Projection-proximal methods for general variational inequalities

In this paper, we consider and analyze some new projection-proximal methods for solving general variational inequalities. The modified methods converge for pseudomonotone operators which is a weaker condition than monotonicity. The proposed methods include several new and known methods as special cases. Our results can be considered as a novel and important extension of the previously known results. Since the general variational inequalities include the quasi-variational inequalities and implicit complementarity problems as special cases, results proved in this paper continue to hold for these problems.     

Merit functions for general variational inequalities

In this paper, we consider some classes of merit functions for general variational inequalities. Using these functions, we obtain error bounds for the solution of general variational inequalities under some mild conditions. Since the general variational inequalities include variational inequalities, quasivariational inequalities and complementarity problems as special cases, results proved in this paper hold for these problems. In this respect, results obtained in this paper represent a refinement of previously known results for classical variational inequalities.     

A class of new iterative methods for general mixed variational inequalities

In this paper, we use the auxiliary principle technique to suggest a class of predictor-corrector 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. As special cases, we obtain a number of known and new results for solving various classes of variational inequalities and related problems.     

广义集值变分不等式的一类新的外梯度算法

考虑和分析了一类求解广义集值变分不等式的一类新的外梯度算法,该方法包含几个新的和已知的算法作为特例.改进了求解变分不等式及其相关的优化问题的已有的许多结果.     

New extragradient-type methods for solving variational inequalities

In this paper, we propose new methods for solving variational inequalities. The proposed methods can be viewed as a refinement and improvement of the method of He et al. [B.S. He, X.M. Yuan, J.J. Zhang, Comparison of two kinds of prediction–correction methods for monotone variational inequalities, Comp. Opt. Appl. 27 (2004) 247–267] by performing an additional projection step at each iteration and another optimal step length is employed to reach substantial progress in each iteration. Under certain conditions, the global convergence of the both methods is proved. Preliminary numerical experiments are included to illustrate the efficiency of the proposed methods.     

An extragradient-type algorithm for non-smooth variational inequalities

In this paper we study extragradient-type methods to solve variational inequality problems involving maximal monotone point-to-set operators. We will show, using an example, that, in this case, to achieve convergence to the solution set of sequences generated by extragradient methods it is necessary, in the search of the auxiliary point and its suitable image by the involved operator, to enlarge the set of possible search directions. Such an enlargement, a generalization of the ε-subgradient, is known and we use it to overcome this failure of the extragradient-methods. We propose a new algo-rithm and prove that it generates a sequence convergent to a solution point of the problem also when the operator is point-to-set     

Two-sided approximations for unilateral variational inequalities by multi-grid methods

For the numerical solution of unilateral variational inequalities two iterative schemes are developed which provide approximations from below resp. from above. Both schemes are based on some kind of active set strategy and require the solution of an algebraic system of equations at each iteration step which is done by means of multigrid techniques. Convergence results are established and illustrated by some numerical results for the elastic-plastic torsion problem     

Wiener-hopf equations and variational inequalities

In this paper, we show that the general variational inequality problem is equivalent to solving the Wiener-Hopf equations. We use this equivalence to suggest and analyze a number of iterative algorithms for solving general variational inequalities. We also discuss the convergence criteria for these algorithms.     

General variational inequalities and nonexpansive mappings

In this paper, we suggest and analyze some three-step iterative schemes for finding the common elements of the set of the solutions of the Noor variational inequalities involving two nonlinear operators and the set of the fixed points of nonexpansive mappings. We also consider the convergence analysis of the suggested iterative schemes under some mild conditions. Since the Noor variational inequalities include variational inequalities and complementarity problems as special cases, results obtained in this paper continue to hold for these problems. Results obtained in this paper may be viewed as an refinement and improvement of the previously known results.     

Convergence analysis of non-quadratic proximal methods for variational inequalities in Hilbert spaces

We consider a general approach for the convergence analysis of proximal-like methods for solving variational inequalities with maximal monotone operators in a Hilbert space. It proves to be that the conditions on the choice of a non-quadratic distance functional depend on the geometrical properties of the operator in the variational inequality, and –- in particular –- a standard assumption on the strict convexity of the kernel of the distance functional can be weakened if this operator possesses a certain `reserve of monotonicity'. A successive approximation of the `feasible set' is performed, and the arising auxiliary problems are solved approximately. Weak convergence of the proximal iterates to a solution of the original problem is proved.     

