A New Extragradient Method for Strongly Pseudomonotone Variational Inequalities

This article proposes a new extragradient solution method for strongly pseudomonotone variational inequalities. A detailed analysis of the iterative sequences’ convergence and of the range of applicability of the method is provided. Moreover, an interesting class of strongly pseudomonotone infinite dimensional variational inequality problems is considered.     

A New Decomposition Method for Variational Inequalities with Linear Constraints

We propose a new decomposition method for solving a class of monotone variational inequalities with linear constraints. The proposed method needs only to solve a well-conditioned system of nonlinear equations, which is much easier than a variational inequality, the subproblem in the classic alternating direction methods. To make the method more flexible and practical, we solve the sub-problems approximately. We adopt a self-adaptive rule to adjust the parameter, which can improve the numerical performance of the algorithm. Under mild conditions, the underlying mapping be monotone and the solution set of the problem be nonempty, we prove the global convergence of the proposed algorithm. Finally, we report some preliminary computational results, which demonstrate the promising performance of the new algorithm.     

A Strongly Convergent Direct Method for Monotone Variational Inequalities in Hilbert Spaces

We introduce a two-step direct method, like Korpelevich's, for solving monotone variational inequalities. The advantage of our method over that one is that ours converges strongly in Hilbert spaces, whereas only weak convergence has been proved for Korpelevich's algorithm. Our method also has the following desirable property: the sequence converges to the solution of the problem that lies closest to the initial iterate.     

Maximum Principles for Optimal Controls for Elliptic Variational Inequalities of the Second Kind

A maximum principle is obtained for optimal controls for someelliptic variational inequalities of the second kind by usingthe penalty method, Ekelans's variational principle, and lowersemicontinuity of some set-valued mappings. It has been shownthat this principle leads to some known optimality conditionsin many cases. It also yields new optimality conditions. Insome cases, it leads to an analogy of pontryagin's principle. Permanent address: Mathematics Department, Centre South Universityof Technology, China.     

Decomposition Techniques for Bilinear Saddle Point Problems and Variational Inequalities with Affine Monotone Operators

The majority of first-order methods for large-scale convex–concave saddle point problems and variational inequalities with monotone operators are proximal algorithms. To make such an algorithm practical, the problem’s domain should be proximal-friendly—admit a strongly convex function with easy to minimize linear perturbations. As a by-product, this domain admits a computationally cheap linear minimization oracle (LMO) capable to minimize linear forms. There are, however, important situations where a cheap LMO indeed is available, but the problem domain is not proximal-friendly, which motivates search for algorithms based solely on LMO. For smooth convex minimization, there exists a classical algorithm using LMO—conditional gradient. In contrast, known to us similar techniques for other problems with convex structure (nonsmooth convex minimization, convex–concave saddle point problems, even as simple as bilinear ones, and variational inequalities with monotone operators, even as simple as affine) are quite recent and utilize common approach based on Fenchel-type representations of the associated objectives/vector fields. The goal of this paper was to develop alternative (and seemingly much simpler) decomposition techniques based on LMO for bilinear saddle point problems and for variational inequalities with affine monotone operators.     

求解结构型单调变分不等式的改进的邻近类分解方法

邻近类分解方法首先是由Chen和Teboulle（Math. Programming,1994,64(1):81-101）提出用来求解凸的极小化问题．在此基础上,该文提出一种新方法求解具有分离结构的单调变分不等式．其主要优点在于放松了算法中对某些参数的限制,使得新方法更加便于计算．在和原分解方法相同的假设下,可以证明新方法是全局收敛的．     

Vector Variational Inequalities with Semi-monotone Operators

In this paper, the concept of a vector semi-monotone operator is introduced. This concept is applied to establish several existence results of vector variational inequalities. These extend some results of others. The work is supported by the Natural Science Foundation of China (Grant No. 19671042) and the Doctoral Program Foundation of the Ministry of Education of China.     

Interior Proximal Method for Variational Inequalities: Case of Nonparamonotone Operators

For variational inequalities characterizing saddle points of Lagrangians associated with convex programming problems in Hilbert spaces, the convergence of an interior proximal method based on Bregman distance functionals is studied. The convergence results admit a successive approximation of the variational inequality and an inexact treatment of the proximal iterations.An analogous analysis is performed for finite-dimensional complementarity problems with multi-valued monotone operators.     

