A hybrid Newton method for solving the variational inequality problem via the D-gap function

The variational inequality problem (VIP) can be reformulated as an unconstrained minimization problem through the D-gap function. It is proved that the D-gap function has bounded level sets for the strongly monotone VIP. A hybrid Newton-type method is proposed for minimizing the D-gap function. Under some conditions, it is shown that the algorithm is globally convergent and locally quadratically convergent. Received May 6, 1997 / Revised version received October 30, 1998?Published online June 11, 1999     

Unconstrained Optimization Reformulations of Variational Inequality Problems

Recently, Peng considered a merit function for the variational inequality problem (VIP), which constitutes an unconstrained differentiable optimization reformulation of VIP. In this paper, we generalize the merit function proposed by Peng and study various properties of the generalized function. We call this function the D-gap function. We give conditions under which any stationary point of the D-gap function is a solution of VIP and conditions under which it provides a global error bound for VIP. We also present a descent method for solving VIP based on the D-gap function.     

Theoretical and numerical investigation of the D-gap function for box constrained variational inequalities

The D-gap function, recently introduced by Peng and further studied by Yamashita et al., allows a smooth unconstrained minimization reformulation of the general variational inequality problem. This paper is concerned with the D-gap function for variational inequality problems over a box or, equivalently, mixed complementarity problems. The purpose of this paper is twofold. First we investigate theoretical properties in depth of the D-gap function, such as the optimality of stationary points, bounded level sets, global error bounds and generalized Hessians. Next we present a nonsmooth Gauss-Newton type algorithm for minimizing the D-gap function, and report extensive numerical results for the whole set of problems in the MCPLIB test problem collection. The work of this author was supported in part by the Scientific Research Grant-in-Aid from the Ministry of Education, Science, Sports and Culture, Japan.     

Error bound results for generalized D-gap functions of nonsmooth variational inequality problems

Solving a variational inequality problem can be equivalently reformulated into solving a unconstraint optimization problem where the corresponding objective function is called a merit function. An important class of merit function is the generalized D-gap function introduced in [N. Yamashita, K. Taji, M. Fukushima, Unconstrained optimization reformulations of variational inequality problems, J. Optim. Theory Appl. 92 (1997) 439-456] and Yamashita and Fukushima (1997) [17]. In this paper, we present new fractional local/global error bound results for the generalized D-gap functions of nonsmooth variational inequality problems, which gives an effective estimate on the distance between a specific point to the solution set, in terms of the corresponding function value of the generalized D-gap function. Numerical examples and a simple application to the free boundary problem are also presented to illustrate the significance of our error bound results.     

Global Method for Monotone Variational Inequality Problems with Inequality Constraints

We consider optimization methods for monotone variational inequality problems with nonlinear inequality constraints. First, we study the mixed complementarity problem based on the original problem. Then, a merit function for the mixed complementarity problem is proposed, and some desirable properties of the merit function are obtained. Through the merit function, the original variational inequality problem is reformulated as simple bounded minimization. Under certain assumptions, we show that any stationary point of the optimization problem is a solution of the problem considered. Finally, we propose a descent method for the variational inequality problem and prove its global convergence.     

Extended Projection Methods for Monotone Variational Inequalities

In this paper, we prove that each monotone variational inequality is equivalent to a two-mapping variational inequality problem. On the basis of this fact, a new class of iterative methods for the solution of nonlinear monotone variational inequality problems is presented. The global convergence of the proposed methods is established under the monotonicity assumption. The conditions concerning the implementability of the algorithms are also discussed. The proposed methods have a close relationship to the Douglas–Rachford operator splitting method for monotone variational inequalities.     

Convergence and Error Bound of a Method for Solving Variational Inequality Problems via the Generalized D-Gap Function

The variational inequality problem (VIP) can be reformulated as an unconstrained minimization problem through the generalized D-gap function. Recently, a hybrid Newton-type method was proposed by Peng and Fukushima for minimizing a special form of the generalized D-gap function. In this paper, the hybrid Newton-type algorithm is extended to minimize the general form g of the generalized D-gap function. It is shown that the algorithm has nice convergence properties. Under some reasonable conditions, it is proved that the algorithm is locally and globally convergent. Moreover, it is proved that the function g has bounded level sets for strongly monotone VIP. An error bound of the algorithm is obtained.     

