求解不可微箱约束变分不等式的下降算法  

SOLUTION OF BOX CONSTRAINED VARIATIONAL INEQUALITY WITH LOCALLY LIPSCHITZIAN FUNCTIONS

作　　者：陈国庆[1] 刘水霞[1] 

机构地区：[1]内蒙古大学理工学院数学系,呼和浩特010021

出　　处：《高等学校计算数学学报》2002年第4期335-348,共14页Numerical Mathematics A Journal of Chinese Universities

基　　金：国家自然科学基金(19701016);内蒙古自然科学基金资助.

摘　　要：A new unconstrained merit function θ(x) for the box constrained vari-ational inequality VI (a,b,F) with locally Lipschitzian function F (need not bedifferentiable) is proposed and its various desirable properties are investigated.Using this merit function the box constrained variational inequality is reformulat-ed as the unconstrained minimization problem min {θ(x) | x∈ Rn }. When every V∈δF(x) is Po-matrix, it is proved that 0∈δθ(x) is necessary and sufficient for xto be a solution to VI(a,b,F). Based on these special properties, for monotonefunctions F a simple derivative-free algorithm with global convergence for solvingmin{θ(x) |x ∈ Rn} is proposed. Moreover, a generalized Newton method is alsopresented. Under the suitable regularity condition it is proved that the method islocally well defined and superlinearly convergent for semismooth functions F ardquadratically convergent for strongly semismooth functions F respectively.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.

关 键 词：不可微箱约束 变分不等式 下降算法 

分 类 号：O221[理学—运筹学与控制论]