A Generalized New ton -Like Method fo r Solving Variational Inequalities

Journal of Liaoning Petrochemical University ›› 2005, Vol. 25 ›› Issue (2): 95-98.

School of Science , Liaoning University of Petroleum & Chemical Technology ,
Fushun Liaoning 113001, P .R .China

Received:2004-10-28 Published:2005-06-20 Online:2017-04-18

解变分不等式的广义拟牛顿法

田秋菊, 宋岱才

辽宁石油化工大学理学院, 辽宁抚顺113001

作者简介:田秋菊(1978 -), 女, 辽宁沈阳市, 在读硕士。

Abstract

Abstract:

　The variational inequality problem , denoted by VIP(X , F), is to find a vector x ∈ X ∈ Rⁿ to make F(x)^T(y -x)≥0,y ∈ X ∈Rⁿ .The problem VIP(X, F)can be reformulated as a mixed nonlinear complementarity problem .A generalized Newton-like method for solving variational inequalities was presented .If (ω^＊)is a solution of VIP(X, F), H^＊₀ ={ h(x ^＊), g_i(x ^＊); i ∈ B(x ^＊)} is of full column rank and Q(ω^＊)+H^＊ H^＊T is apositivedefinite matrix .T_i(ω), i =1 , 2, 4 are continuously differentiable , T′_i(ω), i =1 , 2, 4, satisfy Lipschitz' s condition in the neighbourhood N(ω^＊, δ), then the sequences{ω^k}, generated by algorithm , converges Q -quadratically to VIP(X, F)' s solution ω^＊, and prove Q -superlinear convergence without the strict complementarity slackness condition.

Key words: 　Variational inequality , Generalized New to n-like method , Q -quadratic conv ergence

摘要： 　变分不等式问题(记为VIP(X, F))就是求一个x ∈ X Rⁿ, 使得F(x)^T(y -x)≥0 , y ∈ X Rⁿ 。将VIP(X, F)转化为混合非线性互补问题, 提出了一种解变分不等式的拟牛顿法。若ω^＊是VIP(X, F)的解, H_＊⁰={ h(x ＊), gi(x ^＊);i ∈ B(x ^＊)}列满秩, Q(ω^＊)+H^＊H^＊T 是正定矩阵, T_i(ω), i =1 , 2 , 4 连续可微, T^′_i(ω), i=1, 2, 4 在点ω^＊的邻域N(ω^＊ , δ)内满足李普希兹条件, 那么由算法确定的序列{ω^k}Q-二次收敛到VIP(X , F)的解ω^＊。并在没有严格互补松弛性条件下证明了Q-超线性收敛

关键词: 变分不等式, 　广义拟牛顿法, 　Q-二次收敛性

TIAN Qiu -ju,SONG Dai -cai. A Generalized New ton -Like Method fo r Solving Variational Inequalities[J]. Journal of Liaoning Petrochemical University, 2005, 25(2): 95-98.

田秋菊, 宋岱才. 解变分不等式的广义拟牛顿法[J]. 辽宁石油化工大学学报, 2005, 25(2): 95-98.