解变分不等式的简单牛顿法的收敛性

Convergence of Newton method for solving variational inequalities

牛顿法作为求解变分不等式问题的一个重要方法,它的收敛性一直是各位学者研究的一个核心问题.当变分不等式中的函数F在B(x_0,ρ)内满足γ-条件时,证明了由牛顿法产生的迭代点列是适定的,而且解的序列可以被构造出来的{tn}序列控制收敛到一个变分不等式的最优解.考虑到F在B(x_0,ρ)内满足γ-条件这个区域性条件给进一步的研究带来了困难,因此引入了解析函数,给出了F是解析函数条件下的牛顿法的收敛性结果.数值实验表明算法是有效的.

Newton method is an important method for solving the problem of variational inequalities,the convergence of the algorithm has been a core issue for the scholars.When the function F in variational inequalities satisfiesγ-conditions in B(x_0,ρ),it is proved that the iterative dot sequence produced by the Newton method is valid and can be controlled to converge to an optimal solution of the variational inequalities by a sequence{tn},which is constructed out.Taking into account that the regional conditions of the function F satisfies γ-conditions in B(x_0,ρ)makes more difficult for further study,analytic functions is introduced.The convergence of Newton method under the condition that the F is analytic function is given.Numerical experiments show that the algorithm is effective.