解变分不等式的广义拟牛顿法

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

A Generalized Newton- Like Method for Solving Variational Inequalities

The variational inequality problem, denoted by VIP(X,F), is to find a vector x∈XR~n to make F(x)~T(y-x)≥0,y∈XR~n. 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. Τ_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.