New inexact implicit method for general mixed quasi variational inequalities

In this paper, we suggest and analyze a new self-adaptive inexact implicit method with a variable parameter for general mixed quasi variational inequalities, where the skew-symmetry of the nonlinear bifunction plays a crucial part in the convergence analysis of this method. We use a self-adaptive technique to adjust parameter ρ at each iteration. The global convergence of the proposed method is proved under some mild conditions. Preliminary numerical results indicate that the self-adaptive adjustment rule is necessary in practice. Muhammad Aslam Noor is supported by the Higher Education Commission, Pakistan, through research grant No: 1-28/HEC/HRD/2005/90.     

Inexact proximal point method for general variational inequalities

In this paper, we suggest and analyze a new inexact proximal point method for solving general variational inequalities, which can be considered as an implicit predictor-corrector method. An easily measurable error term is proposed with further relaxed error bound and an optimal step length is obtained by maximizing the profit-function and is dependent on the previous points. Our results include several known and new techniques for solving variational inequalities and related optimization problems. Results obtained in this paper can be viewed as an important improvement and refinement of the previously known results. Preliminary numerical experiments are included to illustrate the advantage and efficiency of the proposed method.     

A self-adaptive method for solving general mixed variational inequalities

The general mixed variational inequality containing a nonlinear term φ is a useful and an important generalization of variational inequalities. The projection method cannot be applied to solve this problem due to the presence of the nonlinear term. To overcome this disadvantage, Noor [M.A. Noor, Pseudomonotone general mixed variational inequalities, Appl. Math. Comput. 141 (2003) 529-540] used the resolvent equations technique to suggest and analyze an iterative method for solving general mixed variational inequalities. In this paper, we present a new self-adaptive iterative method which can be viewed as a refinement and improvement of the method of Noor. Global convergence of the new method is proved under the same assumptions as Noor's method. Some preliminary computational results are given.     

A modified inexact operator splitting method for monotone variational inequalities

The Douglas–Peaceman–Rachford–Varga operator splitting methods (DPRV methods) are attractive methods for monotone variational inequalities. He et al. [Numer. Math. 94, 715–737 (2003)] proposed an inexact self-adaptive operator splitting method based on DPRV. This paper relaxes the inexactness restriction further. And numerical experiments indicate the improvement of this relaxation.      

A modified inexact implicit method for mixed variational inequalities

In this paper, we suggest and analyze an inexact implicit method with a variable parameter for mixed variational inequalities by using a new inexactness restriction. Under certain conditions, the global convergence of the proposed method is proved. Some preliminary computational results are given to illustrate the efficiency of the new inexactness restriction. The results proved in this paper may be viewed as improvement and refinement of the previously known results.     

An inexact implicit method for general mixed variational inequalities

In the present paper, we present an inexact implicit method with a variable parameter for general mixed variational inequalities. We use a self-adaptive technique to adjust parameter ρ$\rho$ at each iteration. The main advantage of this technique is that the method can adjust the parameter automatically and the numbers of iteration are not very sensitive to different initial parameter _ρ0.${\rho }_0.$     

Descent method with inexact linesearch for mixed variational inequalities

We propose a descent method with respect to a merit function for the mixed variational inequality involving a general nonlinear mapping and a convex, but not necessarily differentiable function. The method utilizes an inexact linesearch procedure. Its convergence is proved under the additional assumptions of continuity and strong monotonicity of the cost mapping.     

解一类结构变分不等式问题的非精确并行交替方向法

带线性约束的具有两分块结构的单调变分不等式问题, 出现在许多现代应用中, 如交通和经济问题等. 基于该问题良好的可分结构, 分裂型算法被广泛研究用于其求解. 提出新的带回代的非精确并行交替方向法解该类问题, 在每一步迭代中,首先以并行模式通过投影得到预测点, 然后对其校正得到下一步的迭代点. 在压缩型算法的理论框架下, 在适当条件下证明了所提算法的全局收敛性. 数值结果表明了算法的有效性. 此外, 该算法可推广到求解具有多分块结构的问题.     

An approximate proximal-extragradient type method for monotone variational inequalities

Proximal point algorithms (PPA) are attractive methods for monotone variational inequalities. The approximate versions of PPA are more applicable in practice. A modified approximate proximal point algorithm (APPA) presented by Solodov and Svaiter [Math. Programming, Ser. B 88 (2000) 371–389] relaxes the inexactness criterion significantly. This paper presents an extended version of Solodov–Svaiter's APPA. Building the direction from current iterate to the new iterate obtained by Solodov–Svaiter's APPA, the proposed method improves the profit at each iteration by choosing the optimal step length along this direction. In addition, the inexactness restriction is relaxed further. Numerical example indicates the improvement of the proposed method.     

Inexact implicit methods for monotone general variational inequalities

Solving a variational inequality problem is equivalent to finding a solution of a system of nonsmooth equations. Recently, we proposed an implicit method, which solves monotone variational inequality problem via solving a series of systems of nonlinear smooth (whenever the operator is smooth) equations. It can exploit the facilities of the classical Newton–like methods for smooth equations. In this paper, we extend the method to solve a class of general variational inequality problems Moreover, we improve the implicit method to allow inexact solutions of the systems of nonlinear equations at each iteration. The method is shown to preserve the same convergence properties as the original implicit method. Received July 31, 1995 / Revised version received January 15, 1999? Published online May 28, 1999     

Global convergence of descent processes for solving non strictly monotone variational inequalities

This paper addresses the question of global convergence of descent processes for solving monotone variational inequalities defined on compact subsets ofR ⁿ . The approach applies to a large class of methods that includes Newton, Jacobi and linearized Jacobi methods as special cases. Furthermore, strict monotonicity of the cost mapping is not required.Research supported by NSERC grant A5789.     

