Convergence of inexact Newton methods for generalized equations
Convergence of inexact Newton methods for generalized equations
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DOI：
10.1007/s10107-013-0664-x
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发表时间：
2013-03
影响因子：
2.7
通讯作者：
A. Dontchev;R. Rockafellar
中科院分区：
文献类型：
--
作者：
A. Dontchev;R. Rockafellar
对于广义方程，其中是一个光滑函数，是Banach空间之间的一个集值映射，我们研究了$$\开始{aligned} \left（f（x_k）+Df（x_k））描述的不精确Newton方法（x_{k+1}-x_k）+ F（x_{k+1}）\right）\cap R_k（x_k，x_{k+1}）\ne \empty set，\end{aligned}$$其中是的导数，映射序列表示不精确性。我们展示了如何正则性质的mappingsandare能够保证每一个序列产生的方法是收敛的q-线性，q-超线性，或q-二次，根据特定的假设。我们还表明，在某些情况下，至少有一个收敛序列肯定会产生。作为一个副产品，我们得到了收敛性结果的不精确牛顿方法求解方程，变分不等式和非线性规划问题。
For solving the generalized equation, whereis a smooth function andis a set-valued mapping acting between Banach spaces, we study the inexact Newton method described by $$\begin{aligned} \left( f(x_k)+ D f(x_k)(x_{k+1}-x_k) + F(x_{k+1})\right) \cap R_k(x_k, x_{k+1}) \ne \emptyset , \end{aligned}$$whereis the derivative ofand the sequence of mappingsrepresents the inexactness. We show how regularity properties of the mappingsandare able to guarantee that every sequence generated by the method is convergent either q-linearly, q-superlinearly, or q-quadratically, according to the particular assumptions. We also show there are circumstances in which at least one convergence sequence is sure to be generated. As a byproduct, we obtain convergence results about inexact Newton methods for solving equations, variational inequalities and nonlinear programming problems.