Tikhonov Regularization Methods for Variational Inequality Problems

Motivated by the work of Facchinei and Kanzow (Ref. 1) on regularization methods for the nonlinear complementarity problem and the work of Ravindran and Gowda (Ref. 2) for the box variational inequality problem, we study regularization methods for the general variational inequality problem. A sufficient condition is given which guarantees that the union of the solution sets of the regularized problems is nonempty and bounded. It is shown that solutions of the regularized problems form a minimizing sequence of the D-gap function under a mild condition.     

Gap functions and error bounds for quasi variational inequalities

The paper aims to obtain new local/global error bounds for quasi variational inequality problems in terms of the regularized gap function and the D-gap function. These bounds provide effective estimated distances between a specific point and the exact solution of quasi variational inequality problem.     

Unconstrained optimization reformulations of equilibrium problems

We generalize the D-gap function developed in the literature for variational inequalities to a general equilibrium problem （EP）. Through the D-gap function, the equilibrium problem is cast as an unconstrained minimization problem. We give conditions under which any stationary point of the D-gap function is a solution of EP and conditions under which it provides a global error bound for EP. Finally, these results are applied to box-constrained EP and then weaker conditions are established to obtain the desired results for box-constrained EP.     

求解不可微箱约束变分不等式的下降算法

1 引 论 设X(?)Rn是非空闭集,F:Rn→Rn连续映射,变分不等式问题VI(X,F)是指:求x∈X,使 F(x)T(y-x)≥0,　 (?)y∈X,(1)记指标集N=(1,2,…,n},当 X=[a,b]≡{x∈Rn|a≤xi≤bi,i∈N},(2)其中a={a1,a2,…,an}T,b={b1,b2,…,bn}T∈Rn时,VI(X,F)化为箱约束变分不等式VI(a,b,F).若ai=0,bi=+∞,i∈N,即X=R+n≡{x∈Rn|x≥0}时,VI(a,b,F)化为非线性     

Strong convergence of a double projection-type method for monotone variational inequalities in Hilbert spaces

We introduce a projection-type algorithm for solving monotone variational inequality problems in real Hilbert spaces without assuming Lipschitz continuity of the corresponding operator. We prove that the whole sequence of iterates converges strongly to a solution of the variational inequality. The method uses only two projections onto the feasible set in each iteration in contrast to other strongly convergent algorithms which either require plenty of projections within a step size rule or have to compute projections on possibly more complicated sets. Some numerical results illustrate the behavior of our method.     

Viscosity Approximation Methods for a Class of Generalized Split Feasibility Problems with Variational Inequalities in Hilbert Space

Strong convergence theorem of viscosity approximation methods for nonexpansive mapping have been studied. We also know that CQ algorithm for solving the split feasibility problem (SFP) has a weak convergence result. In this paper, we use viscosity approximation methods and some related knowledge to solve a class of generalized SFP’s with monotone variational inequalities in Hilbert space. We propose some iterative algorithms based on viscosity approximation methods and get strong convergence theorems. As applications, we can use algorithms we proposed for solving split variational inequality problems (SVIP), split constrained convex minimization problems and some related problems in Hilbert space.     

Error bounds and strong upper semicontinuity for monotone affine variational inequalities

Global error bounds for possibly degenerate or nondegenerate monotone affine variational inequality problems are given. The error bounds are on an arbitrary point and are in terms of the distance between the given point and a solution to a convex quadratic program. For the monotone linear complementarity problem the convex program is that of minimizing a quadratic function on the nonnegative orthant. These bounds may form the basis of an iterative quadratic programming procedure for solving affine variational inequality problems. A strong upper semicontinuity result is also obtained which may be useful for finitely terminating any convergent algorithm by periodically solving a linear program.This material is based on research supported by Air Force Office of Scientific Research Grant AFOSR-89-0410 and National Science Foundation Grants CCR-9101801 and CCR-9157632.     

