Some New Iterative Methods for General Mixed Variational Inequalities

In this paper, we use the auxiliary principle technique to suggest a class of predictorcorrector methods for solving general mixed variational inequalities. The convergence of the proposed methods only requires the partially relaxed strongly monotonicity of the operator, which is weaker than co-coercivity. From special cases, we obtain various known and new results for solving various classes of variational inequalities and related problems.AMS Subject Classification (1991): 49J40, 90C33.     

Projection-Splitting Algorithms for General Monotone Variational Inequalities

We suggest and analyze some new splitting type projection methods for solving general variational inequalities by using the updating technique of the solution. The convergence analysis of these new methods is considered and the proof of convergence is very simple. These new methods are versatile.     

Extragradient Methods for Pseudomonotone Variational Inequalities

We consider and analyze some new extragradient-type methods for solving variational inequalities. The modified methods converge for a pseudomonotone operator, which is a much weaker condition than monotonicity. These new iterative methods include the projection, extragradient, and proximal methods as special cases. Our proof of convergence is very simple as compared with other methods.     

Iterative Methods for General Mixed Quasivariational Inequalities

It is well known that mixed quasivariational inequalities are equivalent to implicit fixed-point problems. We use this alternative equivalent formulation to suggest and analyze a new self-adaptive resolvent method for solving mixed quasivariational inequalities in conjunction with a technique updating the solution. We show that the convergence of this method requires pseudomonotonicity, which is a weaker condition than monotonicity. Since mixed quasivariational inequalities include various classes of variational inequalities as special cases, our results continue to hold for these problems.     

Some Predictor-Corrector Algorithms for Multivalued Variational Inequalities

In this paper, we use the auxiliary principle technique to suggest a new class of predictor-corrector algorithms for solving multivalued variational inequalities. The convergence of the proposed methods requires only the partially-relaxed strong monotonicity of the operator, which is weaker than cocoercivity. As special cases, we obtain a number of known and new results for solving various classes of variational inequalities.     

A Class of Combined Iterative Methods for Solving Variational Inequalities

A general approach to constructing iterative methods that solve variational inequalities is proposed. It is based on combining, modifying, and extending ideas contained in various Newton-like methods. Various algorithms can be obtained with this approach. Their convergence is proved under weak assumptions. In particular, the main mapping need not be monotone. Some rates of convergence are also given.     

On a General Projection Algorithm for Variational Inequalities

Let H be a real Hilbert space with norm and inner product denoted by and . Let K be a nonempty closed convex set of H, and let f be a linear continuous functional on H. Let A, T, g be nonlinear operators from H into itself, and let be a point-to-set mapping. We deal with the problem of finding uK such that g(u)K(u) and the following relation is satisfied: , where >0 is a constant, which is called a general strong quasi-variational inequality. We give a general and unified iterative algorithm for finding the approximate solution to this problem by exploiting the projection method, and prove the existence of the solution to this problem and the convergence of the iterative sequence generated by this algorithm.     

Proximal Methods for Mixed Variational Inequalities

A proximal point method for solving mixed variational inequalities is suggested and analyzed by using the auxiliary principle technique. It is shown that the convergence of the proposed method requires only the pseudomonotonicity of the operator, which is a weaker condition than monotonicity. As special cases, we obtain various known and new results for solving variational inequalities and related problems. Our proof of convergence is very simple as compared with other methods.     

Generalized Variational Inequalities with Pseudomonotone Operators Under Perturbations

Variational inequalities and generalized variational inequalities with perturbed operators and constraints are considered and convergence of solutions to such problems is proved under an assumption of pseudomonotonicity. The paper extends previous results given by the authors proved in the setting of monotone operators.     

Existence Theorems for Variational Inequalities in Banach Spaces

In this paper, by employing the notion of generalized projection operators and the well-known Fan’s lemma, we establish some existence results for the variational inequality problem and the quasivariational inequality problem in reflexive, strictly convex, and smooth Banach spaces. We propose also an iterative method for approximate solutions of the variational inequality problem and we establish some convergence results for this iterative method. L. C. Zeng, His research was partially supported by the Teaching and Research Award Fund for Outstanding Young Teachers in Higher Education Institutions of MOE, China and by the Dawn Program Foundation, Shanghai, China. J. C. Yao, His research was partially supported by the National Science Council of the Republic of China     

Splitting Algorithms for General Pseudomonotone Mixed Variational Inequalities

In this paper, we suggest and analyze a number of resolvent-splitting algorithms for solving general mixed variational inequalities by using the updating technique of the solution. The convergence of these new methods requires either monotonicity or pseudomonotonicity of the operator. Proof of convergence is very simple. Our new methods differ from the existing splitting methods for solving variational inequalities and complementarity problems. The new results are versatile and are easy to implement.     

