变分不等式的新的外梯度方法(英文)

A New Extragradient Method for Variational Inequalities

本文引入了一个新的求解非扩张映射的不动点集和具有单调及Lipschitz连续映射的变分不等式的解集的公共元素的近似算法。这一算法是建立在外梯度方法和粘性逼近方法基础上的。在Hilbert空间上得到了这一算法产生序列的强收敛性定理。其内容如下:设C是实Hilbert空间H中的非空闭凸集,映射A∶C→H是单调和k-Lipschitz连续的,S∶C→H是非扩张映射满足Fix(S)∩VI(C,A)≠,其中Fix(S)和VI(C,A)分别是S的不动点集和变分不等式的解集,f∶H→H是压缩映射,序列{xn}和{yn}由下列算法产生的:x1=x∈Cyn=PC(xn-γnAxn)xn+1=αnf(xn)+βnxn+(1-αn-βn)SPC(xn-γnAyn),n=1,2,…,其中{γn},{αn}和{βn}是满足条件limn→∞αn=0和∑n∞=1αn=∞,1>limn→s∞upβn≥limn→∞infβn>0和nl→im∞γn=0的数列,则{xn}和{yn}强收敛到w=PFix(S)∩VI(C,A)f(w),这里PFix(S)∩VI(C,A)f(w)表示f(w)在Fix(S)∩VI(C,A)上的投影。本文结果推广了文献中的一些著名结果。

In this paper, we introduce a new approximation scheme based on the extragradient method and viscosity method for finding a common element of the set of solutions of the set of fixed points of a nonexpansive mapping and the set of the variational inequality for a monotone, Lipschitz continuous mapping. We obtain a strong convergence theorem for the sequences generated by these processes in Hilbert spaces as follows ： Let C be a nonempty closed convex subset of a real Hilbert space H. Let .4 be a monotone and k-Lipschitz continuous mapping of C into H. Let S be a nonexpansive ）napping of C into H such that Fix（S）∩VI（C,A）≠Ф, where Fix（S） and VI（C,A） , respectively, denote the set of fixed point of S and the solution set of a variational inequality. Letf be a contraction of H into itself and { xn } and { yn } be sequences generated by {x1=x∈C γn=Pc（xn-γnAxn） xn＋1=αnf（xn）＋βnxn＋（1-αn-βn）SPc（xn-γnAγn）for every n=1,2,…,where{γ},{αn} and {βn}are sequences of numbers salisfying limαn n→∞=0 and ∑n=1^∞αn=∞,1〉lim n→∞ sup βn≥lim n→∞ inf βn〉0 and limγn n→∞=0.Then,{xn} and {yn} converge strongly to w=PFix（S）∩VI（C,A）f（w）.The results in this paper improves some well-known results in the literature.