Sensitivity analysis framework for variational inequalities

In this paper a sensitivity analysis framework is developed for variational inequalities. The perturbed solution to a parametric variational inequality problem is shown to be continuous and directionally differentiable under appropriate second order and regularity assumptions. Moreover, this solution is once continuously differentiable if strict complementarity also holds.     

The iterative methods for monotone generalized variational inequalities

A new class of iterative methods are presented for monotone generalized variational inequality problems. These methods, which base on an equivalent formulation of the original problem, can be viewed as the extension of the symmetric projection rnethod for monotone variational inequalities. The global convergence of the methods is estab-lished under the monotonicity assumption on the functions associated the problem.Specialization of the proposed algorithms and related results to several special cases are also discussed. Moreover, two combination methods are presented for affine monotone problems. and their global and Q-linear convergence are also established     

一般混合似变分不等式的隐式迭代算法

对一般混合似变分不等式的若干隐式迭代算法进行了研究;利用一般混合似变分不等式与不动点问题和预解方程的等价关系,采用分裂技巧和自适应迭代技巧结合,提出了一个求解一般混合似变分不等式的新的隐式迭代算法;并证明了该算法在算子T是g-单调连续的条件下收敛．     

Interior projection-like methods for monotone variational inequalities

We propose new interior projection type methods for solving monotone variational inequalities. The methods can be viewed as a natural extension of the extragradient and hyperplane projection algorithms, and are based on using non Euclidean projection-like maps. We prove global convergence results and establish rate of convergence estimates. The projection-like maps are given by analytical formulas for standard constraints such as box, simplex, and conic type constraints, and generate interior trajectories. We then demonstrate that within an appropriate primal-dual variational inequality framework, the proposed algorithms can be applied to general convex constraints resulting in methods which at each iteration entail only explicit formulas and do not require the solution of any convex optimization problem. As a consequence, the algorithms are easy to implement, with low computational cost, and naturally lead to decomposition schemes for problems with a separable structure. This is illustrated through examples for convex programming, convex-concave saddle point problems and semidefinite programming.The work of this author was partially supported by the United States–Israel Binational Science Foundation, BSF Grant No. 2002-2010.     

Relaxed extragradient iterative methods for variational inequalities

In this paper, we suggest and analyze some new relaxed extragradient iterative methods for finding a common element of the solution set of a variational inequality, the solution set of a general system of variational inequalities and the set of fixed points of a strictly pseudo-contractive mapping defined on a real Hilbert space. Strong convergence of the proposed methods under some mild conditions is established.     

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.     

A general descent framework for the monotone variational inequality problem

We present a framework for descent algorithms that solve the monotone variational inequality problem VIP _v which consists in finding a solutionv ^* _v satisfyings(v ^*)^T(v–v ^*)0, for allv _v. This unified framework includes, as special cases, some well known iterative methods and equivalent optimization formulations. A descent method is developed for an equivalent general optimization formulation and a proof of its convergence is given. Based on this unified logarithmic framework, we show that a variant of the descent method where each subproblem is only solved approximately is globally convergent under certain conditions.This research was supported in part by individual operating grants from NSERC.     

General algorithm for variational inequalities

In this paper, we consider a general auxiliary principle technique to suggest and analyze a novel and innovative iterative algorithm for solving variational inequalities and optimization problems. We also discuss the convergence criteria.     

Family of perturbation methods for variational inequalities

In recent years, the so-called auxiliary problem principle has been used to derive many iterative type algorithms for solving optimal control, mathematical programming, and variational inequality problems. In the present paper, we use this principle in conjunction with the epiconvergence theory to introduce and study a general family of perturbation methods for solving nonlinear variational inequalities over a product space of reflexive Banach spaces. We do not assume that the monotone operator involved in our general variational inequality problem is of potential type. Several known iterative algorithms, which can be obtained from our theory, are also discussed.This work was completed while the second author was visiting the Department of Mathematics of the University of Washington, Seattle, Washington under financial support from the Belgian Fonds National de la Recherche Scientifique, Grant FNRS: B8/5-JS-9. 549.     

A relaxed projection method for variational inequalities

This paper presents a modification of the projection methods for solving variational inequality problems. Each iteration of the proposed algorithm consists of projection onto a halfspace containing the given closed convex set rather than the latter set itself. The algorithm can thus be implemented very easily and its global convergence to the solution can be established under suitable conditions.This work was supported in part by Scientific Research Grant-in-Aid from the Ministry of Education, Science and Culture, Japan.