Solving Monotone Variational Inequality Problems Research Articles

Golden ratio method for solving monotone variational inequality problems in Hadamard spaces

In this article, a generalized variational inequality problem in the setting of Hadamard spaces is introduced and analyzed. For approximating a solution of the problem when the underlined mapping is monotone, an adaptive algorithm that requires the computation of only one proximal operator at each iteration is proposed. The work gives a step towards investigating monotone variational inequality problems in geodesic settings.

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.

An inexact proximal decomposition method for variational inequalities with separable structure

This paper presents an inexact proximal method for solving monotone variational inequality problems with a given separable structure. The proposed algorithm is a natural extension of the Proximal Multiplier Algorithm with Proximal Distances (PMAPD) proposed by Sarmiento et al. [Optimization 65 (2016) 501–537], which unified the works of Chen and Teboulle (PCPM method), and Kyono and Fukushima (NPCPMM) developed for solving convex programs with a particular separable structure. The resulting method combines the recent proximal distances theory introduced by Auslender and Teboulle [SIAM J. Optim. 16 (2006) 697–725] with a decomposition method given by Chen and Teboulle for convex problems and extends the results of the Entropic Proximal Decomposition Method proposed by Auslender and Teboulle, which used to Logarithmic Quadratic proximal distances. Under some mild assumptions on the problem we prove a global convergence of the primal–dual sequences produced by the algorithm.

Two modifications of the inertial Tseng extragradient method with self-adaptive step size for solving monotone variational inequality problems

AbstractIn this work, we introduce two new inertial-type algorithms for solving variational inequality problems (VIPs) with monotone and Lipschitz continuous mappings in real Hilbert spaces. The first algorithm requires the computation of only one projection onto the feasible set per iteration while the second algorithm needs the computation of only one projection onto a half-space, and prior knowledge of the Lipschitz constant of the monotone mapping is not required in proving the strong convergence theorems for the two algorithms. Under some mild assumptions, we prove strong convergence results for the proposed algorithms to a solution of a VIP. Finally, we provide some numerical experiments to illustrate the efficiency and advantages of the proposed algorithms.

A Novel Inertial Projection and Contraction Method for Solving Pseudomonotone Variational Inequality Problems

In this paper, we introduce a new algorithm which combines the inertial contraction projection method and the Mann-type method (Mann in Proc. Am. Math. Soc. 4:506–510, 1953) for solving monotone variational inequality problems in real Hilbert spaces. The strong convergence of our proposed algorithm is proved under some standard assumptions imposed on cost operators. Finally, we give some numerical experiments to illustrate the proposed algorithm and the main result.

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.

Accelerated Subgradient Extragradient Methods for Variational Inequality Problems

In this paper, we introduce two new iterative algorithms for solving monotone variational inequality problems in real Hilbert spaces, which are based on the inertial subgradient extragradient algorithm, the viscosity approximation method and the Mann type method, and prove some strong convergence theorems for the proposed algorithms under suitable conditions. The main results in this paper improve and extend some recent works given by some authors. Finally, the performances and comparisons with some existing methods are presented through several preliminary numerical experiments.

Self-adaptive subgradient extragradient method with inertial modification for solving monotone variational inequality problems and quasi-nonexpansive fixed point problems

In this paper, we introduce a new algorithm with self-adaptive method for finding a solution of the variational inequality problem involving monotone operator and the fixed point problem of a quasi-nonexpansive mapping with a demiclosedness property in a real Hilbert space. The algorithm is based on the subgradient extragradient method and inertial method. At the same time, it can be considered as an improvement of the inertial extragradient method over each computational step which was previously known. The weak convergence of the algorithm is studied under standard assumptions. It is worth emphasizing that the algorithm that we propose does not require one to know the Lipschitz constant of the operator. Finally, we provide some numerical experiments to verify the effectiveness and advantage of the proposed algorithm.

A self-adaptive Armijo-like step size method for solving monotone variational inequality problems in Hilbert spaces

A self-adaptive Armijo-like step size method for solving monotone variational inequality problems in Hilbert spaces

Strong convergence of a double projection-type method for monotone variational inequalities in Hilbert spaces

We introduce a projection-type algorithm for solving monotone variational inequality problems in real Hilbert spaces without assuming Lipschitz continuity of the corresponding operator. We prove that the whole sequence of iterates converges strongly to a solution of the variational inequality. The method uses only two projections onto the feasible set in each iteration in contrast to other strongly convergent algorithms which either require plenty of projections within a step size rule or have to compute projections on possibly more complicated sets. Some numerical results illustrate the behavior of our method.