Jump and Variational Inequalities for Rough Operators

In this paper, we systematically study jump and variational inequalities for rough operators, whose research have been initiated by Jones et al. More precisely, we show some jump and variational inequalities for the families \(\mathcal T:=\{T_\varepsilon \}_{\varepsilon >0}\) of truncated singular integrals and \(\mathcal M:=\{M_t\}_{t>0}\) of averaging operators with rough kernels, which are defined respectively by

$$\begin{aligned} T_\varepsilon f(x)=\int _{|y|>\varepsilon }\frac{\Omega (y')}{|y|^n}f(x-y)dy \end{aligned}$$

and

$$\begin{aligned} M_t f(x)=\frac{1}{t^n}\int _{|y|<t}\Omega (y')f(x-y)dy, \end{aligned}$$

where the kernel \(\Omega \) belongs to \(L\log ^+\!\!L(\mathbf S^{n-1})\) or \(H^1(\mathbf S^{n-1})\) or \(\mathcal {G}_\alpha (\mathbf S^{n-1})\) (the condition introduced by Grafakos and Stefanov). Some of our results are sharp in the sense that the underlying assumptions are the best known conditions for the boundedness of corresponding maximal operators.

     

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。     

增广Lagrangian对偶分解方法求解变分不等式系统

本文考虑带约束的变分不等式系统.提出一个基于增广Lagrangian对偶的分解算法,本文给出了算法的收敛性分析.     

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.     

模糊映射的完全广义混合型强变分不等式

研究关于模糊映射的一类新的变分不等式－模糊映射的完全广义混合型强变分不等式,得到此类变分不等式解的存在定理分解的一个逼近算法,推广了文[4]和文[8]的主要结果。     

A Logarithmic-Quadratic Proximal Method for Variational Inequalities

We present a new method for solving variational inequalities on polyhedra. The method is proximal based, but uses a very special logarithmic-quadratic proximal term which replaces the usual quadratic, and leads to an interior proximal type algorithm. We allow for computing the iterates approximately and prove that the resulting method is globally convergent under the sole assumption that the optimal set of the variational inequality is nonempty.     

Enlargement of Monotone Operators with Applications to Variational Inequalities

Given a point-to-set operator T, we introduce the operator T defined as T(x)= {u: u – v, x – y – for all y Rn, v T(y)}. When T is maximal monotone T inherits most properties of the -subdifferential, e.g. it is bounded on bounded sets, T(x) contains the image through T of a sufficiently small ball around x, etc. We prove these and other relevant properties of T, and apply it to generate an inexact proximal point method with generalized distances for variational inequalities, whose subproblems consist of solving problems of the form 0 H(x), while the subproblems of the exact method are of the form 0 H(x). If k is the coefficient used in the kth iteration and the k's are summable, then the sequence generated by the inexact algorithm is still convergent to a solution of the original problem. If the original operator is well behaved enough, then the solution set of each subproblem contains a ball around the exact solution, and so each subproblem can be finitely solved.     

Homotopy Method for Variational Inequalities

Solving a finite-dimensional variational inequality is to find a vector x ∈X Rn such that(x - x)TF(x) ≥0, x ∈X, (1)where X is a nonempty, closed and convex subset of Rn and F is a mapping from Rn to itself,denoted by Ⅵ(X, F).The variational inequality problem (VIP) has had many successful practical applications inthe last three decades. It has been used to formulate and investigate equilibrium models arisingin economics, transportation, regional science and operations research. So fa…     

On the Directional Differentiability of the Solution Mapping for a Class of Variational Inequalities of the Second Kind

The directional differentiability of the solution mapping for a class of variational inequalities of the second kind inspired by applications in fluid mechanics and moving free boundary problems is investigated. The result is particularly relevant for the model predictive control or optimal control of such variational inequalities in that it can be used to derive stationarity conditions and efficient numerical methods.     

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.     

Penalty Method for Variational Inequalities

In this paper we deal with the existence theory for a problem and give the proof of the existence by a penalty argument. We shall treat the problem for a variational inequality by introducing the penalized differential equation and then taking the limits of the equation resulting from the penalized approximation. We also discuss the error estimate for the difference of the two solutions in an appropriate norm.