有界约束单调变分不等式的内点连续算法

讨论变分不等式问题VIP(X，F)，其中F是单调函数，约束集X为有界区域．利用摄动技术和一类光滑互补函数将问题等价转化为序列合两个参数的非线性方程组，然后据此建立VIP(X，F)的一个内点连续算法．分析和论证了方程组解的存在性和惟一性等重要性质，证明了算法很好的整体收敛性，最后对算法进行了初步的数值试验。     

A new inexact alternating directions method for monotone variational inequalities

The alternating directions method (ADM) is an effective method for solving a class of variational inequalities (VI) when the proximal and penalty parameters in sub-VI problems are properly selected. In this paper, we propose a new ADM method which needs to solve two strongly monotone sub-VI problems in each iteration approximately and allows the parameters to vary from iteration to iteration. The convergence of the proposed ADM method is proved under quite mild assumptions and flexible parameter conditions. Received: January 4, 2000 / Accepted: October 2001?Published online February 14, 2002     

Interior-point algorithms for monotone affine variational inequalities

Given ann×n matrixM, a vectorq in ⁿ , a polyhedral convex setX={x|Axb, Bx=d}, whereA is anm×n matrix andB is ap×n matrix, the affinne variational inequality problem is to findxX such that (Mx+q) ^T (y–x)0 for allyX. IfM is positive semidefinite (not necessarily symmetric), the affine variational inequality can be transformeo to a generalized complementarity problem, which can be solved in polynomial time using interior-point algorithms due to Kojima et al. We develop interior-point algorithms that exploit the particular structure of the problem, rather than direictly reducing the problem to a standard linear complemntarity problem.This work was partially supported by the Air Force Office of Scientific Research, Grant AFOSR-89-0410 and the National Science Foundation, Grant CCR-91-57632.The authors acknowledge Professor Osman Güler for pointing out the valoidity of Theorem 2.1 without further assumptions and the proof to that effect. They are also grateful for his comments to improve the presentation of this paper.     

Optimality conditions for strongly monotone variational inequalities

Necessary conditions for optimal controls have been obtained for strongly monotone variational inequalities by the penalty method, Ekeland's Variational Principle, and lower-semicontinuity of set-valued mappings. It has been shown that these conditions are easy to apply and can imply some known necessary conditions. They also yield new optimality conditions.     

An inexact logarithmic-quadratic proximal augmented Lagrangian method for a class of constrained variational inequalities

The augmented Lagrangian method is attractive in constraint optimizations. When it is applied to a class of constrained variational inequalities, the sub-problem in each iteration is a nonlinear complementarity problem (NCP). By introducing a logarithmic-quadratic proximal term, the sub-NCP becomes a system of nonlinear equations, which we call the LQP system. Solving a system of nonlinear equations is easier than the related NCP, because the solution of the NCP has combinatorial properties. In this paper, we present an inexact logarithmic-quadratic proximal augmented Lagrangian method for a class of constrained variational inequalities, in which the LQP system is solved approximately under a rather relaxed inexactness criterion. The generated sequence is Fejér monotone and the global convergence is proved. Finally, some numerical test results for traffic equilibrium problems are presented to demonstrate the efficiency of the method.      

A partial inexact alternating direction method for structured variational inequalities

In this article, a new method is proposed for solving a class of structured variational inequalities (SVIs). The proposed method is referred to as the partial inexact proximal alternating direction (piPAD) method. In the method, two subproblems are solved independently. One is handled by an inexact proximal point method and the other is solved directly. This feature is the major difference between the proposed method and some existing alternating direction-like methods. The convergence of the piPAD method is proved. Two examples of the modern convex optimization problem arising from engineering and information sciences, which can be reformulated into the encountered SVIs, are presented to demonstrate the applicability of the piPAD method. Also, some preliminary numerical results are reported to validate the feasibility and efficiency of the piPAD method.     

An additive Schwarz method for variational inequalities

This paper proposes an additive Schwarz method for variational inequalities and their approximations by finite element methods. The Schwarz domain decomposition method is proved to converge with a geometric rate depending on the decomposition of the domain. The result is based on an abstract framework of convergence analysis established for general variational inequalities in Hilbert spaces.

     

On a system of general mixed variational inequalities

In this paper, we introduce and consider a new system of general mixed variational inequalities involving three different operators. Using the resolvent operator technique, we establish the equivalence between the general mixed variational inequalities and the fixed point problems. We use this equivalent formulation to suggest and analyze some new explicit iterative methods for this system of general mixed variational inequalities. We also study the convergence analysis of the new iterative method under certain mild conditions. Since this new system includes the system of mixed variational inequalities involving two operators, variational inequalities and related optimization problems as special cases, results obtained in this paper continue to hold for these problems. Our results can be viewed as a refinement and improvement of the previously known results for variational inequalities.     

单调混合变分不等式的若干新的迭代算法

In this paper,some new iterative algorithms for monotone mixed variational inequalities and the convergence in real Hilbert spaces are studied.