An inexact proximal algorithm for variational inequalities

An inexact proximal algorithm for variational inequalities O. Sarmiento, E. A. Papa Quiroz and P. R. Oliveira Programa de Ingeniería de Sistemas y Computación - COPPE, Universidad Federal de Rio de Janeiro, 68511 CEP: 21941-972, Rio de Janeiro, Brasil. DOI: https://doi.org/10.33017/RevECIPeru2015.0018/ Abstract This paper presents a new inexact proximal method for solving monotone variational inequality problems with a given separable structure. The resulting method combines the recent proximal distances theory introduced by Auslender and Teboulle (2006) with a decomposition method given by Chen and Teboulle that was proposed to solve convex optimization problems. This method extends and generalizes proximal methods using Bregman, Phi-divergences and Quadratic logarithmic distances. Taking mild assumptions we prove that the primal-dual sequences produced by algorithm is well-defined and converge to optimal solution of the variational inequality problem. Furthermore, we show some numerical experiments, for the particular case to solve convex optimization problem, showing that the algorithm is perfectly implementable. Keywords: Inexact proximal method, variational inequality, separable structure, proximal distances. Resumen En este artículo presentamos un nuevo método proximal inexacto para resolver problemas de desigualdad variacional monótono con una estructura separable. El método resultante combina la reciente teoria de distancias proximales introducidas por Auslender y Teboulle (2006) con un método de descomposición proximal dado por Chen y Teboulle que fue propuesto para resolver problemas de optimización convexa. Este método extiende y generaliza métodos proximales usando distancias de Bregman, Phi-divergencias y logaritmo cuadrático, Asumiendo hipotesis adecuadas probamos que la sucesión primal-dual generada por el algoritmo está bien definido y converge a la solución óptima de un problema de desigualdad variacional. Además presentamos algunos resultados computacionales para el caso particular de resolver problemas de optimización convexa, mostrando asi que el algoritmo es perfectamente implementable. Descriptores: Método proximal inexacto, desigualdad variacional, estructura separable, distancias proximales.

Two Classes of Iterative Algorithms for Solving Monotone Variational Inequality Problems

Two classes of iterative algorithms for solving monotone variational inequality problems were proposed.Via solving strongly monotone variational inequality problem,the algorithms generated two iterative sequences which weakly converged to a solution of the variational inequality problems.In the end,the convergence of these two new classes of iterative algorithms was analyzed.

New cooperative projection neural network for nonlinearly constrained variational inequality

This paper proposes a new cooperative projection neural network (CPNN), which combines automatically three individual neural network models with a common projection term. As a special case, the proposed CPNN can include three recent recurrent neural networks for solving monotone variational inequality problems with limit or linear constraints, respectively. Under the monotonicity condition of the corresponding Lagrangian mapping, the proposed CPNN is theoretically guaranteed to solve monotone variational inequality problems and a class of nonmonotone variational inequality problems with linear and nonlinear constraints. Unlike the extended projection neural network, the proposed CPNN has no limitation on the initial point for global convergence. Compared with other related cooperative neural networks and numerical optimization algorithms, the proposed CPNN has a low computational complexity and requires weak convergence conditions. An application in real-time grasping force optimization and examples demonstrate good performance of the proposed CPNN.

Comparison of Two Proximal Point Algorithms for Monotone Variational Inequalities

Since proximal point algorithms (abbreviated as PPA) are attractive for solving monotone variational inequality problems, various approximate versions of PPA (APPA) are developed for practical applications. In this paper, we make a comparison between two different versions of APPAs (APPA I and APPA II) in the literature which share some common properties. Both of the algorithms use the same inexactness restriction and the same step length. The only difference is that they use different search directions. Through theoretical analysis and numerical experiment, we can see that APPA II usually performs better than APPA I.

A new class of projection and contraction methods for solving variational inequality problems

This paper presents a new class of projection and contraction methods for solving monotone variational inequality problems. The methods can be viewed as combinations of some existing projection and contraction methods and the method of shortest residuals, a special case of conjugate gradient methods for solving unconstrained nonlinear programming problems. Under mild assumptions, we show the global convergence of the methods. Some preliminary computational results are reported to show the efficiency of the methods.

An Extended Projection Neural Network for Constrained Optimization

Recently, a projection neural network has been shown to be a promising computational model for solving variational inequality problems with box constraints. This letter presents an extended projection neural network for solving monotone variational inequality problems with linear and nonlinear constraints. In particular, the proposed neural network can include the projection neural network as a special case. Compared with the modified projection-type methods for solving constrained monotone variational inequality problems, the proposed neural network has a lower complexity and is suitable for parallel implementation. Furthermore, the proposed neural network is theoretically proven to be exponentially convergent to an exact solution without a Lipschitz condition. Illustrative examples show that the extended projection neural network can be used to solve constrained monotone variational inequality problems.

A proximal decomposition algorithm for variational inequality problems

In this paper, we propose a new decomposition algorithm for solving monotone variational inequality problems with linear constraints. The algorithm utilizes the problem's structure conductive to decomposition. At each iteration, the algorithm solves a system of nonlinear equations, which is structurally much easier to solve than variational inequality problems, the subproblems of classical decomposition methods, and then performs a projection step to update the multipliers. We allow to solve the subproblems approximately and we prove that under mild assumptions on the problem's data, the algorithm is globally convergent. We also report some preliminary computational results, which show that the algorithm is encouraging.

An Infeasible Interior-Point Method with Nonmonotonic Complementarity Gaps

This article describes an infeasible interior-point (IP) method for solving monotone variational inequality problems with polyhedral constraints and, as a particular case, monotone nonlinear complementarity problems. The method determines a search direction by solving, possibly in an inexact way, the Newton equation for the central path. Then, a curvilinear search is used to meet classical centering conditions and an Armijo rule. The novelty with respect to classical IP methods consists in relaxing the requirement of monotonic decrease in the complementarity gaps. Global convergence results are proved and numerical experiments are presented. The experiments confirm the effectiveness of nonmonotonicity in a number of test problems.