anal 50(2):271–277, 2014) introduced a new extragradient method for variational inequality problem. This paper extends the new extragradient method to equilibrium problems for pseudomonotone and Lipschitz-type continuous bifunctions. We have replaced naturally two projections in the new extragradient method by two optimization programs and proved the weak convergence of the proposed algorithm. The advantage of the algorithm is the computation of only one value of the bifunction in each iteration. In the numerical experiment, we see that the proposed algorithm has a competitive advantage over the time of execution and the iterative number.

On a modified extragradient method for variational inequality problem with application to industrial electricity production

<p style='text-indent:20px;'>In this paper, we present a modified extragradient-type method for solving the variational inequality problem involving uniformly continuous pseudomonotone operator. It is shown that under certain mild assumptions, this method is strongly convergent in infinite dimensional real Hilbert spaces. We give some numerical computational experiments which involve a comparison of our proposed method with other existing method in a model on industrial electricity production.

Modified subgradient extragradient method for variational inequality problems

In this paper, we introduce an algorithm as combination between the subgradient extragradient method and inertial method for solving variational inequality problems in Hilbert spaces. The weak convergence of the algorithm is established under standard assumptions imposed on cost operators. The proposed algorithm can be considered as an improvement of the previously known inertial extragradient method over each computational step. The performance of the proposed algorithm is also illustrated by several preliminary numerical experiments.

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.

Halpern subgradient extragradient method extended to equilibrium 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 operators. This paper extends the Halpern subgradient extragradient method (HSEM) to equilibrium problems for pseudomonotone and Lipschitz-type continuous bifunctions. We have replaced naturally two projections in HSEM by two optimization programs and proved the strong convergence of the obtained algorithm. We also present a modification of the proposed algorithm for finding a common solution of an equilibrium problem and a fixed point problem. In several of numerical experiments, we see that the proposed algorithm seems to have a competitive advantage over the extragradient method for equilibrium problems.

On the $O(1/k)$ Convergence Rate of He's Alternating Directions Method for a Kind of Structured Variational Inequality Problem

The alternating directions method for a kind of structured variational inequality problem (He, 2001) is an attractive method for structured monotone variational inequality problems. In each iteration, the subproblemsare convex quadratic minimization problem with simple constraintsand a well-conditioned system of nonlinear equations that can be efficiently solvedusing classical methods. Researchers have recently described the convergence rateof projection and contraction methods for variational inequality problems andthe original ADM and its linearized variant. Motivated and inspired by researchinto the convergence rate of these methods, we provide a simple proof to show the $O(1/k)$ convergencerate of alternating directions methods for structured monotone variational inequality problems (He, 2001).

Applying Projection-Based Methods to the Asymmetric Traffic Assignment Problem

This article examines the application of a path-based algorithm to the static and fixed demand asymmetric traffic assignment problem. The algorithm is of the simplicial decomposition type and it solves the equilibration or master problem step by means of five existing projection methods for variational inequality problems to evaluate their performance on real traffic networks. The projection methods evaluated are: (1) a cost approximation-based method for minimizing the Fukushima's gap function, (2) the modified descent method of Zhu and Marcotte (1988), (3) the double projection method of Khobotov (1987) and three of its recently developed variants (Nadezhkina and Takahashi, 2006; Wang et al., 2010; and He et al., 2012); (4) the method of Solodov and Svaiter (1999); and (5) the method of Solodov and Tseng (1996). These projection methods do not require evaluation of the Jacobians of the path cost functions. The source for asymmetries are link costs with interactions, as in the case of priority ruled junctions. The path-based algorithm has been computationally tested using the previous projection methods on three medium to large networks under different levels of congestion and the computational results are presented and discussed. Comparisons are also made with the basic projection algorithm for the fixed demand asymmetric traffic assignment problem. Despite the lack of monotonicity properties of the test problems, the only method that failed to converge under heavy congestion levels was the basic projection algorithm. The fastest convergence was obtained in all cases solving the master problem step using the method of He et al. (2012), which is a variant of Khobotov's method.

An iterative method for variational inequality problems

In this paper, we present some properties of generalized proximity operators andpropose an iterative method of approximating solutions for a class ofgeneralized variational inequalities and show its convergence in uniformlyconvex and smooth Banach spaces. MSC: 47J20, 46B20, 46N10, 47N10, 49J40.

Newton’s method for variational inequality problems: Smale’s point estimate theory under the γ-condition

In the present paper, we study Newton’s method for variational inequality problems. Under the -condition, the generalized -theory and -theory for Newton’s method are established. As an application, Smale’s point estimate theory for analytic mappings are given.

A hybrid splitting method for variational inequality problems with separable structure

Alternating direction method (ADM), which decomposes a large-scale original variational inequality (VI) problem into a series of smaller scale subproblems, is very attractive for solving a class of VI problems with a separable structure. This type of method can greatly improve the efficiency, but cannot avoid solving VI subproblems. In this paper, we propose a hybrid splitting method with variable parameters for separable VI problems. Specifically, the proposed method solves only one strongly monotone VI subproblem and a well-posed system of nonlinear equations in each iteration. The global convergence of the new method is established under some standard assumptions as those in classical ADMs. Finally, some preliminary numerical results show that the proposed method performs favourably in practice.

New step lengths in projection method for variational inequality problems

In this paper, we consider projection method for variational inequality problems. First, we give a new modification of recently proposed self-adaptive step length rules, with possibly minimal parameters. Then, the resulting self-adaptive projection method is proven to converge globally at a R-linear rate provided that the underlying mapping is strongly monotone and Lipschitz continuous. Preliminary numerical results confirm its flexibility and effectiveness.