Modified Extragradient Method for Variational Inequalities and Verification of Solution Existence

In this paper, we propose a modified extragradient method for solving variational inequalities (VI) which has the following nice features: (i) The generated sequence possesses an expansion property with respect to the starting point; (ii) the existence of the solution to a VI problem can be verified through the behavior of the generated sequence from the fact that the iterative sequence diverges to infinity if and only if the solution set is empty. Global convergence of the method is guaranteed under mild conditions. Our preliminary computational experience is also reported.     

Variational inequalities with cone constraints

The present paper considers variational inequalities with cone and operator constraints. The characterization of solutions is given. An algorithm for numerical solution of the problem is proposed, and the convergence of the generated sequence of points is proved.This research was supported in part by the Science Foundation of the Republic of Serbia.The author thanks two anonymous referees for their helpful remarks.     

Unified Framework of Extragradient-Type Methods for Pseudomonotone Variational Inequalities

In this paper, we propose a unified framework of extragradient-type methods for solving pseudomonotone variational inequalities, which allows one to take different stepsize rules and requires the computation of only two projections at each iteration. It is shown that the modified extragradient method of Ref. 1 falls within this framework with a short stepsize and so does the method of Ref. 2 with a long stepsize. It is further demonstrated that the algorithmic framework is globally convergent under mild assumptions and is sublinearly convergent if in addition a projection-type error bound holds locally. Preliminary numerical experiments are reported.     

Iterative algorithms for systems of extended regularized nonconvex variational inequalities and fixed point problems

This paper points out some fatal errors in the equivalent formulations used in Noor 2011 [Noor MA. Projection iterative methods for solving some systems of general nonconvex variational inequalities. Applied Analysis. 2011;90:777–786] and consequently in Noor 2009 [Noor MA. System of nonconvex variational inequalities. Journal of Advanced Research Optimization. 2009;1:1–10], Noor 2010 [Noor MA, Noor KI. New system of general nonconvex variational inequalities. Applied Mathematics E-Notes. 2010;10:76–85] and Wen 2010 [Wen DJ. Projection methods for a generalized system of nonconvex variational inequalities with different nonlinear operators. Nonlinear Analysis. 2010;73:2292–2297]. Since these equivalent formulations are the main tools to suggest iterative algorithms and to establish the convergence results, the algorithms and results in the aforementioned articles are not valid. It is shown by given some examples. To overcome with the problems in these papers, we consider a new system of extended regularized nonconvex variational inequalities, and establish the existence and uniqueness result for a solution of the aforesaid system. We suggest and analyse a new projection iterative algorithm to compute the unique solution of the system of extended regularized nonconvex variational inequalities which is also a fixed point of a nearly uniformly Lipschitzian mapping. Furthermore, the convergence analysis of the proposed iterative algorithm under some suitable conditions is studied. As a consequence, we point out that one can derive the correct version of the algorithms and results presented in the above mentioned papers.     

Vector Variational Inequalities Involving Vector Maximal Points

This paper considers the existence of solutions and the equivalence of four kinds of vector variational inequalities (VVI). More precisely, a sufficient condition is provided under which the solution sets of these VVIs are nonempty and equal. An example is given, showing that such a sufficient condition is essential to ensure the results. Actually, the main theorems in this paper can be regarded as a suitable correction and a refinement of recent results due to Chang et al. (Ref. 1).     

Variational Inequalities and the Elastic-Plastic Torsion Problem

We show that, under suitable conditions, the variational inequality that expresses the elastic-plastic torsion problem is equivalent to a variational inequality on a convex set which depends on (x)=d(x, ). Such an equivalence allows us to find the related Lagrange multipliers and to exhibit a computational procedure based on the subgradient method.     

Modified Goldstein–Levitin–Polyak Projection Method for Asymmetric Strongly Monotone Variational Inequalities

In this paper, we present a modified Goldstein–Levitin–Polyak projection method for asymmetric strongly monotone variational inequality problems. A practical and robust stepsize choice strategy, termed self-adaptive procedure, is developed. The global convergence of the resulting algorithm is established under the same conditions used in the original projection method. Numerical results and comparison with some existing projection-type methods are given to illustrate the efficiency of the proposed method.     

一类椭圆型变分不等式离散问题的迭代算法

根据一类椭圆型变分不等式离散问题所具有的非线性特征，提出了一种简明快速的迭代算法，该方法在解决障碍问题及流体润滑油膜破裂自然边值问题等工程应用问题时具有较高的效率。     

Banach空间中集值混和变分不等式问题解的迭代算法

介绍了集值映象的伪单调定义，并在Banach空间中构造了集值混和变分不等式问题近似解的迭代算法．应用伪单调映象定义，证明了该迭代算法收敛于集值混和变分不等式问题的近似解．特别值得注意的是：在文章中对集值映象没有Lipschitz连续性假设．