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.     

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.     

On General Mixed Quasivariational Inequalities

In this paper, we suggest and analyze several iterative methods for solving general mixed quasivariational inequalities by using the technique of updating the solution and the auxiliary principle. It is shown that the convergence of these methods requires either the pseudomonotonicity or the partially relaxed strong monotonicity of the operator. Proofs of convergence is very simple. Our new methods differ from the existing methods for solving various classes of variational inequalities and related optimization problems. Various special cases are also discussed.     

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.     

伪单调广义混合变分不等式的改进k步预解算法

在算子的分裂技巧基础上介绍了求解伪单调广义混合变分不等式的改进五步预解算法，算法的收敛性只要求算子的g-伪单调和g—Lipschitz连续性，算子的伪单调比单调更弱．本文的新算法推广了文献中某些已有的结果．     

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.     

Solving Mixed Variational Inequalities Beyond Convexity

Journal of Optimization Theory and Applications - We show that Malitsky’s recent Golden Ratio Algorithm for solving convex mixed variational inequalities can be employed in a certain...     

On the Restricted Parameter Identification Problems for General Variational Inequalities

A restricted parnmeter identification problem for general variational inequalities is studied using the so-called asymptolic regularization method. Some applications, including the evolution dam problem, are also briefly discussed.     

广义混合变分不等式的近似点-投影算法

引入了求解广义混合变分不等式的近似点-投影算法,证明了由算法所生成迭代序列强收敛于非扩张映射不动点集合与广义混合变分不等式解集合的公共元素.方法和结果是新的,且推广了这一领域内许多已知结果.     

Set-Valued Resolvent Equations and Mixed Variational Inequalities

In this paper, we introduce and study a new class of variational inequalities, which is called the generalized set-valued mixed variational inequality. The resolvent operator technique is used to establish the equivalence among generalized set-valued variational inequalities, fixed point problems, and the generalized set-valued resolvent equations. This equivalence is used to study the existence of a solution of set-valued variational inequalities and to suggest a number of iterative algorithms for solving variational inequalities and related optimization problems. The results proved in this paper represent a significant refinement and improvement of the previously known results in this area.     

Generalized Mixed Variational Inequalities and Resolvent Equations

In this paper, we introduce and study a new class of variational inequalities, which is called the generalized mixed variational inequality. Using essentially the resolvent operator concept, we establish the equivalence between the generalized mixed variational inequalities and the system of resolvent equations. This equivalence is used to suggest a number of new iterative algorithms for solving the variational inequalities. Several special cases are discussed which can be obtained from the main results of this paper.     

On Quasimonotone Variational Inequalities

In this paper, we study variational inequalities with multivalued mappings. By employing Fan's lemma, we establish the existence result for the dual formulation of variational inequalities with semistrictly quasimonotone mappings. We also show that similar results for quasimonotone variational inequalities do not hold.     

On Quasimonotone Variational Inequalities

The purpose of this paper is to prove the existence of solutions of the Stampacchia variational inequality for a quasimonotone multivalued operator without any assumption on the existence of inner points. Moreover, the operator is not supposed to be bounded valued. The result strengthens a variety of other results in the literature.     

Proximal Point Algorithms for General Variational Inequalities

This paper presents a unified framework of proximal point algorithms (PPAs) for solving general variational inequalities (GVIs). Some existing PPAs for classical variational inequalities, including both the exact and inexact versions, are extended to solving GVIs. Consequently, several new PPA-based algorithms are proposed. M. Li was supported by NSFC Grant 10571083 and SRFDP Grant 200802861031. L.Z. Liao was supported in part by grants from Hong Kong Baptist University and the Research Grant Council of Hong Kong. X.M. Yuan was supported in part by FRG/08-09/II-40 from Hong Kong Baptist University and NSFC Grant 10701055.     

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.     

模糊映射的广义混合型强变分不等式

研究了新的一类模糊映射的广义混合型强变分不等式问题。证明了这类问题解的存在定理和收敛定理,给出解的带误差的Ishikawa型迭代算法。     

一个关于混合变分不等式问题的投影算法

作者提出了混合变分不等式的一个新的投影算法. 混合变分不等式在弹性塑料学领域有实际应用, 而且形式上比经典的变分不等式更一般. 假设映射具有某种伪单调性, 作者证明了所提出的新算法是全局收敛的. 如果某种误差届成立, 算法的收敛率也被分析.     

On Some Noncoercive Variational Inequalities

We study existence and regularity of solutions of noncoercive variational inequalities.     

On Implicit Vector Variational Inequalities

In this paper, we study the existence of solutions of implicit vector variational inequalities for multifunctions. Generalized pseudomonotonicity concepts are introduced. Our results extend and unify corresponding earlier existence results of many authors for vector variational inequalities under the Hausdorff topological vector space setting.     

Local Uniqueness of Solutions of General Variational Inequalities

We use the concept of Fréchet approximate Jacobian matrices to establish the local uniqueness of solutions to general variational inequalities which involve continuous, not necessarily locally Lipschitz continuous data.