﻿ 求解拟变分不等式问题的一种外梯度算法 An Extragradient Algorithm for Quasi-Variat-Ional Inequality Problem

Vol.04 No.01(2015), Article ID:14884,5 pages
10.12677/AAM.2015.41009

An Extragradient Algorithm for Quasi-Variat-Ional Inequality Problem

Yuanyuan Yuan, Wenwei Zhang, Biao Qu

School of Management, Qufu Normal University, Rizhao Shandong

Email: shuijing_0622@126.com

Received: Feb. 8th, 2015; accepted: Feb. 21st, 2015; published: Feb. 27th, 2015

ABSTRACT

In this paper, we present a projection-like algorithm for solving the quasi-variational inequality problem. In the second projection step of the algorithm, we replace the orthogonal projection onto a general closed convex set with a projection onto a halfspace, which reduces the difficulty of calculation to some extent. The global convergence of the algorithm is given.

Email: shuijing_0622@126.com

1. 引言

(1.1)

2. 预备知识

(1)

(2)

(3).

(1) 在处上半连续，若

(2) 在处下半连续，若对任意满足的点列，，及满足，均存在一点列，使得当，且成立；

(3) 在处连续，若在处既上半连续，又下半连续；

(4) 在集合上连续，当且仅当在上的每一点都连续。

(a)，对，有

(b) 若的一个解，则对，有

(c)上连续；

(d) 函数连续的，即对，满足：

3. 算法及其收敛性分析

(2) 若，则的一个解。

，有，由假设条件(b)得，

(3.1)

(3.2)

(3.3)

，由的有界性及(3.3)，我们得到

4. 数值实验

Table 1. The resulting data of numerical experiment

An Extragradient Algorithm for Quasi-Variat-Ional Inequality Problem. 应用数学进展,01,70-75. doi: 10.12677/AAM.2015.41009

1. 1. Harker, P. (1991) Generalized Nash games and quasi-variational inequalities. European Journal of Operational Research, 54, 81-94.

2. 2. Pang, J.S. and Fukushima, M. (2005) Quasi-variational inequalities, generalized Nash equilibria and multileader-Fol- lower game. Computational Management Science, 1, 21-56.

3. 3. Zhang, J.Z., Qu, B. and Xiu, N.H. (2010) Some projection-like methods for the generalized Nash equilibria. Computational Optimization and Applications, 45, 89-109.

4. 4. Han, D., Zhang, H., Qian, G. and Xu, L. (2012) An improved two-step method for solving generalized Nash equilibrium problems. European Journal of Operational Research, 216, 613-623.

5. 5. 屈彪, 张善美 (2008) 求解拟变分不等式问题的一种投影算法. 应用数学学报, 5, 922-928.

6. 6. Censor, Y., Gibali, A. and Reich, S. (2011) The subgradient extragradient method for solving variational inequlities in Hilbert space. Journal of Optimization Theory and Applications, 148, 318-335.

7. 7. Zarantonello, E.H. (1971) Projections on convex sets in Hilbert space and spectral theory. In: Zarantonello, E.H., Ed., Contributions to Nonlinear Functional Analysis, American Academic Press, New York, 19-32.

8. 8. Gafni, E.M. and Bertsekas, D.P. (1984) Two-metric projection problems and descent methods for asymmetric variational inequality problems. Math. Program, 53, 99-110.

9. 9. Haker, P.T. (1991) Generalized Nash games and quasi-variational inequalities. European Journal of Operational Research, 54, 81-94.

10. 10. Outrata, J. and Zowe, J. (1995) A numerical approach to optimization problems with variational inequality constraints. Mathematical Programming, 68, 105-130.