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

SOLUTION OF BOX CONSTRAINED VARIATIONAL INEQUALITY WITH LOCALLY LIPSCHITZIAN FUNCTIONS

A new unconstrained merit function (x) for the box constrained variational inequality VI(a,b,F) with locally Lipschitzian function F (need not be differentiable) is proposed and its various desirable properties are investigated. Using this merit function the box constrained variational inequality is reformulated as the unconstrained minimization problem min{(x) |xRn}. When every V F(x) is P0-matrix, it is proved that 0(x) is necessary and sufficient for x to be a solution to VI(a,b,F). Based on these special properties, for monotone functions F a simple derivative-free algorithm with global convergence for solving min{0(x) |xRn} is proposed. Moreover, a generalized Newton method is also presented. Under the suitable regularity condition it is proved that the method is locally well defined and superlinearly convergent for semismooth functions F and quadratically convergent for strongly semismooth functions F respectively.