求解广义互补问题

1 引言 设为一闭凸锥,f是R~n到自身的一映射.广义互补问题,记作GCP(K,f),即找一向量x满足 GCP(K,f) x∈K,f(x)∈且x~Tf(x)=0,(1) 其中,是K的对偶锥(即对任一K中向量x,满足x~Ty≤0的所有y的集合).该问题首先 由Habetler和Price提出.当K=R_+~n(R~n空间的正卦限),此问题就是一般的互补问题.许多作者已经提出了很多求解线性或非线性互补问题的方法.例如:Dafermos,Fukushima,Harker和Price以及其它如参考文献所列.近年来,何针对单调线性变分不等式提出了一些投影收缩算法. Fang在函数是Lipschitz连续及强单调的条件下,在[3]给出一简单的迭代投影法,在[4]中给出一线性化方法去求解广义互补问题(1).在[3]中,他的迭代模式是     

Accelerated schemes for a class of variational inequalities

We propose a novel stochastic method, namely the stochastic accelerated mirror-prox (SAMP) method, for solving a class of monotone stochastic variational inequalities (SVI). The main idea of the proposed algorithm is to incorporate a multi-step acceleration scheme into the stochastic mirror-prox method. The developed SAMP method computes weak solutions with the optimal iteration complexity for SVIs. In particular, if the operator in SVI consists of the stochastic gradient of a smooth function, the iteration complexity of the SAMP method can be accelerated in terms of their dependence on the Lipschitz constant of the smooth function. For SVIs with bounded feasible sets, the bound of the iteration complexity of the SAMP method depends on the diameter of the feasible set. For unbounded SVIs, we adopt the modified gap function introduced by Monteiro and Svaiter for solving monotone inclusion, and show that the iteration complexity of the SAMP method depends on the distance from the initial point to the set of strong solutions. It is worth noting that our study also significantly improves a few existing complexity results for solving deterministic variational inequality problems. We demonstrate the advantages of the SAMP method over some existing algorithms through our preliminary numerical experiments.     

单调变分不等式可行与非可行点组合的连续算法

本文给出了单调变分不等式问题一个新的连续型求解方法，方法的实现依赖于一系列含有四个参数的摄动单调变分不等式的求解．其中摄动参数要求的条件较为温和，这使得本文方法成为可行点与非可行点算法的有机组合和统一．在适当的假设条件下，我们分析和证明了摄动变分不等式问题解的存在性，唯一性和算法的强收敛性．     

A Modified Alternating Direction Method for Variational Inequality Problems

The alternating direction method is an attractive method for solving large-scale variational inequality problems whenever the subproblems can be solved efficiently. However, the subproblems are still variational inequality problems, which are as structurally difficult to solve as the original one. To overcome this disadvantage, in this paper we propose a new alternating direction method for solving a class of nonlinear monotone variational inequality problems. In each iteration the method just makes an orthogonal projection to a simple set and some function evaluations. We report some preliminary computational results to illustrate the efficiency of the method. Accepted 4 May 2001. Online publication 19 October, 2001.     

D-Gap Functions for Nonsmooth Variational Inequality Problems

We study the Clarke generalized gradient of the D-gap functions for the variational inequality problem (VIP) defined by a locally Lipschitz, but not necessarily differentiable, function in an Euclidean space. Using these results, we study the relationship between minimizing sequences and stationary sequences of the D-gap function, regardless of the existence of solutions of (VIP).     

The Tikhonov regularization extended to equilibrium problems involving pseudomonotone bifunctions

We extend the Tikhonov regularization method widely used in optimization and monotone variational inequality studies to equilibrium problems. It is shown that the convergence results obtained from the monotone variational inequality remain valid for the monotone equilibrium problem. For pseudomonotone equilibrium problems, the Tikhonov regularized subproblems have a unique solution only in the limit, but any Tikhonov trajectory tends to the solution of the original problem, which is the unique solution of the strongly monotone equilibrium problem defined on the basis of the regularization bifunction.