基于变分不等式KKT条件等价形式的阻尼牛顿算法

BASED ON VARIATION INEQUALITY KKT CONDITION EQUAL FROM DAMPING NEWTON ALGORITHM

变分不等式是应用数学中一个十分重要的研究领域,对于该问题快速有效地求解仍是一个焦点。本文首先建立了变分不等式KKT条件与非光滑方程组之间的等价关系,然后给出了阻尼牛顿算法。与采用Fischer函数的方法作对应的比较,该等价关系不仅克服了非负约束的条件而且形式更简单算法更易实施。该算法在一定条件下全局收敛并且是局部超线性或二次收敛的。

Variational inequality is a very important field in applied mathematics and it is a key problem to obtain its soluton effciently and rapidly.We first obtain the equivalent relation of variational inequality problem KKTconditions and nonsmooth equations and present the corresponding damped Newton method for it.Comparing with the numerical method of adopting Fischer function to solving variational inequality KKT conditions,not only this equivalent formulation gets rid of nonnegative constrained condition,but also the form is simpler and the algorithm is implemented easier.Under the certain condition,the algorithms is global convergent,local superlinear convergent or quadratically convergent.